On the Propagation of Regularity in Some Non-Local Dispersive Models

INSTITUTO NACIONAL DE MATEMATICA´ PURA E APLICADA DOUTORADO EM MATEMATICA´

ON THE PROPAGATION OF REGULARITY IN SOME NON-LOCAL DISPERSIVE MODELS

Thesis presented to the Postgraduate Program in Mathematics at Instituto Nacional de Matematica´ Pura e Aplicada-IMPA as partial fulﬁllment of the requirements for the degree of Doctor in Mathematics. Author: Argenis J Mendez G Advisor: Felipe Linares

Rio de Janeiro, Brazil March 12th, 2019 ii

Dedicated to

To all of you ”Soldados de franela” my deepest respect and admiration. iii

Acknowledgments

Firstly, I would like to thank to God and to my family: my parents Argenis and Mirna, for en- couraging me constantly to pursuit this goal as well as to my brother Gustavo, who constantly supported me in this part of my life. I would like to express my sincere gratitude to my advisor Prof. Felipe Linares for the continuous support of my Ph.D study and related research, for his patience, motivation, and im- mense knowledge. His guidance helped me in all the time of research and writing of this thesis. Much´ısimas gracias profesor. Besides my advisor, I would like to thank the rest of my thesis committee: Prof. Ademir Pastor, Prof. Adan´ Corcho, Prof. Hermano Frid, Prof. Luis Gustavo Farah, and Prof. Alexei Mailybaev, for their insightful comments and suggestion that improved in great measure the presentation of this work. To Prof. Juan Guevara, that without doubts was the person who guided me in Venezuela to pursuit this goal, thank you very much professor. To Prof. Mercedes Arriojas and Yamilet Quintana who undoubtedly gave me their vote of conﬁdence to get here, thanks for all. To Marco Mendez,´ for your incredible help at my arrive to Rio de Janeiro, as well as the price- less advice at IMPA. To IMPA, that without doubts became my home for the last 4 years, for providing me with an adequate environment that has allowed me to grow professionally. Also, I would like to thanks my fellows from the Analysis & PDE group, Oscar´ and Ricardo for the stimulating discussions we always have about mathematics. To Anabel for all your support, thank you. I thank my friends at IMPA. In particular, I am grateful to Daniel, Clarena, Bely, Mateus, Miguel, Lazaro, Gil, Diego, Makson etc. To CNPq, for the ﬁnancial support.

Brasil, Muito obrigado... Contents

Chapter 1. Preliminary 11 1. Notation 11 2. Commutators and Inequalities 12 3. Commutator Expansions 16

Chapter 2. Dispersive generalized Benjamin-Ono equation: local well-posedness & propagation of regularity 18 1. The Linear Problem. 18 2. The Nonlinear Problem 20 3. Reﬁned Strichartz estimate 23 4. Proof of TheoremA 25 5. Existence of Solutions 30 6. Continuity of the Flow 33 7. Propagation of regularity 34

Chapter 3. Fractional Korteweg-de Vries equation: propagation of regularity 68 1. Proof of TheoremC 69

iv CONTENTS 1

Abstract

In this thesis we study some properties of propagation of regularity of solutions associated to some nonlocal dispersive models. The first analysis, corresponds to the dispersive generalized Benjamin-Ono (DGBO) equation, that can be seen as a dispersive interpolation between Benjamin- Ono (BO) equation and the Korteweg-de Vries (KdV) equation. The second one, is the fractional Korteweg-de Vries (fKdV) equation, that unlike the DGBO, this can be interpreted as a dispersive perturbation of the Burger’s equation. Recently, it has been shown that solutions of the KdV equation and BO equation satisfy the following property: if the initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In the case that attain us, the nonlocal term present in the dispersive part in both equations, the DGBO and the fKdV equations, are more challenging than for KdV and BO equation. In the case of the DGBO, we firstly prove a local well-posedness result associated to the initial value problem, by using a compactness argument. As a second part of the thesis we study the propagation of regularity for solutions of the DGBO equation. To deal with this a new argument is employed. In fact, the approach used to deal with the later equations cannot be applied directly and some new ideas have to be introduced. The new ingredient combines commutator expansions into the weighted energy estimate that allow us to obtain the property of propagation. Finally, the technique used in the propagation of regularity in solutions of the DGBO allow us to verify that the solutions of the fKdV also satisfy the propagation of regularity phenomena. Here the argument is more involved since the dispersion is very weak. CONTENTS 2

Resumo

Nessa tese, nos´ estudamos algumas propriedades de propagaçao˜ de regularidade de soluçoes˜ associadas a modelos dispersivos nao˜ locais. O primeiro a ser considerado corresponde a equaçao˜ dispersiva generalizada Benjamin-Ono (DGBO), que pode ser observada como a interpolaçao˜ dispersiva entre a equaçao˜ de Benjamin-Ono (BO) e a equaçao˜ de Korteweg-de Vries (KdV). Para a segunda analise,´ trataremos da equaçao˜ fracionaria´ de KdV, a qual, diferente da DGBO, pode ser interpretado como uma perturbaçao˜ dispersiva da equaçao˜ de Burger. Recentemente, foi mostrado que as soluoes˜ das equaçoes˜ KdV e BO satisfazem a propriedade: se o dado inicial tem alguma regularidade predeterminada do lado direito da reta, entao˜ essa regularidade e´ propagada com velocidade infinita pelo fluxo da soluao.˜ Os casos os quais es- tamos tratando, os termos nao˜ locais apresentados na parte dispersiva em ambas as equaçoes,˜ a DGBO e a fKdV, sao˜ mais desafiadoras que as apresentadas para as equaçoes˜ BO e KdV. No caso da DGBO, nos,´ primeiramente, provamos um resultado de boa colocaçao˜ local associado ao problema de valor inicial, utilizando um argumento de compacidade. Na segunda parte da tese, nos´ estudamos a propagaçao˜ de regularidade nas soluçoes˜ da DGBO. A fim de lidar com esse problema, introduzimos uma nova abordagem. De fato, a abordagem utilizada para lidar com estas equaçoes˜ nao˜ pode ser aplicada diretamente e novas ideias´ precisam ser introduzidas. Esse novo ingrediente combinado com expansoes˜ do comutador em estimativas de energia com peso permitem-nos obter as propriedades de propagaçao˜ desejadas. Finalmente, a tecnica´ usada na propagaçao˜ de regularidade nas soluçoes˜ da DGBO permite-nos verificar que as soluçoes˜ da fKdV tambem´ satisfazem tal fenomenoˆ de propagaçao˜ da regularidade. Nessa situaçao,˜ o argumento e´ mais elaborado haja vista que a dispersao˜ e´ mais fraca. Introduction

The aim of this work is to study some special regularity properties of solutions to the initial value problem (IVP) associated to some nonlocal dispersives models. The ﬁrst to be considered is the dispersion generalized Benjamin-Ono equation # B u ¡ Dα+1B u + uB u = 0, x, t P R, 0 α 1, (0.1) t x x x u(x, 0) = u0(x), s P where Dx, denotes the homogeneous derivative of order s R, q s ¡B2 s/2 s | |s p Dx = ( x) thus Dx f = cs ξ f (ξ) ,

s B s which in its polar form is decomposed as Dx = (H x) , where H denotes the Hilbert transform, » 1 f (x ¡ y) p q H f (x) = lim dy = (¡i sgn(ξ) f (ξ)) (x), Ñ + π e 0 |y|¥e y wherep¤ denotes the Fourier transform and q denotes its inverse. These equations model vorticity waves in the coastal zone, see Molinet, Saut and Tzvetkov [36] and the references therein. Our starting point is a property established by Isaza, Linares and Ponce [17] concerning the solutions of the IVP associated to the k¡generalized KdV equation # B u + B3u + ukB u = 0, x, t P R, k P N, (0.2) t x x u(x, 0) = u0(x). It was shown by Isaza, Linares and Ponce [17] that the unidirectional dispersion of the k¡generalized KdV equation entails the following propagation of regularity phenomena.

3/4+ THEOREM 0.1 ([17]). If u0 P H (R) and for some l P Z, l ¥ 1 and x0 P R » 8 § § 2 § §2 Bl §Bl § 8 (0.3) xu0 = xu0(x) dx , L2((x ,8)) 0 x0 then the solution of the IVP associated to (0.2) satisﬁes that for any v ¥ 0 and e ¡ 0 » 8 j 2 sup Bxu (x, t) dx c, ¡ 0¤t¤T x0+e vt

3 CONTENTS 4

l = = } } + }B } 2 8 P ( ] for j 0, 1, 2, . . . , l with c c l; u0 H3/4 (R); xu0 L ((x0, )); v; e; T . In particular, for all t 0, T , l the restriction of u(¤, t) to any interval (x0, 8) belongs to H ((x0, 8)). Moreover, for any v ¥ 0, e ¡ 0 and R ¡ e » » ¡ T x0+R vt 2 Bl+1 x u (x, t) dx dt c, ¡ 0 x0+e vt

l = } } + }B } 2 8 with c c l; u0 H3/4 (R); xu0 L ((x0, )); v; e; R; T . The proof of Theorem 0.1 is based on weighted energy estimates. In detail, the iterative process in the induction argument is based in a property discovered originally by T. Kato [19] in the context of the KdV equation. More precisely, he showed that solution of the KdV equation satisﬁes » » T R 2 (B u) (x, t) dx dt ¤ c R; T; }u } 2 , x 0 Lx 0 ¡R being this the fundamental fact in his proof of existence of the global weak solutions of (0.2), for k = 1 and initial data in L2(R). This result was also obtained for the Benjamin-Ono equation by Isaza, Linares and Ponce [18] but it does not follow as the KdV case because of the presence of the Hilbert transform. Later on, Kenig et al. [22] extended the results in Theorem 0.1 to the case when the local regularity of the initial data u0 in (0.3) is measured with a fractional indices. The scope to this case is quite more involved, and its proof is mainly based in weighted energy estimates combined with techniques involving pseudo-differential operators and singular integrals. The property described in Theorem 0.1 is intrinsic to suitable solutions of some nonlinear dispersive models (see also Linares, Ponce and Smith [35]). In the context of 2D models, analogous results for the Kadomtsev- Petviashvili II equation (see Isaza, Linares and Ponce [16]) and Zakharov-Kuznetsov (see Linares and Ponce [32]) equations were proved. Before state our main result we will give an overview of the local well-posedness of the IVP (0.1). Following Kato [19] we have that the initial value problem IVP (0.1) is locally well-posed (LWP) in the Banach space X if for every initial condition u0 P X, there exists T ¡ 0 and a unique solution u(t) satisfying

(0.4) u P C ([0, T] : X) X AT where AT is an auxiliary function space. Moreover, the solution map u0 ÞÝÑ u, is continuous from X into the class (0.4). If T can be taken arbitrarily large, one says that the IVP (0.1) is globally well-posed (GWP) in the space X. It is natural to study the IVP (0.1) in the Sobolev space s ¡B2 ¡s/2 2 P H (R) = (1 x) L (R), s R. There exist remarkable differences between the KdV (0.2) and the IVP (0.1). In case of KdV e.g. it posses infinite conserved quantities, define a Hamiltonian system, have multi-soliton solutions and is a completely integrable system by the inverse scattering method see Coifman and Wicker- hauser [7] and Fokas and Ablowitz [9]. Instead, in the case of the IVP (0.1) there is no integrability, CONTENTS 5 but three conserved quantities (see Sidi, Sulem and Sulem[42]), specifically » » I[u](t) = u dx,M[u](t) = u2 dx, R » § § » R § 1+α §2 1 § 2 § 1 3 H[u](t) = §Dx u§ dx ¡ u dx, 2 R 6 R are satisfied at least for smooth solutions. Another property in which these two models differ, relies in the fact that one can obtain a local existence theory for the KdV equation in Hs(R), based on the contraction principle. On the contrary, this cannot be done in the case of the (IVP) (0.1). This is a consequence of the fact that dispersion is not enough to deal with the nonlinear term. In this direction, Molinet, Saut and s Tzvetkov [36] showed that for 0 ¤ α 1 the IVP (0.1) with the assumption u0 P H (R) is not enough to prove local well-posedness by using fixed point arguments or Picard iteration method. Nevertheless, Herr, Ionescu, Kenig and Koch [15] show that the IVP (0.1) is globally well-posed in the space of the real-valued L2(R)¡functions, by using a renormalization method to control the strong low-high frequency interactions. It is not clear that this theory can be used to establish our main result. We need to have a local theory obtained by using energy estimates in addition to dispersive properties of the smooth solutions. In the first step, we obtain the following a priori estimate for solutions of IVP (0.1)

c}B u} 8 x L1 L }u} 8 s }u } s e T x , LT Hx . 0 Hx part of this estimate is based on the Kato-Ponce commutator estimate (see Kato and Ponce [21]). The inequality above reads as follows: in order to the solution u abide in the Sobolev space s H (R), continuously in time, we require to control the term }Bxu} 1 8 . LT Lx First, we use Kenig, Ponce and Vega in [27] results concerning oscillatory integrals, in order to α+1B obtain the classical Strichartz estimates associated to the group S(t) = etDx x , corresponding to the linear part of the equation in (0.1). In second place, the technique introduced by Koch and Tzvetkov [29] related to reﬁned Strichartz estimate are fundamentals in our analysis. More precisely, their method is mainly based in a decomposition of the time interval in small pieces whose length depends on the spatial frequencies of the solution. This approach allowed to Koch and Tzvetkov to prove local well-posedness, for the + Benjamin-Ono equation in H5/4 (R). Succeeding, Kenig and Koenig [23] enhanced this estimate, + which led to prove local well-posedness for the Benjamin-Ono equation in H9/8 (R). Several issues arise when handling the nonlinear part of the equation in (0.1), nevertheless, following the work of Kenig, Ponce and Vega [24], we manage the loss of derivatives by means of a combination of the local smoothing effect and a maximal function estimate of the group α+1B S(t) = etDx x . These observations lead us to present our ﬁrst result.

THEOREM A. Let 0 α 1. Set s(α) = 9 ¡ 3α and assume that s ¡ s(α). Then, for any 8 8 P s } } ¡ u0 H (R), there exists a positive time T = T u0 Hs(R) 0 and a unique solution u satisfying (0.1) such that s 1 8 (0.5) u P C ([0, T] : H (R)) and Bxu P L ([0, T] : L (R)) . CONTENTS 6

! ) ¡ ÞÑ P s } } Moreover, for any r 0, the map u0 u(t) is continuous from the ball u0 H (R) : u0 Hs(R) to C ([0, T] : Hs(R)) .

TheoremA is the base result to describe the propagation of regularity phenomena. As we mentioned above the propagation of regularity phenomena is satisfied by the BO and KdV equations respectively. These two models correspond to particular cases of the IVP (0.1), specifically by taking α = 0 and α = 1. A question that arises naturally is to determine whether the propagation of regularity phenomena is satisfied for a model with an intermediate dispersion between these two models mentioned above. Our second main result give answer to this problem and it is summarized in the following:

P s 3¡α THEOREM B. Let u0 H (R) with s = 2 , and u = u(x, t), be the corresponding solution of the IVP (0.1) provided by TheoremA. + If for some x0 P R and for some m P Z , m ¥ 2, Bm P 2 t ¥ u (0.6) x u0 L ( x x0 ) , then for any v ¥ 0, T ¡ 0, e ¡ 0 and τ ¡ e » » » 8 + ¡ 2 2 T x0 τ vt α+1 j 2 j sup Bxu (x, t)dx + Dx Bxu (x, t)dx dt ¡ ¡ 0¤t¤T x0+e vt 0 x0+e vt » » ¡ T x0+τ vt α+1 2 2 j + Dx HBxu (x, t)dx dt ¤ c ¡ 0 x0+e vt

m for j = 1, 2, . . . , m with c = c T; e; v; α; }u } s ; }B u } 2 8 ¡ 0. 0 H x 0 L ((x0, )) + If in addition to (0.6) there exists x0 P R

1¡α 2 Bm P 2 t ¥ u (0.7) Dx x u0 L ( x x0 ) then for any v ¥ 0, e ¡ 0 and τ ¡ e » » » 8 2 + ¡ 1¡α T x0 τ vt 2 2 Bm Bm+1 sup Dx x u (x, t) dx + x u (x, t) dx dt ¡ ¡ 0¤t¤T x0+e vt 0 x0+e vt (0.8) » » ¡ T x0+τ vt 2 Bm+1 ¤ + x Hu (x, t) dx dt c ¡ 0 x0+e vt ! 1¡α } } 2 Bm ¡ with c = c T; e; v; α; u0 Hs ; Dx x u0 0. 2 8 L ((x0, ))

REMARK 0.2. (I) It will be clear from our proof that the requirement on the initial data, 3¡α 9¡3α + that is u0 P H 2 (R) in TheoremB, can be lowered to H 8 (R). (II) Also it is worth highlighting that the proof of TheoremB can be extended to solutions of the IVP # B ¡ α+1B kB P P + tu Dx xu + u xu = 0, x, t R, 0 α 1, k Z , u(x, 0) = u0(x). CONTENTS 7

(III) The results in TheoremB still holds for solutions of the defocussing generalized dispersive Benjamin-Ono equation # B ¡ α+1B ¡ B P tu Dx xu u xu = 0, x, t R, 0 α 1, u(x, 0) = u0(x).

This can be seen applying TheoremB to the function v(x, t) = u(¡x, ¡t), where u(x, t) is a solution of (0.1). In short, TheoremB remains valid, backward in time for initial data

u0, satisfying (0.6) and (0.7).

Although the argument of the proof of TheoremB follows in spirit that of KdV i.e. an induction process combined with weighted energy estimates. The presence of the non-local operator α+1B Dx x, in the term providing the dispersion, makes the proof much harder. More precisely, two difficulties appear, in the first place and the most important is to obtain explicitly the Kato smoothing effect as in Kato [19], that as in the proof of Theorem 0.1 it is fundamental. In contrast to KdV equation, the gain of the local smoothing in solutions of the dispersive gen- α+1 eralized Benjamin-Ono equation is just 2 derivatives, so as occurs in the case of the Benjamin- Ono equation Isaza, linares and Ponce [18], the iterative argument in the induction process is 1¡α carried out in two steps, one for positive integers m and another one for m + 2 derivative. In the case of the BO equation Isaza, linares and Ponce [18], the authors obtain the smoothing effect basing their analysis on several commutator estimates, such as the extension of the first Calderon’s commutator for the Hilbert transform (see B. Bajsanskiˇ and Coifman [1]). However, their method of proof do not allow them to obtain explicitly the local smoothing as in Kato [19]. The advantage of our method is that it allows obtain explicitly the smoothing effect for any α P (0, 1) in the IVP (0.1). Roughly, we rewrite the term modeling the dispersive part of the h α+2 2 i equation in (0.1), in terms of an expression involving HDx ; χe,b . At this point, we incorporate Ginibre and Velo [13] results about commutator decomposition. This, allows us to obtain explicitly the smoothing effect as in Kato [19], at every step of the induction process in the energy estimate. Concerning the nonlinear part of IVP (0.1) into the weighted energy estimate, several issues arises. Nevertheless, following Kenig et al. [22] approach, combined with the works of Kato-Ponce [21], and the recent work D. Li [30] on the generalization of several commutators estimate, allow us to overcome these difficulties. Next, we present some immediate consequences of TheoremB.

3¡α COROLLARY 0.3. Let u P C [¡T, T] : H 2 (R) be a solution of the equation in (0.1) described by P + ¤ P TheoremB. If there exist n , m Z with m n such that for some τ1, τ2 R with τ1 τ2

» 8 |Bn |2 8 Bm R 2 8 x u0(x) dx but x u0 L ((τ1, )), τ2 then for any t P (0, T) and any v ¡ 0 and e ¡ 0

» 8 |Bn |2 8 x u(x, t) dx , ¡ τ2+e vt CONTENTS 8

and for any t P (¡T, 0) and any τ3 P R » 8 |Bm |2 8 x u(x, t) dx = . τ3 The second part of the results we will present here are based mainly on verifying if propagation of regularity phenomena is also satisfied by real solutions when considering a model similar to that one in (0.1), but taken into consideration less dispersion than that in (0.1). More precisely, we studied the IVP # B u ¡ DαB u + uB u = 0, x, t P R, 0 α 1, (0.9) t x x x u(x, 0) = u0(x), where u = u(x, t) represents a real value function. This model is known in the literature as the fractional Korteweg-de Vries (fKdV) equation. Unlike the DGBO this new model can be seen as a ”dispersive” interpolation between the Burger’s equation (α = 0) and the Benjamin-Ono equation (α = 1). The IVP (0.9) has been the focus of attention of recent studies where the local well posedness theory has been approached as well as the existence of solitary wave solutions, the stability properties of ground states and numerical simulations, these issues have been largely carried out by Linares et al. [34], [33], Frank and Lenzmann [11], Klein and Saut [28] and Angulo [39]. Since, the applicability of the method used in the proof of TheoremB depends strongly on local well-posedness results in the Sobolev scale with the minimal regularity that ensures Bxu P 8 L1([0, T] : L (R)), our first task is to fix the space where will take place the solutions. To our knowledge the best result that fits with our requirements was obtained by Linares, Pilod and Saut [33] when studying weakly dispersive perturbation of Burger’s. More precisely, they obtain the following:

3 ¡ 3α ¡ THEOREM 0.4. Let 0 α 1. Deﬁne s(α) := 2 8 and assume that s s(α). Then, for every s u0 P H (R), there exists a positive time T = T(}u0}s,2) (which can be chosen as a non-increasing functions of its argument) and a unique solution u to (0.9) such that

s 1 8 (0.10) u P C ([0, T] : H (R)) and Bxu P L ([0, T] : L (R)) .

It is also important to point out that several improvements have been made in the well- posedness theory of the IVP (0.9). In this sense, we shall mention the recent work announced by Molinet, Pilod and Vento [38], where the authors prove that the IVP (0.9) is locally well-posedness s ¡ 3 ¡ 5α in the Sobolev space H (R), with s 2 4 . Additionally, Molinet, Pilod and Vento [38] proved α/2 ¡ 6 Global well-posedness (GWP) in the energy space H (R), as long as α 7 . Most of these results are based in a method introduced by Molinet and Vento [37] to obtain energy estimates at low regularity for strongly nonresonant dispersive equations. However, the s 3 5α LWP of the IVP (0.9) in the space H (R), s ¡ ¡ , does not ensure the boundeness of }Bxu} 1 8 2 4 LT Lx which is crucial in our analysis as we mentioned above. Since the space where we will consider solutions has already been described, we will proceed to describe our third main result. CONTENTS 9

P sα+ ¡ α THEOREM C. Let u0 H (R) where sα := 2 2 , and u = u(x, t) be the corresponding solution of the IVP (0.9) provided by Theorem 0.4. Suppose that for some x0 P R and some m P Z+, m ¥ 2, Bm P 2 t ¥ u (0.11) x u0 L ( x x0 ) . Then for any v ¥ 0, T ¡ 0, e ¡ 0 and τ ¡ e are satisﬁed the following relations: » » » 8 T x +τ¡vt 2 αn 2 0 j+α n+1 j 2 ( 2 ) sup BxDx u (x, t) dx + Dx u (x, t) dx dt ¡ ¡ 0¤t¤T x0+e vt 0 x0+e vt » » ¡ T x0+τ vt n+1 2 j+α( 2 ) + HDx u (x, t) dx dt ¤ c, ¡ 0 x0+e vt » » » 8 + ¡ ¡ α 2 T x0 τ vt 2 j 1 2 j+1 sup BxDx u (x, t) dx + Bx u (x, t) dx dt ¡ ¡ 0¤t¤T x0+e vt 0 x0+e vt » » ¡ T x0+τ vt j+1 2 + HBx u (x, t) dx dt ¤ c, ¡ 0 x0+e vt and » » » 8 T x +τ¡vt 0 m+ α 2 Bm 2 2 sup ( x u) (x, t) dx + Dx u (x, t) dx dt 0¤t¤T x +e¡vt 0 x +e¡vt » » 0 0 + ¡ T x0 τ vt α 2 m+ 2 + HDx u (x, t) dx dt ¤ c, ¡ 0 x0+e vt ¤ ¤ ¤ ¡ ¤ ¤ ¤ r 2 s ¡ r¤s where j = 2, 3, , m 1 and n = 0, 1, , α 1, where denotes the greatest integer. If in addition to (0.11)

1¡ α αj 2 Bm P 2 t ¥ u 2 Bm P 2 t ¥ u (0.12) Dx x u0 L ( x x0 ) and Dx x u0 L ( x x0 ) , ¤ ¤ ¤ r 2 s ¡ ¥ ¡ ¡ ¡ for j = 1, 2, , α 1, then for any v 0, T 0, e 0 and τ e

» » » + !2 αj 2 T m+α j 1 Bm 2 2 sup x Dx u (x, t) dx + Dx u (x, t) dx dt 0¤t¤T R 0 R » » 2 T j+1 ! m+α 2 + HDx u (x, t) dx dt ¤ c, 0 R and » » » 8 T x +τ¡vt 1¡ α 2 0 2 Bm 2 Bm+1 sup x Dx u (x, t) dx + x u (x, t) dx dt ¡ ¡ 0¤t¤T x0+e vt 0 x0+e vt » » ¡ T x0+τ vt 2 Bm+1 ¤ + H x u (x, t) dx dt c. ¡ 0 x0+e vt

REMARK 0.5. We shall remark the difference between the index s(α) and sα used in Theorem 0.4 and TheoremB, respectively.

The method of proof used in TheoremC follows in spirit the same lines than that used in the proof of TheoremB. Nevertheless, the main differences relay on the inductive argument. In the case of the DGBO equation, the induction is carried out in two steps one for a positive integer m CONTENTS 10

1¡α and another for m + 2 , 0 α 1. In contrast, in the fKdV is required a bi-induction argument, the first for a positive integer m, followed by a a sequence of steps of the form m + αj/2, j = ¤ ¤ ¤ r 2 s ¡ ¡ 1, , α 1, and a final one step for m + 1 α/2. A direct consequence of the Theorems B and C is that for an appropriated class of initial data, the regularity of the solutions is propagated with infinite speed to the left as time evolves. Also, the time reversibility property implies that the solution cannot have had some regularity in the past. As well as in the case of the DGBO equation, the propagation of regularity, also is satisfied in some related models to the (0.9). More precisely, the following properties are verified:

REMARK 0.6. (I) It will be clear from our proof that the requirement on the initial data, ¡ α + P (2 2 ) sα+ 3 ¡ 3α that is u0 H (R) in TheoremC can be lowered to H (R), where sα = 2 8 . (II) Also, it is worth highlighting that the proof of TheoremC can be extended to solutions of the IVP # B ¡ αB kB P P + tu Dx xu + u xu = 0, x, t R, 0 α 1, k Z , u(x, 0) = u0(x). (III) The results in TheoremC still holds for solutions of the defocussing fractional Korteweg-de Vries equation # B ¡ α+1B ¡ B P tu Dx xu u xu = 0, x, t R, 0 α 1, u(x, 0) = u0(x). This can be seen applying TheoremC to the function v(x, t) = u(¡x, ¡t), where u(x, t) is a solution of (0.9). In short, TheoremC remains valid, backward in time for initial data

u0, satisfying (0.11) and (0.12).

The document is organized as follows: the first chapter is focused in fix the notation used throughout all the document as well as to present several known results concerning commutators estimates joint with a particular decomposition of these. In chapter 2 we make a review of several issues concerning oscillatory integrals that will allow us to obtain refined Strichartz estimates s ¡ 9¡3α required to proof the local well-posedness of the DGBO equation in the space H (R), s 8 . In the second part of chapter 2 we present the proof of TheoremB. Finally, in the chapter 3 we present the proof of TheoremC. As a direct consequence of the TheoremsB andC, one has that for an appropriate class of initial data, the singularity of the solution travels with infinity speed to the left as time evolves. Also, the time reversibility property implies that the solution cannot have had some regularity in the past. Chapter 1 Preliminary

This chapter is devoted to ﬁx the notation used throughout all the document as well as to make a review of some commutators estimates of Kato-Ponce type, interpolation inequalities and ﬁnally a pointwise commutator decomposition that will be fundamental of Theorems B and C.

1. Notation

s The following notation will be used extensively throughout this work. The operators Dx = ¡B2 1/2 s ¡B2 s/2 ¡ ( x) and J = (1 x) denote the Riesz potential and the Bessel potentials of order s, respectively. + m For m P Z the Sobolev space H ((x0, 8)) coincides with the space of the functions f P 2 j 2 L ((x0, 8)) whose derivatives B f (in the distributional sense) belongs to L ((x0, 8)) for every j P Z+ with j ¤ m. p For 1 ¤ p ¤ 8, L (R) is the usual Lebesgue space with the norm indicated by } ¤ }Lp . In addition, for s P R, we consider the Sobolev space Hs(R) defined as the set of tempered distributions 2 s/2 p P 2 } } } s } ( f , such that (1 + ξ ) f (ξ) L (R) with its norm defined as f Hs = J f 2 , and the space H˙ R) p L consist of tempered distributions such that |ξ|s f (ξ) P L2(R). In this context, we define £ 8 H (R) = Hs(R). s¥0 Let f = f (x, t) be a function defined for every x P R and t in the interval [0, T], with T ¡ 0 or in p the hole line R. Then, if A denotes any of the spaces defined above, we define the spaces LT Ax p and Lt Ax by the norms

» 1/p » T ! 1/p p p } f } p = } f (¤, t)} dt and } f } p = } f (¤, t)} dt , L Ax A L Ax A T 0 t R with the natural modiﬁcation in the case p = 8. Moreover, we use similar deﬁnitions for the mixed q p q p ¤ ¤ 8 spaces Lx Lt and Lx LT with 1 p, q . For A, B operators we indicate the commutator between A and B as [A; B] = AB ¡ BA.

11 2. COMMUTATORS AND INEQUALITIES 12

The set of the even and odd numbers will be denoted by Q1 and Q2 respectively. Additionally, P + for m Z the sets Q1(m) and Q2(m) will denote the set of the even and odd numbers between 1 and m, respectively. For two quantities A and B, we denote by A . B whenever A ¤ cB, for some positive constant c. Similarly, A & B if A ¥ cB, for some c ¡ 0. We say that A and B are comparable when A . B and B . A and it will be indicated as A B. The dependence on the constant c on other parameters are usually clear from the context and we will often suppress this dependence whenever it be possible. For a real number a we will denote by a+ instead of a + e, whenever e is a positive number whose value is small enough.

2. Commutators and Inequalities

In this section, we state several inequalities to be used in the next sections. First, we have an extension of the Calderon commutator theorem (see Calderon [6]) established by B. Bajsanskiˇ et al. [1].

+ THEOREM 1.1. For any p P (1, 8) and any l, m P Z Y t0u there exists c = c(p; l; m) ¡ 0 such that Bl Bm ¤ Bm+l } } (1.1) x [H; ψ] x f c x ψ f Lp . Lp L8 For a different proof see [8] Lemma 3.1. In our analysis the Leibniz rule for fractional derivatives, established by Grafakos and Se- ungly [14], Kato and Ponce [21] and Kenig, Ponce and Vega [25]; will be crucial. Even though most of these estimates are valid in several dimensions, we will restrict our attention to the one- dimensional case.

LEMMA 1.2. For s ¡ 0, p P [1, 8) } s } } } } s } } } } s } (1.2) D ( f g) Lp . f Lp1 D g Lp2 + g Lp3 D f Lp4 with 1 1 1 1 1 P 8 = + = + , pj (1, ], j = 1, 2, 3, 4. p p1 p2 p3 p4 Also, we will state the fractional Leibniz rule proved by Kenig, Ponce and Vega [24]. P P P 8 LEMMA 1.3. Let s = s1 + s2 (0, 1) with s1, s2 (0, s), and p, p1, p2 (1, ) satisfy 1 1 1 = + . p p1 p2 Then,

} s ¡ s ¡ s } } s1 } } s2 } (1.3) D ( f g) f D g gD f Lp . D f Lp1 D g Lp2 .

Moreover, the case s2 = 0 and p2 = 8 is allowed. A natural question about Lemma 1.3 is to investigate the possible generalization of the estimate (1.3) when s ¥ 1. The answer to this question was given recently by Li [30], where he establishes new fractional Leibniz rules for the nonlocal operator Ds, s ¡ 0, and related ones, including various end-point situations. 2. COMMUTATORS AND INEQUALITIES 13

THEOREM 1.4. Case 1: 1 p 8. ¡ 8 ¥ P Let s 0 and 1 p . Then for any s1, s2 0 with s = s1 + s2, and any f , g S(R), the following hold: 8 1 1 1 (1) If 1 p1, p2 with p = p + p , then 1 2 ¸ ¸ 1 1 β s ¡ Bα s,α ¡ B s,β } s1 } } s2 } D ( f g) f D g gD f D f p1 D g p2 . α! x β! x . L L α¤s β¤s 1 2 Lp (2) If p = p, p = 8, then 1 2 ¸ ¸ 1 1 β s ¡ Bα s,α ¡ B s,β } s1 } } s2 } D ( f g) x f D g x gD f . D f Lp D g BMO, α! β! α s1 β¤s 2 Lp } ¤ } 1 where BMO denotes the norm in the BMO space . (3) If p = 8, p = p, then 1 2 ¸ ¸ s 1 α s,α 1 β s,β s s D ( f g) ¡ B f D g ¡ B gD f }D 1 f } }D 2 g} p . α! β! . BMO L α¤s β s 1 2 Lp The operator Ds,α is deﬁned via Fourier transform2 z y Ds,αg(ξ) = Ds,α(ξ)gp(ξ),

ys,α ¡αBα | |s D (ξ) = i ξ ( ξ ) . 1 ¤ Case 2: 2 p 1. If 1 p ¤ 1, s ¡ 1 ¡ 1 or s P 2N, then for any 1 p , p 8 with 1 = 1 + 1 , any s , s ¥ 0 2 p 1 2 p p1 p2 1 2 with s + s = s, 1 2 ¸ ¸ 1 1 β s ¡ Bα s,α ¡ B s,β } s1 } } s2 } D ( f g) f D g gD f D f p1 D g p2 . α! x β! x . L L α¤s β¤s 1 2 Lp ° REMARK 1.5. As usual empty summation (such as 0¤α 0) is deﬁned as zero.

PROOF. For a detailed proof of this Theorem and related results, see Li [30]. Next we have the following commutator estimates involving non-homogeneous fractional derivatives, established by Kato and Ponce [21]. ¡ P 8 P 8 LEMMA 1.6. Let s 0 and p, p2, p3 (1, ) and p1, p4 (1, ] be such that 1 1 1 1 1 = + = + . p p1 p2 p3 p4

1For any f P L1 (Rn), the BMO semi-norm is given by loc » } } 1 | ¡ | f BMO = sup | | f (y) ( f )Q dy, Q Q Q

n where ( f )Q is the average of f on Q, and the supreme is taken over all cubes Q in R . 2The precise form of the Fourier transform does not matter. 2. COMMUTATORS AND INEQUALITIES 14

Then, } s } }B } } s¡1 } } s } } } (1.4) [J ; f ]g Lp . x f Lp1 J g Lp2 + J f Lp3 g Lp4 and } s } } s } } } } s } } } J ( f g) Lp . J f Lp1 g Lp2 + J g Lp3 f Lp4 . There are many other reformulations and generalizations of the Kato-Ponce commutator inequalities (cf. [2] and the references therein). Recently Li [30], has obtained a family of reﬁned Kato-Ponce type inequalities for the operator Ds. In particular he showed that 8 ¤ 8 LEMMA 1.7. Let 1 p . Let 1 p1, p2, p3, p4 satisfy 1 1 1 1 1 = + = + . p p1 p2 p3 p4 Therefore, (a) If 0 s ¤ 1, then } s ¡ s } } s¡1B } } } D ( f g) f D g Lp . D x f Lp1 g Lp2 . (b) If s ¡ 1, then } s ¡ s } } s¡1B } } } }B } } s¡1 } (1.5) D ( f g) f D g Lp . D x f Lp1 g Lp2 + x f Lp3 D g Lp4 . For a more detailed exposition on these estimates see section 5 in Li [30]. In addition, a non sharp commutator estimate is required in our analysis.

8 1 8 LEMMA 1.8. Let φ P C (R) with φ P C (R). If f P Hs(R), s ¡ 0, then for any l ¡ s + 1 0 2 s 1 }[ ] } } } ¡ Dx; φ f 2 . φ l,2 f s 1,2.

PROOF. We will give an outline of the proof. s ¡ First, we use the decomposition of the operator Dx for s 0, given by Bourgain and Li [5] in the study of commutator estimates. In fact, ¸ s s ¡ s¡2j Dx = Jx cs,j Jx + Ks ¤ ¤ s 1 j 2 p p where Ks is a bounded integral operator satisfying Ks : L (R) ÝÑ L (R), 1 ¤ p ¤ 8. By using this decomposition and a proof similar to that found in [10, Chapter 6, Lemma 6.16 ], the lemma follows. In addition, we have the following inequality of Gagliardo-Nirenberg type:

LEMMA 1.9. Let 1 q, p 8, 1 r ¤ 8 and 0 α β. Then, } α } } }1¡θ} β }θ D f Lp . c f Lr D f Lq with 1 1 1 ¡ α = (1 ¡ θ) + θ ¡ β , θ P [α/β, 1]. p r q PROOF. See chapter 4 in Berg and Loftr¨ om¨ [3]. 2. COMMUTATORS AND INEQUALITIES 15

Now, we present a result that will help us to establish the propagation of regularity of solutions of (0.1). A previous result was proved by Kenig et al.(c.f [22], Corollary 2.1) using the fact that Jr (r P R) can be seen as a pseudo-differential operator. Thus, this approach allows to obtain an expression for Jr in terms of a convolution with a certain kernel k(x, y) which enjoys some properties on localized regions in R. In fact, this is known as the singular integral realization of a pseudo-differential operator, whose proof can be found in Stein [43] Chapter 4. The estimate we consider here involves the non-local operator Ds instead of Js.

+ LEMMA 1.10. Let m P Z and s ¥ 0. If f P L2(R) and g P Lp(R), 2 ¤ p ¤ 8, with

(1.6) dist (supp( f ), supp(g)) ¥ δ ¡ 0.

Then } Bm s } } } } } g x D f Lp . g Lp f L2 .

PROOF. Let f , g be functions in the Schwartz class satisfying (1.6). Notice that » g(x) y g(x) (Ds Bm f ) (x) = ixξ | |sBm f ( ) x x 1/2 e ξ x ξ dξ (2π) R (1.7) » g(x) s { = | | ( ¡ Bm f )( ) 1/2 ξ τ x x ξ dξ. (2π) R

3 where τh is the translation operator. Moreover, the last expression in (1.7) deﬁnes a tempered distribution for s in a suitable class, that will be speciﬁed later. Indeed, for z P C with ¡1 Re(z) 0 » » Bm 1 z { (τ¡x x ϕ) (y) | | ( ¡ Bm )( ) = c(z) y @ P S(R) (1.8) 1/2 ξ τ x x ϕ ξ dξ 1+z d , ϕ (2π) R R |y|

¡ 2 with c(z) is independent of ϕ. In fact, evaluating ϕ(x) = e x /2 in (1.8) yields

2z Γ z+1 c(z) = 2 . 1/2 ¡ z π Γ 2 Thus, for every ϕ P S(R) the right hand side in (1.8) deﬁnes a meromorphic function for every test function, which can be extended analytically to a wider range of complex numbers z’s, speciﬁcally z with Im(z) = 0 and Re(z) = s ¡ 0 that is the case that attains us. By an abuse of notation, we will denote the meromorphic extension and the original as the same. Thus, combining (1.6), (1.7) and (1.8) it follows that » g(x) (τ¡ Bm f ) (y) ( ) ( s Bm ) ( ) = ( ) x x g x Dx x f x c s 1+s dy R |y| 1t|y|¥δu = c(s)g(x) f ¦ (x) |y|s+m+1

3 P ¡ For h R the translation operator τh is deﬁned as (τh f ) (x) = f (x h). 3. COMMUTATOR EXPANSIONS 16

Notice that the kernel in the integral expression is not anymore singular due to the condition (1.6). In fact, in the particular case that m is even, we obtain after apply integration by parts 1t|y|¥δu g(x) (Ds Bm f ) (x) = c(s, m)g(x) f ¦ (x) x x |y|s+m+1 and in the case m being odd y1t|y|¥δu g(x) (Ds Bm f ) (x) = c(s, m)g(x) f ¦ (x). x x |y|s+m+2

Finally, in both cases combining Young’s inequality and Holder’s¨ inequality one gets 1t| |¥ u m s y δ }g B D f } 2 }g} p } f } 2 x x L . L L | ¤ |s+m+1 Lr } } } } . g Lp f L2 where the index p satisﬁes 1 1 1 = + , 2 p r which clearly implies p P [2, 8], as was required.

Further, we will use extensively some results about commutator additionally to those presented in previous section. Next, we will study the smoothing effect for solutions of the dispersive generalized Benjamin-Ono equation (0.1) following Kato’s ideas [19].

3. Commutator Expansions

In this section we present several tools obtained by Ginibre and Velo [12], [13] which will be the cornerstone in the proof of TheoremB. They include commutator expansions together with their estimates. The basic problem is to handle the non-local operator Ds for non-integer s and in particular to obtain representations of its commutator with multiplication operators by functions that exhibit as much locality as possible. Let a = 2µ + 1 ¡ 1, let n be a non-negative integer and h be a smooth function with suitable 1 P 8 decay at inﬁnity, for instance with h C0 (R). We deﬁne the operator 1 (1.9) R (a) = [HDa; h] ¡ (P (a) ¡ HP (a)H) , n 2 n n

¸ ¡ j ¡j µ¡j (2j+1) µ¡j (1.10) Pn(a) = a c2j+1( 1) 4 D h D 0¤j¤n where ¹ 1 2 2 c = 1, c + = a ¡ (2k + 1) and H = ¡H. 1 2j 1 (2j + 1)! 0¤k j 3. COMMUTATOR EXPANSIONS 17

4 It was shown in [12] that the operator Rn(a) can be represented in terms of anti-commutators as follows 1 (1.11) R (a) = [H; Q (a)] + [Da; [H; h]] , n 2 n + + where the operator Qn(a) is represented in the Fourier space variables by the integral kernel 1 p 1 1 a (1.12) Qn(a) ÝÑ (2π) 2 h(ξ ¡ ξ )|ξξ | 2 2aqn(a, t), 1 with |ξ| = |ξ | e2t, and » t 1 2 ¡ 2 2n+1 ¡ qn(a, t) = (a (2n + 1) )c2n+1 sinh τ sinh((a(t τ))) dτ. a 0 Based on (1.11) and (1.12), Ginibre and Velo [13] obtain the following properties of boundedness and compactness of the operator Rn(a).

PROPOSITION 1.11. Let n be a non-negative integer, a ¥ 1, and σ ¥ 0, be such that (1.13) 2n + 1 ¤ a + 2σ ¤ 2n + 3. Then (a) The operator DσR (a)Dσ is bounded in L2 with norm n σ σ ¡1/2 {a+2σ (1.14) }D R (a)D f } 2 ¤ C(2π) (D h) } f } 2 . n L 1 L Lξ If a ¥ 2n + 1, one can take C = 1. (b) Assume in addition that 2n + 1 ¤ a + 2σ 2n + 3. σ σ 2 Then the operator D Rn(a)D is compact in L (R).

PROOF. See Proposition 2.2 in Ginibre and Velo [13]. In fact the Proposition 1.11 is a generalization of a previous result, where the derivatives of operator Rn(a) are not considered (cf. Proposition 1 in Ginibre and Velo [12]).

4 For any two operators P and Q we denote the anti-commutator by [P; Q]+ = PQ + QP. Chapter 2 Dispersive generalized Benjamin-Ono equation: local well-posedness & propagation of regularity

In this chapter we will establish the local well-posedness result that will be used later to prove our main result concerning the DGBO equation. Our argument of proof uses energy estimates. The key point behind this argument is to establish an estimate of the type }Bxu} 1 8 ¤ cT. To reach this goal we prove a reﬁned Strichartz LT Lx estimate for solutions associated to the linear problem. Once we establish the local well-posedness in a suitable Sobolev space that guarantee us that

}Bxu} 1 8 ¤ cT, we proceed to show the arguments behind the proof of Theorem B. LT Lx

1. The Linear Problem.

The aim of this section is to obtain Strichartz estimates associated to solutions of the IVP (0.1). First, consider the linear problem # B u ¡ Dα+1B u = 0, x, t P R, 0 α 1, (2.1) t x x u(x, 0) = u0(x), whose solution is given by

α+1 it |ξ| ξ p (2.2) u(x, t) = S(t)u0 = e u0 q.

We begin studying estimates of the unitary group obtained in (2.2).

2 1 1 ¤ ¤ 8 PROPOSITION 2.1. Assume that 0 α 1. Let q, p satisfy q + p = 2 with 2 p . Then α q } } (2.3) Dx S(t)u0 u0 L2 , q p . x Lt Lx 2 for all u0 P L (R).

18 1. THE LINEAR PROBLEM. 19

PROOF. The proof follows as an application on Theorem 2.1 in Kenig, Ponce and Vega [27].

REMARK 2.2. Notice that the condition in p implies q P [4, 8], which in one of the extremal (p q) = (8 ) cases , , 4 yields α 4 D S(t)u }u } 2 x 0 4 8 . 0 Lx Lt Lx α which shows the gain of 4 derivatives globally in time for solutions of (2.1). 8 LEMMA 2.3. Assume that 0 α 1. Let ϕk be a C (R) function supported in the interval k¡1 k+1 P + α [2 , 2 ] where k Z . Then, the function Hk deﬁned as $ &' 2k i f |x| ¤ 1, α k | |¡ 1 ¤ | | ¤ k(α+1) Hk (x) = ' 2 2 x 2 i f 1 x c2 , % ¡1 1 + x2 i f |x| ¡ c2k(α+1) satisﬁes §» § § 8 § § i(tξ|ξ|α+1+xξ) § α (2.4) § e ψk(ξ)dξ§ . Hk (x) ¡8 for |t| ¤ 2, where the constant c does not depend on t nor k. Moreover, we have that ¸8 α+1 α | | k( 2 ) (2.5) Hk ( l ) . 2 . l=¡8

PROOF. The proof of estimate (2.4) is given in Proposition 2.6 Kenig, Ponce and Vega [26], and it uses arguments of localization and the classical Van der Corput’s Lemma. Meanwhile, (2.5) follows exactly that of Lemma 2.6 in [33].

¡ 1 THEOREM 2.4. Assume 0 α 1. Let s 2 . Then,

 ¸8 1/2 2 } ( ) } 8 ¤ | ( ) ( )| } } s S t u0 L2 L ([¡1,1])  sup sup S t u0 x  . u0 Hx x t | |¤ ¤ j=¡8 t 1 j x j+1

s for any u0 P H (R).

PROOF. See Theorem 2.7 in Kenig, Ponce and Vega [26].

Next, we recall a maximal function estimate proved by Kenig, Ponce and Vega [26].

¡ 1 ¡ 3 COROLLARY 2.5. Assume that 0 α 1. Then, for any s 2 and any η 4

 ¸8 1/2 2 η | ( ) | ( + ) } } s  sup sup S t v0  . 1 T v0 Hx . | |¤ ¤ j=¡8 t T j x j+1

PROOF. See Corollary 2.8 in Kenig, Ponce and Vega [26]. 2. THE NONLINEAR PROBLEM 20

2. The Nonlinear Problem

This section is devoted to study general properties of solutions of the non-linear problem # B u ¡ Dα+1B u + uB u = 0, x, t P R, 0 α 1, (2.6) t x x x u(x, 0) = u0(x). We begin this section stating the following local existence theorem proved by Kato [20] and Saut, Teman [40].

P s ¡ 3 THEOREM 2.6. (1) For any u0 H (R) with s 2 there exists a unique solution u to (2.6) in s the class C([¡T, T] : H (R)) with T = T(}u0}Hs ) ¡ 0. 1 s r r (2) For any T T there exists a neighborhood V of u0 in H (R) such that the map u0 ÞÝÑ u(t) from V into C ([¡T, T] : Hs(R)) is continuous. s1 1 (3) If u0 P H (R) with s ¡ s, then the time of existence T can be taken to depend only on }u0}Hs .

Our first goal will be obtain some energy estimates satisfied by smooth solutions of the IVP (2.6). We firstly present a result that arises as a consequence of commutator estimates.

8 LEMMA 2.7. Suppose that 0 α 1. Let u P C([0, T] : H (R)) be a smooth solution of (2.6). If s ¡ 0 is given, then

c}B u} 8 x L1 L (2.7) }u} 8 s }u } s e T x . LT Hx . 0 Hx

PROOF. Let s ¡ 0. By a standard energy estimate argument we have that » » » 1 d s 2 s B s s s B (Jxu) dx + [Jx; u] xu Jxu dx + uJxuJx xu dx = 0. 2 dt R R R Hence integration by parts, Gronwall’s inequality and commutator estimate (1.4) lead to (2.7).

REMARK 2.8. In view of the energy estimate (2.7), the key point to obtain a priori estimates in s s H (R) is to control }Bxu} 1 8 at the H (R)¡level. x LT Lx x Additionally to this estimate, we will present the smoothing effect provided by solutions of dispersive generalized Benjamin-Ono equation. In fact, the smoothing effect was ﬁrst observed by Kato in the context of the Korteweg-de Vries equation (see [19]). Following Kato’s approach joint with the commutator expansions, we present a result proved by Kenig-Ponce-Vega [26] (see Lemma 5.1). The estimative (1.14) yields the following identity of localization of derivatives. P 8 1 P 8 LEMMA 2.9. Assume 0 α 1. Let be ϕ C (R) with ϕ C0 (R). Then, » » § § § § 2 2 α+1 α + 2 § α+1 § § α+1 § 1 ϕ f D B f dx = §D 2 f § + §D 2 H f § ϕ dx x 4 (2.8) R » R 1 + f R0(α + 2) f dx, 2 R where R0(α + 2) is given as in (1.11) . 2. THE NONLINEAR PROBLEM 21

PROOF. First notice that h i α+1B α+1B ¡ α+1B (2.9) Dx x; ϕ f = Dx x(ϕ f ) ϕDx x f .

Using identity (2.9) and integration by parts we obtain » » » h i α+1 B ¡ α+1B α+1B ϕ f Dx ( x f ) dx = f Dx x; ϕ f dx + f Dx x (ϕ f ) dx (2.10) R »R »R h i ¡ α+1B ¡ B α+1 = f Dx x; ϕ f dx x f Dx (ϕ f ) dx, R R applying Plancherel’s identity to the second term in the right-hand side of (2.10) we ﬁnd » » B α+1 α+1B (2.11) x f Dx (ϕ f ) dx = ϕ f Dx x f dx. R R Thus, gathering (2.10) and (2.11) we obtain » » 1 h i ϕ f Dα+1(B f ) dx = ¡ f Dα+1B ; ϕ f dx x 2 x (2.12) R » R 1 = f HDα+2; ϕ f dx. 2 R α + 1 Now, we employ (1.9), with µ = and n = 0, which yields 2 1 1 [HDs+2; ϕ] = P (α + 2) ¡ HP (α + 2)H + R (α + 2) 2 0 2 0 0 (2.13) α + 2 α+1 1 α+1 α + 2 α+1 1 α+1 = D 2 ϕ D 2 ¡ HD 2 ϕ D 2 H + R (α + 2). 2 x x 2 x x 0 Hence, inserting (2.13) into (2.12) and using Plancherel’s identity we obtain » » § § » § § § + §2 § + §2 + α + 2 § α 1 § 1 α + 2 § α 1 § 1 f [HDα 2; ϕ] f dx = §D 2 f § ϕ (x) dx + §D 2 H f § ϕ (x) dx x 2 x 2 x (2.14) R » R R

+ f R0(α + 2) f dx. R Therefore, gathering (2.12) and (2.14) follows the identity (2.8).

This identity will be useful in establishing the smoothing effect in solutions of the DGBO equation and the fKdV equation.

1 PROPOSITION 2.10. Let ϕ denote a non-decreasing smooth function such that supp ϕ (¡1, 2) 1| P ¤ ¤ ¡ P 8 and ϕ [0,1) = 1. For j Z, we deﬁne ϕj( ) = ϕ( j). Let u C([0, T] : H (R)) be a real smooth solution of (0.1) with 0 α 1. Assume also, that s ¥ 0 and r ¡ 1/2. Then, » » § § § § ! !1/2 T § α+1 §2 § α+1 §2 § s+ 2 § § s+ 2 § 1 §Dx u(x, t)§ + §Dx Hu(x, t)§ ϕj(x) dx dt (2.15) 0 R 1/2 }B } } } 8 } } 8 1 + T + xu 1 8 + T u L Hr u L Hs . . LT Lx T x T x 2. THE NONLINEAR PROBLEM 22

s s PROOF. Apply the operator Dx to the equation in (0.1), multiply by Dxuϕj, and integrate in space to obtain » » » 1 d | s |2 ¡ s 1+α+sB s B s (2.16) Dxu ϕj dx ϕjDxu Dx xu dx + Dx(u xu)Dxu ϕj dx = 0. 2 dt R R R Now the term corresponding to the nonlinear part of equation in (0.1), can be handled through Mihlin’s Theorem (see [31] Theorem 2.8 ) and the commutator estimate in (1.4), » » » s B s ¡1 B 1 | s |2 s B s Dx(u xu)Dxu ϕj dx = xuϕj + uϕj Dxu dx + [Dx; u] xu Dxuϕj dx R 2 R R 2 }B u} 8 + }u} 8 }u} s . . x Lx Lx Hx However, Lemma 2.9 allow us to write the second term in the left-hand side of (2.16) as » » § § § § § §2 § §2! α + 2 s+ α+1 s+ α+1 1 s α+1 B s § 2 § § 2 § ϕjDxu D ( xDxu) dx = §Dx u§ + §Dx Hu§ ϕj dx R 4 R (2.17) » 1 s s + DxuR0(α + 2)Dxu dx. 2 R Replacing (2.17) into (2.16) we obtain » » d s 2 2 s s |D u| ϕ dx + 1 + }B u} 8 + }u} 8 }u} s ¡ D uR (α + 2)D u dx x j x Lx Lx Hx x 0 x dt R R » § § § § (2.18) 2 2! + § α+1 § § α+1 § α 2 § s+ 2 § § s+ 2 § 1 & ( ) §Dx u§ + §Dx Hu§ ϕj dx. 4 R

2 Since the operator R0(α + 2) satisﬁes (1.14) i.e., R0(α + 2) is bounded in the L -norm; we deduce after integrating in time in (2.18) that » » § § § § ! T § α+1 §2 § α+1 §2 § s+ 2 § § s+ 2 § §Dx u(x, t)§ + §Dx Hu(x, t)§ ϕ(x ¡ j) dx dt (2.19) 0 R }B } } } 8 8 } } 8 1 + T + xu 1 8 + T u L L u L Hs . . LT Lx T x T x Finally, (2.19) and Sobolev’s embedding yield estimate (2.15).

REMARK 2.11. Using a similar argument as the employed in the proof of Proposition 2.10 is possible obtain the smoothing effect provided by DGBO, but instead localized over any interval ¡ symmetric on the real line. This can be achieved replacing the function ϕj by φR, where R 0 and 1 ¡ 1 φR denotes a nondecreasing smooth function such that φR is supported in ( 2R, 2R) with φR = 1 on [¡R, R].

In addition to the smoothing effect presented above, we will need the following localized version of the Hs(R)-norm. For this propose we will consider a cutoff function ϕ, with the same properties that in Proposition 2.10.

PROPOSITION 2.12. Let s ¥ 0. Then, for any f P Hs(R)

 8 1/2 ¸ 2 } } 1 f Hs(R)  f ϕj  . Hs(R) j=¡8 3. REFINED STRICHARTZ ESTIMATE 23

3. Reﬁned Strichartz estimate

Our ﬁrst goal in establishing the local well-posedness of (2.6), will start off in obtaining Strichartz estimates associated to solutions of

B ¡ 1+αB (2.20) tu Dx xu = F.

PROPOSITION 2.13. Assume that 0 α 1, T ¡ 0 and σ P [0, 1]. Let u be a smooth solution to ¤ (2.20) deﬁned on the time interval [0, T]. Then there exist 0 µ1, µ2 1/2 such that ¡ α σ ¡ α ¡ 3σ µ1 1 + +ε µ2 1 +ε (2.21) }Bxu} 2 8 T J 4 4 u + T J 4 4 F LT Lx . 8 2 2 2 LT Lx LT Lx for any ε ¡ 0.

REMARK 2.14. The optimal choice in the parameters present in the estimate (2.21) corresponds 1¡α to σ = 2 . Indeed, as pointed out Kenig and Koenig in the case of the Benjamin-Ono equation (case α = 0) (see Remarks in Proposition 2.8 [23]) given a linear estimate of the form }B } µ1 } a } µ2 b xu 2 8 T J u 8 2 + T J F LT Lx . LT Lx 2 2 LT Lx the idea is to apply the smoothing effect (2.15) and absorb as many as derivatives as possible of the 1¡α function F. Concerning to our case, the approach requires the choice a = b + 2 ; this particular 1¡α ¡ ¡ choice, σ = 2 , in the estimate (2.21) provides the regularity s 9/8 3α/8 in TheoremA. ° PROOFOF PROPOSITION 2.13. Let f = k fk denote the Littlewood-Paley decomposition of a 8 function f . More precisely we choose functions η, χ P C (R) with

supp(η) tξ : 1/2 |ξ| 2u and supp(χ) tξ : |ξ| 2u, such that ¸8 η(ξ/2k) + χ(ξ) = 1 k=1 and fk = Pk( f ), where

{ p { k p ¥ (P0 f )(ξ) = χ(ξ) f (ξ) and (Pk f )(ξ) = η(ξ/2 ) f (ξ) for all k 1.

Recall that for 1 p 8 the Littlewood-Paley theorem

¸8 !1/2 } } | |2 f p Pk f . Lx k=0 p Lx 3. REFINED STRICHARTZ ESTIMATE 24

¡ ¡ 1 Fix ε 0. Let p ε . By Sobolev embedding and Littlewood-Paley Theorem it follows that

} } 8 } ε } f L . J f p x Lx ¸8 !1/2 |JεP f |1/2 k k=0 p L x 8 1/2 ¸ | ε |2 = J Pk f = p/2 k 0 Lx ¸8 !1/2 ε 2 }J P f } p . . k Lx k=0 Therefore, to obtain (2.21) it enough to prove that for p ¡ 2 ¡ α + σ + α¡σ ¡ α ¡ 3σ + α¡σ 1 4 4 2p 1 4 4 2p }B u } 2 p D u + D F , k ¥ 1. x k L Lx . x k x k T 8 2 2 2 LT Lx LT Lx The estimate for the case k = 0 follows using Holder’s¨ inequality and (2.3). For such reason we ﬁx k ¥ 1, and at these level of frequencies we have that

B ¡ α+1B tuk Dx xuk = Fk. Consider a partition of the interval [0, T] = jPJ Ij where Ij = [aj, bj], and T = bj for some j. kσ 1¡µ | | ¡kσ µ Indeed, we choose a quantity 2 T of intervals, with length Ij 2 T , where µ is a positive number to be ﬁxed. Let q be such that 2 1 1 + = . q p 2 Using that u solves the integral equation » t 1 1 1 (2.22) u(t) = S(t)u0 + S(t ¡ t )F(t ) dt , 0 we deduce that }B } xuk 2 p LT Lx ¸ 1/2 }B }2 =  xuk 2 p  LI Lx jPJ j

 1/2 1 ¡ 1 ¸ ¤ µ ¡kσ 2 q }B }2 T 2  xuk q p  LI Lx jPJ j  1/2 » 2 1 ¡ 1 ¸ t µ ¡kσ 2 q  ¡ B 2 ¡ B  . T 2 S(t aj) xuk(aj) q p + S(t s) x Fk(s)ds .  LI Lx  P j aj q p j J L Lx Ij 4. PROOF OF THEOREMA 25

In this sense, it follows from (2.3) that $ , & » !2.1/2 1 ¡ 1 ¸ ¡ α 2 ¸ ¡ α }B } µ ¡kσ 2 q q B q B xuk 2 p T 2 Dx xuk(aj) + Dx x Fk(t) dt LT Lx . % - P L2 P Ij L2 $ j J x j J x &' 1/2 1 ¡ 1 ¸ ¡ α 2 µ ¡kσ 2 q 1 q . T 2 ' Dx uk  % 8 2 jPJ LT Lx ,  » 1/2./ ¸ ¡ α 2 µ ¡kσ 1 q +  T 2 Dx Fk(t) dt / 2 - jPJ Ij Lx 1 ¡ 1 1/2 ¡ α µ ¡kσ 2 q 1¡µ kσ 1 q . T 2 T 2 Dx uk 8 2 LT Lx » !1/2 1 ¡ 1 1/2 T ¡ α 2 µ ¡kσ 2 q µ ¡kσ 1 q + T 2 T 2 Dx Fk(t) dt 0 2 Lx ¡ α + σ ¡ α + σ ¡ 1/2¡µ/q 1 q q µ(1¡1/q) 1 q q σ . T Dx uk + T Dx Fk . 8 2 2 2 LT Lx LT Lx ¡ α σ ¡ α σ α¡σ ¡ α σ ¡ ¡ α ¡ 3σ α¡σ ¡ 1 Since, 1 q + q = 1 4 + 4 + 2p and 1 q + q σ = 1 4 4 + 2p . We recall that ε p , P P α¡σ ¡ α¡σ+2 ¡ 1 ¡ µ ¡ 1 σ [0, 1] and α (0, 1), then ε + 2p 2p 0. Next, we choose µ1 = 2 q , µ2 = µ(1 q ) with the particular choice µ = 1/2. Gathering the inequalities above follows the proposition. Now we turn our attention to the proof of TheoremA. Our starting point will be the energy estimate (2.7), that as was remarked above, the key point is to establish a priori control of }Bxu} 1 8 . LT Lx

4. Proof of TheoremA

4.1. A priori estimates. First notice that by scaling, it is enough to deal with small initial data in the Hs¡norm. Indeed, if u(x, t) is a solution of (0.1) deﬁned on a time interval [0, T], for some ¡ 1+α 2+α positive time T, then for all λ 0, uλ(x, t) = λ u(λx, λ t) is also solution with initial data 1+α 2+α u0,λ(x) = λ u0(λx), and time interval 0, T/λ . ¡ s For any δ 0, we deﬁne Bδ(0) as the ball with center at the origin in H (R) and radius δ. Since 1+2α 1 } } 2 } } } s } 2 +α+s } s } u 2 = λ u0 2 and D u 2 = λ D u0 2 , 0,λ Lx Lx x 0,λ Lx x Lx then 1 +α s } } s 2 ( + )} } s u0,λ Hx . λ 1 λ u0 Hx , ¤ so we can force uλ( , 0) to belong to the ball Bδ(0) by choosing the parameter λ with the condition " * 1+2α ¡ 1+2α 2 } } 2 λ min δ u0 s , 1 . Hx

Thus, the existence and uniqueness of a solution to (0.1) on the time interval [0, 1] for small initial

} } s data u0 Hx will ensure the existence and uniqueness of a solution to (0.1) for arbitrary large initial 4. PROOF OF THEOREMA 26 data on a time interval [0, T] with

" * ¡ 2(2+α) } } 1+2α T min 1, u0 s . Hx

Thus, without loss of generality we will assume that T ¤ 1, and that

} } } s } ¤ Λ := u0 L2 + Dxu0 L2 δ, where δ is a small positive number to be ﬁxed later. We ﬁx s such that

9 3α 3 α s(α) = ¡ s ¡ 8 8 2 2 and set ε = s ¡ s(α) ¡ 0. 1¡α ¡ ¡ B Next, taking σ = 2 0, F = u xu in (2.21) together with (2.7) yields

¡ α ¡ 3σ + }B } µ1 } s } 8 µ2 1 4 4 ε B xu L2 L8 . T J u L L2 + T J (u xu) T x T x L2 L2 T x ¡ ! s+ α 1 µ1 } } } s } 8 µ2 } B } 2 B T u0 2 + D u 2 + T u xu 2 2 + Dx (u xu) . Lx x LT Lx LT Lx L2 L2 T x (2.23) ¡ c}B u} 8 α 1 x L2 L s+ 2 T Λ + Λ e T x +}Bxu} 2 8 }u0} 2 + Dx (uBxu) . LT Lx Lx L2 L2 T x }B } α¡1 c xu 2 8 s+ L Lx 2 B .T Λ + Λ e T + Dx (u xu) . 2 2 LT Lx

Now, to analyze the product coming from the nonlinear term we use the Leibniz rule for fractional derivatives (1.3) joint with the energy estimate (2.7) as follows

¡ ¡ ¡ s+ α 1 s+ α 1 s+ α 1 D 2 (uB u) uD 2 B u + }B u(t)} 8 D 2 u(t) x x . x x x Lx x 2 2 2 2 2 LT Lx LT Lx Lx L2 T α¡1 α¡1 s+ 2 s+ 2 ¤ uD B u + }B u} 2 8 D u x x x L Lx x 2 2 T 8 2 (2.24) LT Lx LT Lx ¡ s+ α 1 } 2 B } }B } } } 8 . uDx xu L2 L2 + xu L2 L8 u L Hs T x T x T x α¡1 }B } s+ c xu 2 8 2 B L Lx . uDx xu + Λ e T . 2 2 LT Lx 4. PROOF OF THEOREMA 27

To handle the ﬁrst term in the right hand side above, we incorporate Kato’s smoothing effect estimate obtained in (2.15) in the following way

» » !1/2 α¡1 T α¡1 s+ 2 s+ 2 2 uDx Bxu = |uDx Bxu| dx dt 2 2 R LT Lx 0  » 1/2 ¸8 T 2 s+ α+1 ¤ } }2 2  u(t) L8 Dx Hu(t) dt 0 [j,j+1) 2 j=¡8 L[j,j+1)  1/2 ¸8 2 s+ α+1 ¤ } }2 2  u L8 L8 Dx Hu  T [j,j+1) 2 2 j=¡8 LT L[j,j+1) (2.25)  1/2 ¸8 s+ α+1 ¤ } }2 2  u 8 8  sup Dx Hu LT L[ + ) j,j 1 jPZ 2 2 j=¡8 LT L[j,j+1)

 8 1/2 ¸ }B } c xu 2 8 } }2 }B } L Lx  u 8 8  1 + Λ + xu 2 8 Λ e T . LT L[j,j+1) LT Lx j=¡8

 ¸8 1/2 c}B u} 8 2 x L2 L  }u} 8 8  (1 + Λ) Λ e T x . . LT L[j,j+1) j=¡8

In summary, after gathering the estimates (2.23)-(2.25) yields

 8 1/2 }B } ¸ }B } c xu 2 8 c xu 2 8 }B } L Lx } }2 L Lx (2.26) xu 2 8 Λ(1 + Λ) e T  u 8 8  + Λ + Λ e T . LT Lx . LT L[j,j+1) j=¡8

Since u is a solution to (2.6), then by Duhamel’s formula it follows that

» t u(t) = S(t)u0 ¡ S(t ¡ s) (uBxu) (s) ds 0

α+1B where S(t) = etDx x . ¡ 1+α Now, we ﬁx η 0 such that η 8 ; this choice implies that

1 α ¡ 1 η + s + . 2 2 4. PROOF OF THEOREMA 28

Hence, Sobolev’s embedding, Holder’s¨ inequality and Corollary 2.5 produce (2.27)  1/2  » 1/2 ¸8 ¸8 t 2 } }2 ¡ B  u L8 L8  =  S(t)u0 S(t¡s)(u xu) ds  T [j,j+1) 0 8 8 j=¡8 j=¡8 LT L[j,j+1)  1/2  » 1/2 ¸8 ¸8 t 2 } }2 ¡ B .  S(t)u0 L8 L8  +  S(t s)(u xu)(s)ds  T [j,j+1) 0 8 8 j=¡8 j=¡8 LT L[j,j+1)

(1 + T)Λ + (1 + T)}uB u} η+1/2 . x L1 H T x } B } η+1/2 B .T Λ + u xu L1 L2 + Dx (u xu) T x L1 L2 T x η+1/2 T Λ + Λ}Bxu} 2 8 + Dx (uBxu) . . LT Lx 2 2 LT Lx Employing an argument similar to the one applied in (2.24) and (2.26) it is possible to bound the last term in the right hand side as follows

 ¸8 1/2 }B } 8 }B } 8 η+1/2 c xu L2 L c xu L2 L B } }2 8 8 T x T x (2.28) Dx (u xu) .  u L L  Λ(Λ + 1) e +Λ e . L2 L2 T [j,j+1) T x j=¡8 Next, we deﬁne »  1/2 T ¸8 2 2 φ(T) = }Bxu(s)} 8 ds +  }u} 8 8  , Lx LT L[j,j+1) 0 j=¡8 which is a continuous, non-decreasing function of T. From obtained in (2.26), (2.27) and (2.28) follows that

c φ(T) c φ(T) c φ(T) φ(T) . Λ(Λ + 1)φ(T) e +Λ e φ(T) + Λ e +Λ + Λφ(T).

Now, if we suppose that Λ ¤ δ ¤ 1 we obtain

φ(T) ¤ cΛ + cΛ ecφ(T) for some constant c ¡ 0. To complete the proof we will show that there exists δ ¡ 0 such that if Λ ¤ δ, then

φ(1) ¤ A, for some constant A ¡ 0.

To do this, we deﬁne the function

(2.29) Ψ(x, y) = x ¡ cy ¡ cy ecx .

First notice that Ψ(0, 0) = 0 and BxΨ(0, 0) = 1. Then the Implicit Function Theorem asserts that there exists δ ¡ 0, and a smooth function ξ(y) such that ξ(0) = 0, and Ψ(ξ(y), y) = 0 for |y| ¤ δ. Notice that the condition Ψ(ξ(y), y) = 0 implies that ξ(y) ¡ 0 for y ¡ 0. Moreover, since

BxΨ(0, 0) = 1, then the function Ψ(¤, y) is increasing close to ξ(y), whenever δ is chosen sufﬁciently small. 4. PROOF OF THEOREMA 29

Let us suppose that Λ ¤ δ, and set λ = ξ(Λ). Then, combining interpolation and Proposition 2.12 we obtain

 ¸8 !21/2 φ(0) =  sup |u(x, 0)|  j=¡8 xP[j,j+1) } } . u0 Hs(R) s ¤ c }u } 2 + c }D u } 2 1 0 L 1 x 0 Lx ¡ where we take c c1. Therefore, ¤ cξ(Λ) φ(0) c1Λ cΛ + cΛ e = λ. Suppose that φ(T) ¡ λ for some T P (0, 1) and deﬁne

T0 = inf tT P (0, 1) | φ(T) ¡ λu . ¡ t u Hence, T0 0 and φ(T0) = λ, besides, there exists a decreasing sequence Tn n¥1 converging to T0 such that φ(Tn) ¡ λ. In addition, notice that (2.29) implies Ψ(φ(T), Λ) ¤ 0 for all T P [0, 1]. Since the function Ψ(¤, Λ) is increasing near λ it implies that

Ψ(φ(Tn), Λ) ¡ Ψ(φ(T0), Λ) = Ψ(λ, Λ) = Ψ(ξ(Λ), Λ) = 0 for n sufﬁciently large. This is a contradiction with the fact that φ(T) ¡ λ. So we conclude φ(T) ¤ A for all T P (0, 1), as was claimed. Thus, φ(1) ¤ A. In conclusion we have proved that

»  1/2 T ¸8 }B }2 } }2 } } @ P (2.30) φ(T) = xu(s) 8 ds +  u 8 8  u0 Hs , T [0, 1]. Lx LT L[j,j+1) . x 0 j=¡8

4.2. Uniqueness. This subsection is devoted to prove the uniqueness of solutions to the IVP (0.1).

Let u(t) and v(t) be two solutions to the equation in (0.1) with initial conditions u(0) = u0 and v(0) = v0, respectively. Set ! )

K = max }Bxu} 1 8 , }Bxv} 1 8 . LT Lx LT Lx We deﬁne w = u ¡ v. Then w satisﬁes the IVP # B w ¡ Dα+1B w + wB v + uB w = 0, x, t P R, 0 α 1, (2.31) t x x x x w|t=0 = u0 ¡ v0,

After multiply the equation in (2.31) by w and integrate in the x¡ variable we obtain d } }2 } }2 }B } 8 }B } 8 (2.32) w(t) 2 . w(t) 2 xu(t) L + xv(t) L dt Lx Lx x x 5. EXISTENCE OF SOLUTIONS 30 which holds for all t P [0, T]. Applying Gronwall’s inequality in (2.32) yields }B u} 8 +}B v} 8 x L1 L x L1 L } }2 } ¡ }2 T x T x w(t) 2 u0 v0 2 e (2.33) Lx . Lx ¤ c K } ¡ }2 e u0 v0 2 Lx which clearly proves the uniqueness.

2¡ REMARK 2.15. In fact, the estimate (2.33) besides granting the uniqueness in the Lx space, it also provides the L2¡Lipschitz bound of the ﬂow.

5. Existence of Solutions

To establish the existence of local solutions of (0.1) we will follow the approach of Bona and Smith [4] (Section 4 ). So, we will recall some results about approximation of solutions. P Let ρ» S(R) such that (i) ρ(x) dx = 1, »R (ii) xkρ(x) dx = 0 for k P Z+ with 0 ¤ k ¤ tsu + 1. R ¡1 ¡1 Let ρe(x) = e ρ(e x) for any e ¡ 0.

s LEMMA 2.16. Let q ¥ 0, φ P H (R) and for e ¡ 0, φe = ρe ¦ φ. Then, } } ¡r} } ¥ (2.34) φe Hq+r . e φ Hq for all r 0 and } ¡ } q Ñ P (2.35) φ φe Hq¡p = o(e ) as e 0, for all p [0, q].

PROOF. See section 4 in [4].

s P (R) } } s We will consider u0 H . As was done previously, we can always assume that u0 Hx δ for some δ ¡ 0. Then, as was proved in section 4.1, it follows that for e ¡ 0, the solution ue is defined in the interval [0, 1] and satisfies the a priori estimate (2.30). s So, in this direction we fix u0 P H (R) with s ¡ 9/8 ¡ 3α/8, 0 α 1 and the regularized initial data, this is u0,e = ρe ¦ u0. 8 Since u0,e P H (R), we can warranty by means of Theorem 2.6, that for any e ¡ 0, there exists 8 a unique solution ue P C([0, 1]; H (R)) satisfying the condition ue(x, 0) = u0,e(x), x P R. The following lemmas summarize some properties associated to the smooth solutions.

LEMMA 2.17. Let e ¡ 0. Then,

} } s } } s (2.36) u0,e Hx . u0 Hx .

} s B } } } ¡s} } (2.37) D xue 2 8 u0,e 2s e u0 Hs . x LT Lx . Hx . x ¡ α 1 α¡1 s+ 2 ¡(s+ ) B } } α¡1 2 } } s (2.38) Dx xue . u0,e 2s+ . e u0 Hx . 8 2 2 Hx LT Lx 5. EXISTENCE OF SOLUTIONS 31

PROOF. By the sake of brevity we will omit the proof of these estimates. In fact, these one’s are obtained by similar arguments as the ones used in section 4.1 and the estimate (2.34). 1 These estimates are the basis to prove the convergence of the smooth solutions when e, e go to zero. Since u satisﬁes e # B ¡ α+1B B P tue Dx xue + ue xue = 0, x, t R, 0 α 1, ue|t=0 = u0,e. 1 Let ε ¡ ε ¡ 0. We set v = u ¡ u 1 . Then v satisﬁes the IVP # ε ε + B v ¡ D1 αB v + vB u + u 1 B v = 0 = 0, x, t P R, 0 α 1, (2.39) t x x x ε e x ¡ 1 v(x, 0) = u0,e(x) u0,e (x). s We will prove that the sequence tueue¡0 is a Cauchy sequence in the space C([0, 1] : H (R)). In this direction, we notice that arguing as in section 4.2 we obtain by Gronwall’s inequality, and (2.35) with q = s, p = 0

} } 8 ¤ cK } ¡ 1 } s (2.40) v L L2 e u0,e u0,e L2 = o(e ). T x xeÑ0 Moreover, combining (2.40) and Lemma 1.9 it follows that

r s¡r s s s¡r } } 8 r } } 8 } } 8 = ( ) ¤ v L Hx . v L Hs v 2 o e for all 0 r s. T T x LT Lx eÑ0 s To prove that tueue¡0 is a Cauchy sequence in C([0, 1] : H (R)), it remains to show the following: 9 ¡ 3α 3 ¡ α PROPOSITION 2.18. Assume that 0 α 1 and 8 8 s 2 2 . Let v be the solution of (2.39). } } ¤ Then, there exists a time T1 = T1 u0 Hs , with 0 T1 T 1 such that x ! )

}v} 8 s ¤ max }v} 8 s , }B v} 2 8 ÝÑ 0. LT Hx LT Hx x L L 1 1 T1 x eÑ0

PROOF. A standard energy estimate shows that » » 1 d 2 s s s s (2.41) }v} s + J (vB u ) J v dx + J (u 1 B v) J v dx = 0. Hx x e e x 2 dt R R The second term in the left side can be handled by using Lemma 1.2 » s s s J (vB u ) J v dx }v} 2 }B u } 8 + }D v} 2 }B u } 8 x e . Lx x e Lx x Lx x e Lx R s +}D B u } 8 }v} 2 }v} s . x x e Lx Lx Hx On the other hand, the estimate (1.4) applied in the third term in the left hand side of (2.41) yields » s s 2 J (u 1 B v)J v dx }B u 1 } 8 }v} s + }u 1 } s }B v} 8 }v} s . e x . x e Lx Hx e Hx x Lx Hx R Thus, after collecting the estimate above d }v} s }B u 1 } 8 + }B u } 8 }v} s dt Hx . x e Lx x e Lx Hx s + }v} 2 }D B u } 8 + }u 1 } s }B v} 8 Lx x x e Lx e Hx x Lx 5. EXISTENCE OF SOLUTIONS 32

Hence, by Gronwall’s inequality and (2.30) one gets that 1/2 }B } 8 +}B 1 } 8 T xue L2 L xue L2 L 1 } } 8 T x T x 1/2} } }B } v L Hs e T u0 Hs xv 2 8 + f (e, e ) T x . x LT Lx 1/2} } 1 T u0 Hs 1/2 e x T }u0}Hs }Bxv} 2 8 + f (e, e ) . x LT Lx where 1 1/2} } 8 } s B } } ¡ 1 } ÝÑ f (e, e ) = T v L L2 Dx xue L2 L8 + u0,e u0,e Hs 0. T x T x x eÑ0 To ﬁnish with the estimates it is necessary to establish control on the term involving the partial derivatives of v. So, as was done previously, the estimate (2.21) will be our starting point, in this 1¡α ¡ case with the particular choice of parameters σ = 2 , e = s s(α), and the function

¡ B ¡ 1 B F = v xue ue xv.

In this way α¡1 α¡1 }B } µ1 } } 8 µ2 s+ 2 B µ2 s+ 2 1 B xv 2 8 T v L Hs + T J (v xue) + T J (ue xv) (2.42) LT Lx . T x 2 2 2 2 LT Lx LT Lx P where µ1, µ2 (0, 1/2). 1¡α Since 0 s + 2 1, we apply inequality (1.3) to obtain ¡ α¡1 s+ α 1 s+ 2 B } B } 2 J (v xue) v xue 2 2 + Dx v 2 2 . LT Lx LT Lx L2 L2 (2.43) T x α¡1 s+ 2 }v} 8 s }B u } 2 8 + D B u }v} 8 2 , . LT Hx x e L Lx x x e L Lx T 2 8 T LT Lx and ¡ α¡1 s+ α 1 s+ 2 1 B } 1 B } 2 1 B J (ue xv) ue xv 2 2 + Dx (ue xv) 2 2 . LT Lx LT Lx L2 L2 T x α¡1 s+ 2 }B v} 2 8 }u 1 } 8 s + u 1 D B v , . x L Lx e LT Hx e x x T 2 2 LT Lx where the last term on the right hand side is handled as follows

 8 1/2 ¡ ¸ 2 s+ α 1 s+ α+1 1 2 B 1 2 ue Dx xv =  ue Dx Hv  2 2 2 2 LT Lx j=¡8 LT L[j,j+1)  1/2 ¸8 (2.44) s+ α+1 ¤ } 1 }2 2  ue 8 8  sup Dx Hv LT L[j,j+1) =¡8 jPZ L2 L2 j T [j,j+1) α+1 η s+ 2 ( + ) } } s H ¡ . 1 T u0 Hx sup Dx v for η 1/2, jPZ 2 2 LT L[j,j+1) being the last inequality a consequence of Duhamel’s formula and Corollary 2.5. Since v is a solution of the equation in (2.39), then we will compare the Kato smoothing effect satisﬁed by this solution with the last term on the right hand side of (2.44). 6. CONTINUITY OF THE FLOW 33

In this way, by using an approach similar to the one in Proposition 2.10 it follows that s+ α+1 1/2 2 }B } }B 1 } } } 8 sup Dx Hv 1 + T + xue 1 8 + xue 1 8 v L Hs . LT Lx LT Lx T x jPZ 2 2 (2.45) LT L[j,j+1)

} s B } } } 8 } 1 } 8 }B } + D xue 2 8 v 2 + ue L Hs xv 1 8 . x LT Lx LT Lx T x LT Lx Gathering the bounds obtained in (2.42) and (2.43)- (2.45) it follows that 1/2 }B } µ1 µ2 } } 1/2} } µ2 η} } } } 8 xv 2 8 T + T u0 Hs + 1 + T + T u0 Hs T (1 + T) u0 Hs v L Hs LT Lx . x x x T x 1 µ2 + T }u0}Hs }Bxv} 2 8 + h(e, e ), x LT Lx where ¡ 1 + α 1 µ2 s 2 µ2 s h(e, e ) = T D B u }v} 8 2 + T }D B u } 2 8 }v} 8 2 ÝÑ 0. x x e L Lx x x e L L L Lx 2 8 T T x T eÑ0 LT Lx in view of (2.37), (2.38), and (2.40). By gathering the estimates above the proposition follows. As was mentioned before the Proposition 2.18 is the basis to conclude that the sequence 8 t ¡ 1 u s ue ue e¡0 deﬁnes a Cauchy sequence in the space L ([0, T] : H (R)). In fact, by using a weak compactness argument it is possible to show that there exists a function u such that

} ¡ } 8 ÝÑ (2.46) u ue L Hs 0. T1 x In fact, from this we deduce that u is a solution of the IVP (0.1) in the distributional sense satisfying the condition (0.5).

6. Continuity of the Flow

To finish with the proof of TheoremA it remains to prove the continuous dependence of the solution upon the initial data. P s 9¡3α Let u0 H (R), with 8 s 2. By using a scaling argument, we can always assume } } s ¤ that u0 Hx δ, being δ a small positive number. In fact, in section5, it was proved that the corresponding solution u to (0.1) is defined in the interval [0, 1] and belongs to u P C([0, 1] : Hs(R)). Thus, the continuous dependence upon the initial data comes down to prove that given ν ¡ 0, s ¡ P (R) } ¡ } s P ([ ] there exists θ 0, such that for any v0 H with v0 u0 Hx ν, then the solution v C 0, 1 : s H (R)) of (0.1) satisfies }v ¡ u} 8 s θ. L1 Hx To achieve this goal we regularize the initial data u0 and v0, by defining v0,e = ρe ¦ v0 and u0,e = ρe ¦ u0, where ρe is the mollifier defined in section5. Next, we consider the solutions 8 ue, ve P C([0, 1] : H (R)) associated to the initial data u0,e and v0,e respectively. Then, it follows from the triangle inequality that

(2.47) }u ¡ v} 8 s ¤ }u ¡ u } 8 s + }u ¡ v } 8 s + }v ¡ v } 8 s . L1 Hx e L1 Hx e e L1 Hx e L1 Hx

Because, in view of (2.46) we can choose e0 ¡ 0 sufﬁciently small such that

(2.48) }u ¡ u } 8 s + }v ¡ v } 8 s 2θ/3. e0 L1 Hx e0 L1 Hx 7. PROPAGATION OF REGULARITY 34

Next, we proceed to estimate the second term on the right hand side of (2.47). For such purpose ¡ we consider the regularization of the elements ue0 and ve0 , speciﬁcally by deﬁning for µ 0, the elements uµ = u ¦ ρ and vµ = v ¦ ρ . The function uµ , respectively vµ , is a solution of 0,e0 0,e0 µ 0,e0 0,e0 µ e0 e0 the equation in (0.1) with initial conditions u0,e0 , respectively v0,e0 . 2 Since u0,e0 and v0,e0 are elements in H (R) by virtue of (2.34) it follows that

s¡2 s¡2 }u } 2 e }u } s e δ 0,e0 Hx . 0 Hx . 0 and s¡2 s¡2 }v } 2 }u ¡ v } 2 + }u } 2 e ν + e δ. 0,e0 Hx . e0 e0 Hx e0 Hx . 0 0

Thus, by choosing δ(e0), ν(e0) ¡ 0 small enough, we obtain by the existence theory of solutions µ µ 8 t u ¡ t u ¡ s that ue0 µ 0 and ve0 µ 0 converge to ue0 and ve0 , respectively in L ([0, 1] : H (R)) as µ goes to zero. 8 s The convergence in L ([0, 1] : H (R)) implies that there exists µ0 ¡ 0 small enough, such that µ0 µ0 µ0 µ0 }u ¡ v } 8 s ¤ u ¡ u 8 + v ¡ v 8 + u ¡ v 8 e0 e0 L1 Hx e0 e0 L Hs e0 e0 L Hs e0 e0 L Hs (2.49) 1 x 1 x 1 x µ0 µ0 ¤ ¡ 8 θ/6 + ue0 ve0 s L1 Hx however, by (2.34) it follows that

}u0,e ¡ v0,e } 8 2 ν. 0 0 L1 Hx Thus, by applying Theorem 2.6, the continuity of the ﬂow solution for initial data in H2(R) implies that for ν ¡ 0 small enough µ0 µ0 ¡ 8 (2.50) ue0 ve0 s θ/6. L1 Hx Finally, combining (2.48), (2.49)) and (2.50) it follows the continuous dependence upon the initial data. This ﬁnish the proof of TheoremA. l

7. Propagation of regularity

The aim of this section is to prove TheoremB. To achieve this goal it is necessary to take into account two important aspects of our analysis. First, the ambient space, that in our case is the Sobolev space where the theorem is valid together with the properties satisﬁed by the real solutions of the dispersive generalized Benjamin-Ono equation. In second place, the auxiliaries weights functions involved in the energy estimates that we will describe in detail. The following is a summary of the local well-posedness and Kato’s smoothing effect presented in the previous sections.

P s ¥ 3¡α P THEOREM D. If u0 H (R), s 2 , α (0, 1), then there exist a positive time T = T(}u0}Hs ) ¡ 0 and a unique solution of the IVP (0.1) such that (a) u P C([¡T, T] : Hs(R)), 1 8 (b) Bxu P L ([¡T, T] : L (R)), (Strichartz), 7. PROPAGATION OF REGULARITY 35

(c) Smoothing effect: for R ¡ 0, » » § § § § ! T R § α+1 §2 § α+1 §2 § r+ 2 § § r+ 2 § §BxDx u§ + §HBxDx u§ dx dt ¤ C ¡T ¡R

9¡3α P = ( } } s ) ¡ with r 8 , s and C C α; R; T; u0 Hx 0.

Since we have set the Sobolev space where we will work, the next step is the description of the cutoff functions to be used in the proof. In this part we consider families of cutoff functions that will be used systematically in the proof of the TheoremsB andC. This collection of weights functions were constructed originally by Isaza, Linares and Ponce [17] in the contex of the KdV equation, for more details see [17].

LEMMA 2.19. Given e ¡ 0 and b ¥ 5e. There exist functions P 8 χe,b, φe,b, φe,b, ψe, ηe,b C (R) satisfying the following properties: 1 ¥ (1) χe,b 0, # 0, x ¤ e (2) χe,b(x) = 1, x ¥ b, 8 (3) supp(χe,b) [e, ); 1 1 (4) χ (x) ¥ 1 ¡ (x), e,b 10(b ¡ e) [2e,b 2e] 1 (5) supp(χe,b) [e, b]; (6) There exists real numbers cj such that § § § § (j) ¤ 1 @ P P + §χ (x)§ cjχ e (x), x R, j Z . e,b 3 ,b+e (7) For x P (3e, 8) 1 e χ (x) ¥ . e,b 2 b ¡ 3e (8) For x P R 1 ¤ e χ e ,b+e(x) . 3 b ¡ 3e ¡ ¥ ¡ (9) Given e 0 and b 5e there exist c1, c2 0 such that 1 ¤ 1 χe,b(x) c1χe/3,b+e(x)χe/3,b+e(x), 1 ¤ χe,b(x) c2χe/5,e(x). (10) For e ¡ 0 given and b ¥ 5e, we deﬁne the function b b 1 1 ηe,b = χe,bχe,b and ϕe,b = χe,b. (11) supp(φe,b), supp(φe,b) [e/4, b], P (12) φe(x) = φe,b(x) = 1, x [e/2, e], (13) supp(ψe) (¡8, e/2], 7. PROPAGATION OF REGULARITY 36

(14) for x P R

χe,b(x) + φe,b(x) + ψe(x) = 1, and 2 2 χe,b(x) + φe,b (x) + ψe(x) = 1. t ¡ ¥ u P 8 ¥ The family χe,b : e 0, b 5e is constructed as follows: let ρ C0 (R), ρ(x) 0, even, with ¡ } } supp(ρ) ( 1, 1) and ρ L1 = 1. Then deﬁning $ &' 0, x ¤ 2e, ν (x) = x ¡ 2e , 2e ¤ x ¤ b ¡ e, e,b %' b¡3e b¡3e 1, x ¥ b ¡ e, and ¦ χe,b(x) = ρe νe,b(x) ¡1 where ρe(x) = e ρ(x/e). Since it has been described all the required estimates and tools necessary, we present the proof of our main result.

PROOFOF THEOREM B. Since the argument is translation invariant, without loss of generality we will consider the case x0 = 0. First, we will describe the formal calculations assuming as much as regularity as possible, later we provide the justiﬁcation using a limiting process. The proof will be established by induction, however in every step of induction we will subdi- vide every case in two steps, due to the non-local nature of the operator involving the dispersive part in the equation in (0.1).

Case j = 1

STEP 1:

B 2 First we apply one spatial derivative to the equation in (0.1), after that we multiply by xu(x, t)χe,b(x + vt), and ﬁnally we integrate in the x¡variable to obtain the identity » » » 1 d B 2 2 ¡v B 2 2 1 ¡ B α+1B B 2 ( xu) χe,b dx ( xu) (χe,b) dx ( xDx xu) xuχe,b dx 2 dt R looooooooooooomooooooooooooon2 R looooooooooooooooomooooooooooooooooonR

» A1(t) A2(t) B B B 2 + x (u xu) xuχe,b dx = 0. loooooooooooooomoooooooooooooonR

A3(t) §.1 Combining the local theory we obtain the following » » » T T | | ¤ v B 2 2 1 A1(t) dt ( xu) (χe,b) dx dt 0 2 0 R

}u} 3¡α . . 8 2 LT Hx 7. PROPAGATION OF REGULARITY 37

§.2 Integration by parts and Plancherel’s identity allow us rewrite the term A as follows » » 2 h i B α+1B 2 B ¡ B 1+αB 2 B A2(t) = xu Dx x; χe,b xu dx xuDx x(χe,b xu) dx »R »R h i B α+1B 2 B 2 B B 1+αB = xu Dx x; χe,b xu dx + χe,b xu xDx xu dx »R R h i B α+1B 2 B ¡ = xu Dx x; χe,b xu dx A2(t). R Consequently, » 1 h i A (t) = B u Dα+1B ; χ2 B u dx 2 2 x x x e,b x (2.51) R» ¡1 B α+2 2 B = xu HDx ; χe,b xu dx. 2 R ¡ α+2 2 Since α + 2 1, we have by (1.9) that the commutator HDx ; χe,b can be decomposed as 1 1 (2.52) HDα+2; χ2 = ¡ P (α + 2) + HP (α + 2)H ¡ R (α + 2) x e,b 2 n 2 n n for some positive integer n, that will be ﬁxed later. Inserting (2.52) into (2.51) » » 1 1 A2(t) = BxuRn(α + 2)Bxu dx + BxuPn(α + 2)Bxu dx 2 R» 4 R 1 ¡ BxuHPn(α + 2)HBxu dx 4 R

= A2,1(t) + A2,2(t) + A2,3(t).

Now, we proceed to ﬁx the value of n present in the terms A2,1, A2,2 and A2,3, according to a determinate condition. First, notice that » » 1 1 A2,1(t) = DxHuRn(α + 2)DxHu dx = HuDxRn(α + 2)DxHu dx. 2 R 2 R Then we ﬁx n such that 2n + 1 ¤ a + 2σ ¤ 2n + 3, that according to the case we are studying (j = 1), corresponds to a = α + 2 and σ = 1. This produces n = 1. 2 2 For this n in particular we have by Proposition 1.11 that R1(α + 2) maps Lx into Lx. Hence, { { } }2 4+α 2 } }2 4+α 2 A2,1(t) Hu(t) 2 Dx χ = c u0 2 Dx χ , . Lx e,b 1 Lx e,b 1 Lξ Lξ which after integrating in time yields » T { | | } }2 4+α 2 A2,1(t) dt u0 2 sup Dx χ . . Lx e,b 1 0 0¤t¤T Lξ

Next, we turn our attention to A2,2. Replacing P1(α + 2) into A2,2 » » 2 2 α + 2 α+1 1 α + 2 α+1 3 2 B 2 ¡ 2 2 A2,2(t) = Dx xu (χe,b) dx c3 Dx Hu (χe,b) dx 4 R 16 R

= A2,2,1(t) + A2,2,2(t). 7. PROPAGATION OF REGULARITY 38

We shall underline that A2,2,1(t) is positive, besides it represents explicitly the smoothing effect for the case j = 1.

Regarding A2,2,2, the local theory combined with interpolation leads to » T (2.53) |A2,2,2(t)| dt }u} 3¡α . . 8 2 0 LT Hx

After replacing (1.10) into A2,3 and using the fact that Hilbert transform is skew-symmetric

» » 2 2 α + 2 1+ α+1 1 α + 2 α+1 3 2 2 ¡ 2 2 A2,3(t) = Dx u (χe,b) dx c3 HDx u (χe,b) dx 4 R 16 R

= A2,3,1(t) + A2,3,2(t).

Notice that the term A2,3,1 is positive and represents the smoothing effect. In contrast, the term A2,3,2 is estimated as we did with A2,2,2 in (2.53). So, after integration in the time variable » T |A2,3,2(t)| dt }u} 3¡α . . 8 2 0 LT Hx Finally, after applying integration by parts » B B B 2 A3(t) = x(u xu) xuχe,b dx R» » 1 B B 2 2 ¡ 1 B 2 2 1 = xu( xu) χe,b dx u( xu) (χe,b) dx 2 R 2 R

= A3,1(t) + A3,2(t). On one hand, » | ( )| }B ( )} 8 (B )2 2 A3,1 t . xu t Lx xu χe,b dx, R where the integral expression on the right-hand side is the quantity to be estimated by means of Gronwall’s inequality. On the other hand, » T 1 | ( ) } ( )} 8 (B )2( 2 ) A3,2 t . u t Lx xu χe,b dx. 0 By Sobolev embedding we have after integrating in time » » » T ! T 2 2 1 |A (t)| dt sup }u(t)} s(α)+ (B u) (χ ) dx dt ¤ c. 3,2 . H x e,b 0 0¤t¤T x 0 R

Since }Bxu} 1 8 8. Then after gathering all estimates above and apply Gronwall’s inequality LT Lx we obtain » » » T 2 α+1 1 B 2 2 2 B 2 sup ( xu) χe,b dx + Dx xu (χe,b) dx dt 0¤t¤T R 0 R (2.54) » » T 2 1+ α+1 1 ¦ 2 2 ¤ + Dx u (χe,b) dx dt c1,1, 0 R ¦ ¦ } } }B } ¡ ¡ ¥ ¥ where c = c α; e; T; u0 3¡α ; xu0χe,b 2 0, for any e 0, b 5e and v 0. 1,1 1,1 2 Lx Hx This estimate ﬁnish the step 1 corresponding to the case j = 1. 7. PROPAGATION OF REGULARITY 39

1+α The local smoothing effect obtained above is just 2 derivative (see Isaza, Linares and Ponce [18]). So, the iterative argument is carried out in two steps, the ﬁrst step for positive integers m 1¡α and the second one for m + 2 .

STEP 2:

1¡α 2 After apply the operator Dx Bx to the equation in (0.1) and multiply the resulting by 1¡α 2 B 2 Dx xuχe,b(x + vt) one gets

1¡α 1¡α 1¡α 1¡α 1¡α 1¡α 2 B B 2 B 2 ¡ 2 B 1+αB 2 B 2 2 B B 2 B 2 Dx x tuDx xuχe,b Dx xDx xuDx xuχe,b + Dx x(u xu)Dx xuχe,b = 0, which after integrating in the spatial variable it gives » » 1 d 1¡α 1¡α 1 2 B 2 2 ¡ 2 B 2 2 (Dx xu) χe,b dx v (Dx xu) (χe,b) dx 2 dt R looooooooooooooooomooooooooooooooooonR

» A1(t) » 1¡α 1¡α 1¡α 1¡α ¡ 2 B 1+αB 2 B 2 2 B B 2 B 2 (Dx xDx xu) Dx xu χe,b dx + Dx x(u xu)Dx xu χe,b dx = 0. looooooooooooooooooooooooomooooooooooooooooooooooooonR looooooooooooooooooooooomooooooooooooooooooooooonR

A2(t) A3(t) §.1 First observe that by the local theory » » » T T ¡ 2 1 α 1 | | ¤ | | 2 B 2 A1(t) dt v Dx xu (χe,b) dx dt 0 0 R

}u} 3¡α . . 8 2 LT Hx

§.2 Concerning to the term A2 integration by parts and Plancherel’s identity yields » 1¡α h i 1¡α 2 B 1+αB 2 2 B ¡ A2(t) = Dx xu Dx x; χe,b Dx xu dx A2(t). R As a consequence » ¡ ¡ 1 1 α +1 1 α +1 ¡ 2 2+α 2 2 (2.55) A2(t) = Dx Hu HDx ; χe,b Dx Hu dx. 2 R ¡ α+2 2 Since 2 + α 1, we have by (1.9) that the commutator HDx ; χe,b can be decomposed as 1 1 (2.56) HDα+2; χ2 + P (α + 2) + R (α + 2) = HP (α + 2)H, x e,b 2 n n 2 n for some positive integer n that as in the previous cases it will be ﬁxed suitably. Replacing (2.56) into (2.55) » » 1 3¡α 3¡α 1 3¡α 3¡α A (t) = D 2 Hu R (α + 2)D 2 Hu dx + D 2 Hu P (α + 2)D 2 Hu dx 2 2 x n x 4 x n x R» R 3¡α 3¡α 1 2 2 ¡ Dx Hu HPn(α + 2)HDx Hu dx 4 R

= A2,1(t) + A2,2(t) + A2,3(t). 7. PROPAGATION OF REGULARITY 40

Fixing the value of n present in the terms A2,1, A2,2 and A2,3 requires an argument almost similar to that one used in step 1. First, we deal with A2,1 where a simple computation produces » 3¡α 3¡α 1 2 2 A2,1(t) = Hu Dx Rn(α + 2)Dx Hu dx. 2 R We ﬁx n P Z+ in such a way 2n + 1 ¤ a + 2σ ¤ 2n + 3, 3¡α where a = α + 2 and σ = 2 in order to obtain obtain n = 1 or n = 2. For the sake of simplicity we choose n = 1. Hence, by construction R1(α + 2) satisﬁes the hypothesis of Proposition 1.11, and {5 2 {5 2 |A (t)| }Hu(t)} 2 D (χ ) }u } 2 D (χ ) . 2,1 . Lx x e,b 1 . 0 Lx x e,b 1 Lξ Lξ

Thus » T {5 2 |A (t)| dt }u } 2 sup D (χ ) . 2,1 . 0 Lx x e,b 1 0 0¤t¤T Lξ

Next, after replacing P1(α + 2) in A2,2 » α + 2 3¡α α+1 1 α+1 3¡α A (t) = D 2 Hu D 2 χ2 D 2 D 2 Hu dx 2,2 4 x x e,b x x R » α + 2 3¡α α¡1 3 α¡1 3¡α ¡ c D 2 Hu D 2 χ2 D 2 D 2 Hu dx 3 16 x x e,b x x » R » 1 3 α + 2 2 α + 2 = D2Hu χ2 dx ¡ c (D Hu)2 χ2 dx 4 x e,b 3 16 x e,b »R » R + 1 + 3 α 2 B2 2 2 ¡ α 2 B 2 2 = H xu (χe,b) dx c3 ( xu) (χe,b) dx 4 R 16 R

= A2,2,1(t) + A2,2,2(t).

The smoothing effect corresponds to the term A2,2,1 and it will be bounded after integrating in time. In contrast, to bound A2,2,2 is required only the local theory, in fact » T |A2,2,2(t)| dt }u} 3¡α . . 8 2 0 LT Hx

Concerning the term A2,3 we have after replacing P1(α + 2) and using the properties of the Hilbert transform that » α + 2 3¡α α+1 1 α+1 3¡α A (t) = ¡ D 2 Hu HD 2 χ2 D 2 HD 2 Hu dx 2,3 4 x x e,b x x » R ¡ ¡ ¡ ¡ α + 2 3 α α 1 (3) α 1 3 α + D 2 Hu HD 2 χ2 D 2 HD 2 Hu dx 16 x x e,b x x » R » α + 2 2 1 α + 2 (3) = D2u χ2 dx ¡ (D u)2 χ2 dx 4 x e,b 16 x e,b »R »R + + α 2 B2 2 2 1 ¡ α 2 2 2 3 = ( xu) (χe,b) dx c3 (Dxu) (χe,b) dx 4 R 16 R

= A2,3,1(t) + A2,3,2(t). 7. PROPAGATION OF REGULARITY 41

¥ As before, A2,3,1 0 and represents the smoothing effect. Besides, the local theory and interpolation yields » T |A2,3,2(t)| dt }u} 3¡α . . 8 2 0 LT Hx

§.3 It only remains to handle the term A3. Since

1¡α 1¡α 1¡α 2 B B ¡ 2 B B 2 B B Dx x(u xu)χe,b = Dx x; χe,b (u xu) + Dx x(χe,bu xu)

1 1¡α 1¡α 1¡α = ¡ D 2 B ; χ B (u2) + D 2 B ; uχ B u + uχ D 2 B2u 2 x x e,b x x x e,b x e,b x x 1 1¡α = ¡ D 2 B ; χ B ((χ u)2 + (φu)2 + (ψ u2)) 2 x x e,b x e,b e,b e

1¡α 1¡α 2 B B 2 B2 + Dx x; uχe,b x((χe,bu) + (uφe,b) + (uψe)) + uχe,bDx xu = A3,1(t) + A3,2(t) + A3,3(t) + A3,4(t) + A3,5(t) + A3,6(t) + A3,7(t). First, we rewrite A3,1 as follows

¡ ¡ 1+ 1 α 1+ 1 α 2 B 2 2 B 2 A3,1(t) = cαH Dx ; χe,b x((uχe,b) ) + cα [H; χe,b] Dx x((χe,bu) ), where cα denotes a non-null constant. Next, combining (1.2), (1.5) and Lemma 1.9 one gets 1¡α 1+ 2 2 }A (t)} 2 = D ; χ B (uχ ) 3,1 Lx x e,b x e,b 2 Lx 1¡α 1+ 2 D (uχ ) }u} 8 + }u } 2 }u} 8 . x e,b Lx 0 Lx Lx 2 Lx and 1¡α 2 1+ 2 }A (t)} 2 = D ; χ B uφ 3,2 Lx x e,b x e,b 2 Lx 1¡α 1+ 2 D (uφ ) }u} 8 + }u } 2 }u} 8 . . x e,b Lx 0 Lx Lx 2 Lx Next, we recall that by construction e dist (supp (χ ) , supp (ψ )) ¥ , e,b e 2 so that, by Lemma 1.10 1¡α 2 2 }A (t)} 2 = D B ; χ B (ψ u ) 3,3 Lx x x e x e,b 2 Lx 1¡α 2 B2 2 = χe,bDx x ψe,bu 2 Lx

}u } 2 }u} 8 . . 0 Lx Lx 7. PROPAGATION OF REGULARITY 42

Rewriting

¡ ¡ 1+ 1 α 1+ 1 α 2 B ¡ B 2 A3,4(t) = cH Dx ; uχe,b x(uχe,b) c [H; uχe,b] xDx (uχe,b), for some non-null constant c. Thus, by the commutator estimates (1.1) and (1.7) 1¡α 1¡α 1+ 2 1+ 2 }A (t)} 2 H D ; uχ B (uχ ) + [H; uχ ] B D (uχ ) 3,4 Lx . x e,b x e,b e,b x x e,b 2 2 Lx Lx ¡ ¡ 1+ 1 α 1+ 1 α ¤ 2 B }B } 2 c Dx ; uχe,b x (uχe,b) + c x (uχe,b) L8 Dx (uχe,b) 2 x 2 Lx Lx ¡ 1+ 1 α }B ( )} 8 2 ( ) . x uχe,b Lx Dx uχe,b . 2 Lx Applying the same procedure to A3,5 yields

¡ ¡ 1+ 1 α 1+ 1 α ¡ 2 B ¡ B 2 A3,5(t) = H Dx ; uχe,b x (uφe,b) [H; uχe,b] xDx (uφe,b) for some constant c. Then, 1¡α 1¡α 1+ 2 1+ 2 }A (t)} 2 ¤ H D ; uχ B (uφ ) + [H; uχ ] B D (uφ ) 3,5 Lx x e,b x e,b e,b x x e,b 2 2 Lx Lx ¡ ¡ 1+ 1 α 1+ 1 α ¤ 2 B }B } 2 c Dx ; uχe,b x (uφe,b) + c x (uχe,b) L8 Dx (uφe,b) 2 x 2 Lx Lx ¡ ¡ 1+ 1 α 1+ 1 α }B ( )} 8 2 ( ) + }B ( )} 8 2 ( ) . x uχe,b Lx Dx uφe,b x uφe,b Lx Dx uχe,b . 2 2 Lx Lx

Since the supports of χe,b and ψe are separated we obtain by Lemma 1.10 1¡α 2 1+ 2 }A (t)} 2 = uχ B D (uψ ) }u } 2 }u} 8 . 3,6 Lx e,b x x e,b . 0 Lx Lx 2 Lx

To ﬁnish with the estimates above we use the relation

@ P χe,b(x) + φe,b(x) + ψe(x) = 1 x R.

Then

1¡α 1¡α 1¡α 1+ 2 1+ 2 1+ 2 Dx (uχe,b) = Dx uχe,b + Dx ; χe,b (uχe,b + uφe,b + uψe)

= I1 + I2 + I3 + I4.

Notice that }I } 2 is the quantity to estimate. In contrast, }I } 2 and }I } 2 can be handled by 1 Lx 2 Lx 3 Lx Lemma 1.7 combined with the local theory. Meanwhile I3 can be bounded by using Lemma 1.10. 7. PROPAGATION OF REGULARITY 43

1+ 1+α 2 8 We notice that the gain of regularity obtained in the step 1 implies that Dx (uφe,b) . 2 Lx To show this we use Theorem 1.4 and Holder’s¨ inequality as follows 1+α 1+α ¸ 1+ 2 1+ 2 1 β s,β D (uφ ) }u} 4 D φ + B φ D u x e,b . Lx x e,b x e,b x 2 4 β! Lx Lx β¤1 L2 x ¡ ¸ 2+ 1 α } }1/2} }1/2 2 Bβ s,β u0 2 u 8 Dx φe,b + cs,β x φe,bDx u . Lx Lx 2 L4 ¤ Lx x β 2 1+ 1+α 1+α } }1/2} }1/2 2 B 2 u0 2 u 8 + φ Dx u + xφ HDx u . L Lx e,b e,b x 2 2 Lx Lx 1¡α 2 + 1[e/4,b+e/4]Dx u 2 Lx 1+α 1+α 1+ 2 2 }u } 2 + }u} 8 + 1 D u + 1 HD u . . 0 Lx Lx [e/4,b] x [e/4,b] x 2 2 Lx Lx The second term on the right hand side after integrate in time is controlled by using Sobolev’s embedding. Meanwhile, the third term can be handled after integrate in time and use (2.54) with (e, b) = (e/24, b + 7e/24). The fourth term in the right hand side can be bounded combining the local theory and interpolation. Hence, after integration in time 1+ 1+α 2 8 (2.57) Dx (uφe,b) , 2 2 LT Lx ¡ 1+ 1 α 2 8 which clearly implies Dx (uφe,b) , as was required. 2 2 LT Lx 1¡α 1+ 2 A similar analysis can be used to estimate Dx (uφe,b) . 2 Lx Finally, » 1¡α r 2 B A3,7(t) = A3,7(t)Dx xu χe,b dx »R 1¡α 1¡α 2 2 B2 2 B = uχe,bDx xuDx xu dx R» 2 1 1¡α 2 B 2 B = uχe,b x Dx xu dx 2 R » » ¡ 2 ¡ 2 1 1 α 1 1 1 α ¡ B 2 2 B ¡ 2 2 B = xuχe,b Dx xu dx u(χe,b) Dx xu dx 2 R 2 R

= A3,7,1(t) + A3,7,2(t).

Since » 1¡α 2 | ( )| }B ( )} 8 2 B 2 A3,7,1 t . xu t Lx Dx xu χe,b dx, R 7. PROPAGATION OF REGULARITY 44 being the last integral the quantity to be estimated by means of Gronwall’s inequality, and by the local theory }Bxu} 1 8 8. LT Lx Sobolev’s embedding led us to » » » T T ¡ 2 1 1 α | | ¤ 2 B A3,7,2(t) dt u χe,b χe,b Dx xu dx dt R 0 0 » » T ¡ 2 1 1 α ¤ } ( )} 8 2 B u t Lx χe,b χe,b Dx xu dx dt 0 R ! » » T 1¡α 2 1 2 sup }u(t)} s(α)+ χ χ D B u dx dt. . H e,b e,b x x 0¤t¤T x 0 R

Gathering all the information corresponding to this step combined with Gronwall’s inequality yields

» » » ¡ 2 T 1 α 2 1 2 B 2 B2 2 sup Dx xu χe,b(x + vt) dx + xu (χe,b) dx dt 0¤t¤T R 0 R (2.58) » » T B2 2 2 1 ¤ ¦ + H xu (χe,b) dx dt c1,2, 0 R ¡ ! ¦ ¦ 1 α } } ¡ 2 B ¡ ¥ ¥ with c1,2 = c1,2 α; e; T; v; u0 3 α ; Dx xu0χe,b for any e 0, b 5e and v 0. H 2 2 x Lx This ﬁnishes the step two corresponding to the case j = 1 in the induction process. Next, we present the case j = 2, to show how we proceed in the case j even.

Case j = 2

STEP 1:

First we apply two spatial derivatives to the equation in (0.1), after that we multiply by B2 2 ¡ xu(x, t)χe,b(x + vt), and ﬁnally we integrate in the x variable to obtain the identity » » » 1 d B2 2 2 ¡ v B2 2 2 1 ¡ B2 α+1B B2 2 xu χe,b dx xu (χe,b) dx xDx xu xuχe,b dx 2 dt R looooooooooooomooooooooooooon2 R looooooooooooooooooomooooooooooooooooooonR

» A1(t) A2(t) B2 B B2 2 + x(u xu) xuχe,b dx = 0. looooooooooooomooooooooooooonR

A3(t)

Similarly as was done in the previous steps we ﬁrst proceed to estimate A1. §.1 By (2.58) it follows that » » » T T | | ¤ B2 2 2 1 ¤ ¦ (2.59) A1(t) dt ( xu) (χe,b) dx dt c1,2. 0 0 R 7. PROPAGATION OF REGULARITY 45

§2. To extract information from the term A2 we use integration by parts and Plancherel’s identity to obtain » » B2 α+1B 2 B2 ¡ B2 1+αB 2 B2 A2(t) = xu[Dx x; χe,b] xu dx xuDx x(χe,b xu) dx »R »R B2 α+1B 2 B2 2 B2 B2 1+αB = xu[Dx x; χe,b] xu dx + χe,b xu xDx xu dx »R R h i B2 α+1B 2 B2 ¡ = xu Dx x; χe,b xu dx A2(t). R

Consequently, » h i 1 B2 α+1B 2 B2 A2(t) = xu Dx x; χe,b xu dx 2 R» ¡1 B2 α+2 2 B2 = xu HDx ; χe,b xu dx. 2 R

Although this stage of the process is related to the one performed in step 1 (for j = 1), we will use again the commutator expansion in (1.9), taking into account in this case that a = α + 2 ¡ 1 and n is a non-negative integer whose value will be ﬁxed later. Then, » » » 1 B2 B2 1 B2 B2 ¡ 1 B2 B2 A2(t) = xu Rn(s + 2) xu dx + xu Pn(s + 2) xu dx xu HPn(s + 2)H xu dx 2 R 4 R 4 R

= A2,1(t) + A2,2(t) + A2,3(t).

Essentially, the key term which allows us to ﬁx the value of n, correspond to A2,1. Indeed, after some integration by parts » 1 B2 B2 A2,1(t) = u xRn(α + 2) xu dx 2 »R 1 2 2 = u DxRn(α + 2)Dxu dx. 2 R

We ﬁx n such that it satisﬁes

2n + 1 ¤ a + 2σ ¤ 2n + 3.

In this case with a = α + 2 ¡ 1 and σ = 2, we obtain n = 2. 2 2 2 Hence by construction the Proposition 1.11 guarantees that DxR2(α + 2)Dx is bounded in Lx. Thereby | | } }2 2 2 A2,1(t) . u(t) 2 DxR2(α + 2)Dxu 2 Lx Lx { } }2 α+6 = c u0 2 Dx χ b . Lx e, 1 Lξ

Since we ﬁxed n = 2, we proceed to handle the contribution coming from A2,2 and A2,3. 7. PROPAGATION OF REGULARITY 46

Next,

» ¸ α + 2 ¡ α+1 ¡k (2k+1) α+1 ¡k ¡ k k B2 2 2 2 B2 A2,2(t) = c2k+1( 1) 4 xu Dx χe,b Dx xu dx 4 ¤ ¤ R »0 k 2 α + 2 α+1 1 α+1 = B2u D 2 χ2 D 2 B2u dx 4 x x e,b x x R » α + 2 α¡1 3 α¡1 ¡ c B2u D 2 χ2 D 2 B2u dx 3 16 x x e,b x x »R ¡ ¡ α + 2 α 3 (5) α 3 B2 2 2 2 B2 + c5 xu Dx χe,b Dx xu dx 64 R » » 2 2 α + 2 α+1 1 α + 2 1+ α+1 = D 2 B2u (χ2 ) dx ¡ c D 2 u (χ2 )(3)dx 4 x x e,b 3 16 x e,b R » R + α+1 α 2 2 2 2 (5) + c5 (Dx u) (χe,b) dx 64 R

= A2,2,1(t) + A2,2,2(t) + A2,2,3(t).

¥ Notice that A2,2,1 0 represents the smoothing effect. We recall that

§ § § (j) § 1 + §χ (x)§ χ e (x) @x P R, j P Z , e,b . 3 ,b+e then

» » » T T 2 1+ 1+α 1 | | 2 A2,2,2(t) dt . Dx u χe/3,b+e dx dt. 0 0 R

Taking (e, b) = (e/9, b + 10e/9) in (2.54) combined with the properties of the cutoff function we have

» T | | ¦ A2,2,2(t) dt . c1,1. 0

To ﬁnish the terms that make A2 we proceed to estimate A2,2,3. As usual the low regularity is controlled by interpolation and the local theory. Therefore

» T |A2,2,3(t)| dt }u} 3¡α . . 8 2 0 LT Hx 7. PROPAGATION OF REGULARITY 47

Next, » (α + 2) A (t) = ¡ B2HP HB2u dx 2,3 4 x n x R » ¸ k+1 (¡1) c + α+1 ¡k (2k+1) α+1 ¡k = (α + 2) 2k 1 B2uH D 2 χ2 D 2 HB2u dx 4k+1 x x e,b x x 0¤k¤2 R » ¸ k 2 α + 2 (¡1) c + α+1 ¡k (2k+1) = 2k 1 D 2 HB2u χ2 (x + vt) dx 4 4k x x e,b 0¤k¤2 R » » 2 2 α + 2 α+1 1 α + 2 α+1 2 B2 2 ¡ 2 B 2 (3) = HDx xu (χe,b) dx c3 Dx xu (χe,b) dx 4 R 16 R » 2 + α+1 α 2 2 2 (5) + c5 Dx Hu (χe,b) dx 64 R

= A2,3,1(t) + A2,3,2(t) + A2,3,3(t).

A2,3,1 is positive and it will provide the smoothing effect after being integrated in time. The terms A2,3,2 and A2,3,3 can be handled exactly in the same way that were treated A2,2,2 and A2,2,3 respectively, so we will omit the proof. §.3 Finally, » » B B2 2 2 B3 B2 2 A3(t) = 3 xu( xu) χe,b dx + u xu xuχe,b dx »R R» 5 B B2 2 2 ¡ 1 B2 2 2 1 = xu( xu) χe,b dx u( xu) (χe,b) dx 2 R 2 R

= A3,1(t) + A3,2(t). First, » | ( )| }B ( )} 8 (B2 )2 2 A3,1 t . xu t Lx xu χe,b dx, R B P 1 8 by the local theory xu L ([0, T] : Lx (R)) (see TheoremD-(b)); and the integral expression is the quantity we want to estimate. Next, » 1 | ( )| } ( )} 8 (B2 )2( 2 ) A3,2 t . u t Lx xu χe,b dx. R After apply the Sobolev embedding and integrate in the time variable we obtain » » » T ! T 2 2 2 1 |A (t)| dt sup }u(t)} s(α)+ (B u) (χ ) dx dt, 3,2 . H x e,b 0 0¤t¤T x 0 R and the integral term in the right hand side was estimated previously in (2.59). Thus, after grouping all the terms and apply Gronwall’s inequality we obtain » » » T 2 2 α+1 1 B2 2 2 B2 2 sup xu χe,b dx + Dx xu χe,b dx dt 0¤t¤T R 0 R (2.60) » » T 2 α+1 1 ¦ 2 B2 2 ¤ + Dx H xu χe,b dx dt c2,1, 0 R 7. PROPAGATION OF REGULARITY 48

¦ ¦ } } ¡ B2 ¡ ¥ ¥ where c = c α; e; T; v; u0 3 α ; u0χe,b 2 , for any e 0, b 5e and v 0. 2,1 2,1 2 x Lx Hx

STEP 2:

1¡α 2 B2 From equation in (0.1) one gets after applying the operator Dx x and multiplying the result by 1¡α 2 B2 2 Dx xuχe,b(x + vt), to obtain

1¡α 1¡α 1¡α 1¡α 1¡α 1¡α 2 B2B 2 B2 2 ¡ 2 B2 1+αB 2 B2 2 2 B2 B 2 B2 2 Dx x tuDx xuχe,b Dx xDx xuDx xuχe,b + Dx x(u xu)Dx xuχe,b = 0, which after integration in the spatial variable it becomes

» » ¡ 2 ¡ 2 1 d 1 α v 1 α 1 2 B2 2 ¡ 2 B2 2 Dx xu χe,b dx Dx xu (χe,b) dx 2 dt R loooooooooooooooooomoooooooooooooooooon2 R

A (t) » 1 » 1¡α 1¡α 1¡α 1¡α ¡ 2 B2 1+αB 2 B2 2 2 B2 B 2 B2 2 Dx xDx xu Dx xu χe,b dx + Dx x(u xu) Dx xu χe,b dx = 0. looooooooooooooooooooooooooooomooooooooooooooooooooooooooooonR loooooooooooooooooooooooooooomoooooooooooooooooooooooooooonR

A2(t) A3(t)

To estimate A1 we will use different techniques to the ones implemented to bound A1 in the previous step. The main difﬁculty we face is to deal with the non-local character of the operator s P +z P s Dx for s R 2N, the case s 2N is less complicated because Dx becomes local, so we can integrate by parts. The strategy to solve this issue will be the following. In (2.60) we proved that u has a gain of α+1 1+α 2 derivatives (local) which in total sum 2 + 2 . This suggests that if we can ﬁnd an appropriated channel where we can localize the smoothing effect, we shall be able to recover all the local ¤ 1+α derivatives r with r 2 + 2 . Henceforth we will employ recurrently a technique of localization of commutator used by Kenig, Linares, Ponce and Vega [22] in the study of propagation of regularity (fractional) for solutions of the k-generalized KdV equation. Indeed, the idea consists in constructing an appropriate system of smooth partition of unit, localizing the regions where is available the information obtained in the previous cases. We recall that for e ¡ 0 and b ¥ 5e b 1 (2.61) ηe,b = χe,bχe,b and χe,b + φe,b + ψe = 1.

§.1 Claim 1+α 2 B2 8 Dx x(uηe,b) . 2 2 LT Lx 7. PROPAGATION OF REGULARITY 49

Combining the commutator estimate (1.5), (2.61), Holder’s¨ inequality and (2.60) yields 1+α 2+ 1+α 2+ 1+α 2 B2 ¤ 2 2 Dx x(uηe,b) Dx uηe,b + Dx ; ηe,b (uχe,b + uφe,b + uψe) L2 L2 L2 L2 L2 L2 T x T x T x 2+ 1+α 1+ 1+α 2 }B } 2 Dx uη b + xη b 8 8 Dx (uχ b) . e, e, L Lx e, L2 L2 T L2 L2 T x T x 2+ 1+α 1+ 1+α 2 } } }B } 2 + Dx η b uχ b 2 2 + xη b 8 8 Dx (uφ b) e, e, L Lx e, L Lx e, L8 L8 T T L2 L2 T x T x 2+ 1+α 2+ 1+α 2 } } 2 + Dx η b uφ b 2 2 + η bDx (uψe) e, e, L Lx e, L8 L8 T L2 L2 T x T x 1+α 1+α ¦ 2 1+ 2 1+ 2 c + D (uχ ) + D (uφ ) +}u } 2 . 2,1 x e,b x e,b 0 Lx 2 2 2 2 looooooooooomooooooooooonLT Lx looooooooooomooooooooooonLT Lx

B1 B2 1+α 2+ 2 + ηe,bDx (uψe) . 2 2 looooooooooooomooooooooooooonLT Lx

Since χe/5,e = 1 on the support of χe,b then @ P χe,b(x)χe/5,e(x) = χe,b(x) x R. Thus, combining Lemma 1.9 and Young’s inequality we obtain ¡ 1+ 1+α 3+α 1 α 2 ( ) }B2( )} 4 } } 4 Dx uχe,b . x uχe,b L2 u0 L2 2 x x Lx 2 }B (uχ )} 2 + }u } 2 . x e,b Lx 0 Lx 2 1 2 = c}B uχ + 2Bxuχ e χ + χ u} 2 + c}u } 2 x e,b 5 ,e e,b e,b Lx 0 Lx 2 }B uχ } 2 + }Bxuχ e } 2 + }u } 2 . . x e,b Lx 5 ,e Lx 0 Lx Then, by an application of (2.60) adapted to every case yields

2 ¦ ¦ B }B uχ } 8 2 + }Bxuχ e } 8 2 + }u } 2 c + c + }u } 2 . 1 . x e,b LT Lx 5 ,e LT Lx 0 Lx . 2,1 1,1 0 Lx

Notice that B2 was estimated in the case j = 1, step 2 see (2.57), so we will omit the proof. Next, we recall that by construction e dist (supp (η ) , supp (ψ )) ¥ . e,b e 2 Hence, by Lemma 1.10 1+α 2+ 2 B = η D (uψ ) }η } 8 8 }u } 2 . 3 e,b x e . e,b LT Lx 0 Lx 2 2 LT Lx The claim follows gathering the calculations above. α+1 At this point we have proved that locally in the interval [e, b] there exists 2 + 2 derivatives. By Lemma 1.9 we get 1¡α 1+α 2+ 2 2+ 2 D (uη ) D (uη ) + }u } 2 8. x e,b . x e,b 0 Lx 2 2 2 2 LT Lx LT Lx 7. PROPAGATION OF REGULARITY 50

As before ¡ ¡ ¡ 2+ 1 α 2+ 1 α 2+ 1 α 2 2 ¡ 2 Dx uηe,b = Dx (uηe,b) Dx ; ηe,b (uχe,b + uφe,b + uψe) .

The argument used in the proof of the claim yields ¡ 2+ 1 α 2 8 Dx uηe,b . 2 2 LT Lx Therefore,

» » » T T ¡ 1 α 1 | | ¤ | | 2 B2 2 2 A1(t) dt v (Dx xu) (χe,b) dx dt 0 0 R 1¡α 2 (2.62) 2+ 2 . Dx uηe,b 2 2 LT Lx 8.

§.2 Now we focus our attention in the term A2. Notice that after integration by parts and Plancherel’s identity » » 1¡α h i 1¡α 1¡α 1¡α 2 B2 1+αB 2 2 B2 ¡ 2 B2 1+αB 2 2 B2 A2(t) = Dx xu Dx x; χe,b Dx xu dx Dx xu Dx x χe,b Dx xu dx »R »R 1¡α h i 1¡α 1¡α 1¡α 2 B2 1+αB 2 2 B2 B 2 B2 1+α 2 2 B2 = Dx xu Dx x; χe,b Dx xu dx + xDx xu Dx χe,b Dx xu dx »R »R 1¡α h i 1¡α 1¡α 1¡α 2 B2 1+αB 2 2 B2 1+αB 2 B2 2 2 B2 = Dx xu Dx x; χe,b Dx xu dx + Dx xDx xu χe,b Dx xu dx »R R 1¡α h i 1¡α 2 B2 1+αB 2 2 B2 ¡ = Dx xu Dx x; χe,b Dx xu dx A2(t). R Consequently » 1 1¡α h i 1¡α 2 B2 1+αB 2 2 B2 A2(t) = Dx xu Dx x; χe,b Dx xu dx 2 »R 1 5¡α 5¡α 2 1+αB 2 2 (2.63) = Dx u[Dx x; χe,b]Dx u dx 2 R» 1 5¡α 5¡α ¡ 2 2+α 2 2 = Dx u HDx ; χe,b Dx u dx. 2 R The procedure to decompose the commutator will be almost similar to the introduced in the previous step, the main difference relies on the fact that the quantity of derivatives is higher in com- parison with the step 1. ¡ α+2 2 Concerning this, we notice that 2 + α 1 and by (1.9) the commutator HDx ; χe,b can be decomposed as 1 1 (2.64) HDα+2; χ2 + P (α + 2) + R (α + 2) = HP (α + 2)H x e,b 2 n n 2 n for some positive integer n. We shall ﬁx the value of n satisfying a suitable condition. 7. PROPAGATION OF REGULARITY 51

Replacing (2.64) into (2.63) produces » » 1 5¡α 5¡α 1 5¡α 5¡α A (t) = D 2 u R (α + 2)D 2 u dx + D 2 u P (α + 2)D 2 u dx 2 2 x n x 4 x n x R» R 5¡α 5¡α 1 2 2 ¡ Dx u HPn(α + 2)HDx u dx 4 R

= A2,1(t) + A2,2(t) + A2,3(t).

Now we proceed to fix the value of n present in A2,1, A2,2 and A2,3. First we deal with the term that determine the value n in the decomposition associated to A2. In this case it corresponds to A2,1. Applying Plancherel’s identity, A2,1 becomes » 5¡α 5¡α 1 2 2 A2,1(t) = u Dx Rn(α + 2)Dx u dx. 2 R We fix n such that it satisfies (1.13) i.e.,

2n + 1 ¤ a + 2σ ¤ 2n + 3,

5¡α with a = α + 2 and σ = 2 , which produces n = 2 or n = 3. Nevertheless, for the sake of simplicity we take n = 2. 2 Hence, by construction R2(α + 2) is bounded in Lx (see Proposition 1.11). Thus, » » T T { | | ¤ } } 7 2 ¤ A2,1(t) dt c u(t) L2 Dx(χ ( + vt)) dt x e,b L1 0 0 ξ {7 2 }u } 2 sup D (χ ) . . 0 Lx x e,b 1 0¤t¤T Lξ

Since we have ﬁxed n = 2, we obtain after replace P2(α + 2) into A2,2 » » α + 2 2 1 α + 2 2 A (t) = HB3u (χ2 ) dx ¡ c B2u (χ2 )(3) dx 2,2 4 x e,b 3 16 x e,b R » R + α 2 B 2 2 (5) + c5 (H xu) (χe,b) dx 64 R

= A2,2,1(t) + A2,2,2(t) + A2,2,3(t).

We underline that A2,2,1 is positive and represents the smoothing effect. On the other hand, by (2.58) with (e, b) = (e/5, e) we have » » » T T | | B2 2 2 2 3 A2,2,2(t) dt = c ( xu) χ e ,e(χe,b) dx dt 0 0 »R 5 B2 2 2 ¦ . sup ( xu) χ e ,edx . c1,2. 0¤t¤T R 5 Next, by the local theory » T (2.65) |A2,2,3(t)| dt }u} 3¡α . . 8 2 0 LT Hx 7. PROPAGATION OF REGULARITY 52

After replacing P2(α + 2) into A2,3, and using the fact that Hilbert transform is skew adjoint » » α + 2 1 α + 2 3 A (t) = (B3u)2(χ2 ) dx ¡ c (HB2u)2(χ2 ) dx 2,3 4 x e,b 3 16 x e,b R » R + α 2 B 2 2 (5) + c5 ( xu) (χe,b) dx 64 R

= A2,3,1(t) + A2,3,2(t) + A2,3,3(t).

¥ Notice that A2,3,1 0 and it represents the smoothing effect. However, A2,3,2 can be handled if we take (e, b) = (e/5, e) in (2.54) as follows » » B2 2 2 2 3 B2 2 2 A2,3,3(t) = ( xu) χ e ,e(χe,b) dx . ( xu) χ e ,e dx, R 5 R 5 thus, » » T | | B 2 2 ¦ A2,3,3(t) dt . sup ( xu) χ e ,e dx . c1,1. 0 0¤t¤T R 5

To ﬁnish the estimate of A2 only remains to bound A2,3,2. To do this we recall that § § § § § (j) § 1 @ P P + χe,b(x) . χe/3,b+e(x), x R, j Z ,

that joint with the property (9) of χe,b yields » » » » T T B2 2 1 B2 2 1 H xu χe/3,b+e dx dt . H xu χe/9,b+10e/9χe/9,b+10e/9 dx dt 0 R 0 R ¦ . c1,2, where the last inequality is obtained taking (e, b) = (e/9, b + 10e/9) in (2.58). The term A2,3,3 can be handled by interpolation and the local theory.

§.3 Finally we turn our attention to A3.We start rewriting the nonlinear part as follows

1¡α 1¡α 1¡α 2 B2 B ¡ 2 B2 B 2 B2 B Dx x (u xu) χe,b = Dx x; χe,b (u xu) + Dx x (χe,bu xu)

1 1¡α 1¡α 1¡α = ¡ D 2 B2; χ B (u2) + D 2 B2; uχ B u + uχ D 2 B3u 2 x x e,b x x x e,b x e,b x x 1 1¡α = ¡ D 2 B2; χ B (uχ )2 + (uφ)2 + (ψ u2) 2 x x e,b x e,b e,b e

1¡α 1¡α 2 B2 B 2 B3 + Dx x; uχe,b x ((uχe,b) + (uφe,b) + (uψe)) + uχe,bDx xu = A3,1(t) + A3,2(t) + A3,3(t) + A3,4(t) + A3,5(t) + A3,6(t) + A3,7(t). 7. PROPAGATION OF REGULARITY 53

Hence, after replacing (2.65) into A3 and apply Holder’s¨ inequality » » ¸ 1¡α 1¡α 2 B2 2 B2 A3(t) = A3,m(t) Dx xu χe,b dx + A3,7(t) Dx xu χe,b dx ¤ ¤ R R 1 m 6 » ¸ 1¡α 1¡α 2+ 2 2 2 ¤ }A (t)} 2 D u(t)χ (¤ + vt) + A (t) D B u χ dx 3,m Lx x e,b 3,7 x x e,b ¤ ¤ L2 R 1 m 6 x » 1¡α ¸ 1¡α 2+ 2 2 2 = D u(t)χ (¤ + vt) }A (t)} 2 + A (t) D B u χ dx x e,b 3,m Lx 3,7 x x e,b 2 R Lx 1¤m¤6 ¡ ¸ 2+ 1 α 2 ¤ = Dx u(t)χe,b( + vt) A3,m(t) + A3,7(t). 2 Lx 1¤m¤6

Notice that the ﬁrst factor in the right hand side is the quantity to be estimated by Gronwall’s inequality. So, we shall focus on establish control in the remaining terms. First, combining (1.2), (1.5) and Lemma 1.9 one gets that 1¡α 2+ 2 2 }A (t)} 2 = D ; χ B (uχ ) 3,1 Lx x e,b x e,b 2 Lx ¡ 2+ 1 α ¤ 2 B 2 c Dx ; χe,b x (uχe,b) L2 x ¡ ¡ 2+ 1 α 1+ 1 α }B } 2 2 2 B B 2 xχ b 8 Dx (uχ b) + Dx xχ b x (uχ b) . e, Lx e, e, e, 2 2 8 Lx Lx Lx (2.66) ¡ 2+ 1 α ¤ 2 } } } } } } c Dx (uχe,b) uχe,b L8 + c Dx(uχe,b) L2 uχe,b L8 2 x x x Lx 2 1¡α 1¡α 5¡α 3¡α 2+ 2+ 5¡α D 2 (uχ ) }u} 8 + D 2 (uχ ) }u } }u} 8 . x e,b Lx x e,b 0 L2 Lx 2 2 x Lx Lx 1¡α 2+ 2 D (uχ ) }u} 8 + }u } 2 }u} 8 , . x e,b Lx 0 Lx Lx 2 Lx and 1¡α 2+ 2 2 }A (t)} 2 = D ; χ B ((uφ ) ) 3,2 Lx x e,b x e,b 2 Lx ¡ ¡ 2+ 1 α 2 1+ 1 α 2 ¤ }B } 2 2 B B c xχ b 8 Dx uφ b + Dx xχ b x uφ b e, Lx e, e, e, L2 L8 L2 x x x 1¡α 2+ 2 ¤ c D uφ uφ + c Dx(uφ ) uφ (2.67) x e,b e,b 8 e,b 2 e,b 8 2 Lx Lx Lx Lx 2 1¡α 1¡α 5¡α 3¡α 2+ 2+ 5¡α D 2 uφ }u} 8 + D 2 (uφ ) }u } }u} 8 . x e,b Lx x e,b 0 L2 Lx 2 2 x Lx Lx 1¡α 2+ 2 D (uφ ) }u} 8 + }u } 2 }u} 8 . . x e,b Lx 0 Lx Lx 2 Lx 7. PROPAGATION OF REGULARITY 54

To ﬁnish with the quadratic terms, we employ Lemma 1.10 1¡α 2+ 2 2 }A (t)} 2 = D ; χ B (ψ u ) 3,3 Lx x e,b x e 2 Lx

}u } 2 }u} 8 . . 0 Lx Lx

Combining (1.1) and (1.5) we obtain 1¡α 2+ 2 }A (t)} 2 = D ; uχ B (uχ ) 3,4 Lx x e,b x e,b 2 Lx ¡ 2+ 1 α }B ( )} 8 2 ( ) . x uχe,b Lx Dx uχe,b . 2 Lx

Meanwhile, 1¡α 2+ 2 }A (t)} 2 = D ; uχ B (uφ ) 3,5 Lx x e,b x e,b 2 Lx ¡ ¡ 2+ 1 α 2+ 1 α }B ( )} 8 2 ( ) + }B ( )} 8 2 ( ) . x uχe,b Lx Dx uφe,b x uφe,b Lx Dx uχe,b . 2 2 Lx Lx

Next, we recall that by construction

e dist (supp (χ ) , supp (ψ )) ¥ . e,b e 2

Thus by Lemma 1.10 1¡α 2+ 2 }A (t)} 2 = uχ B D (uψ ) 3,6 Lx e,b x x e 2 Lx

}u } 2 }u} 8 . . 0 Lx Lx 1¡α 2+ 2 To complete the estimates in (2.66)-(2.67) only remains to bound Dx (uχe,b) , 2 Lx 1¡α 1¡α 2+ 2 2+ 2 Dx uφe,b , and Dx (uφe,b) . 2 2 Lx Lx For the ﬁrst term we proceed by writing

1¡α 1¡α 1¡α 2+ 2 2+ 2 2+ 2 Dx (uχe,b) = Dx uχe,b + Dx ; χe,b (uχe,b + uφe,b + uψe)

= I1 + I2 + I3 + I4.

Notice that }I } 2 is the quantity to be estimated by Gronwall’s inequality. Meanwhile, }I } 2 , 1 Lx 2 Lx }I } 2 and }I } 2 were estimated previously in the case j = 1, step 2. 3 Lx 4 Lx 7. PROPAGATION OF REGULARITY 55

1+α 2+ 2 Next, we focus on estimate the term Dx (uφe,b) which will be treated by means of 2 Lx Holder’s¨ inequality and Theorem 1.4 as follows 1+α 1+α ¸ 2+ 2 2+ 2 1 β s,β D (uφ ) }u} 4 D φ + B φ D u x e,b . Lx x e,b x e,b x 2 4 β! Lx Lx β¤2 L2 x ¡ ¸ 1 1 2+ 1 α } } 2 } } 2 2 Bβ s,β u0 2 u 8 Dx φe,b + cs,β x φe,bDx u . Lx Lx 2 L4 ¤ Lx x β 2 2+ 1+α 1+α } }1/2} }1/2 2 2 B u0 2 u 8 + 1 Dx u + 1 Dx xu . L Lx [e/4,b+e/4] [e/4,b+e/4] x 2 2 Lx Lx 1+α B2 2 + xφe,bDx u 2 Lx 1 2+ 1+α 1 1+α } }1/2} }1/2 2 2 B u0 2 u 8 + χ Dx u + χ Dx xu . L Lx e/8,b+e/4 e/8,b+e/4 x 2 2 Lx Lx 1+α 2 + Dx u 2 Lx 2+ 1+α 1+α } }1/2} }1/2 2 2 B u0 2 u 8 + η + Dx u + η + Dx xu . L Lx e/24,b 7e/24 e/24,b 7e/24 x 2 2 Lx Lx 1+α 2 + Dx u . 2 Lx After integrate in time, the second and third term on the right hand side can be estimated taking (e, b) = (e/24, b + 7e/24) in (2.60) and (2.54) respectively. Hence, after integrate in time follows ¡ 2+ 1 α 2 8 by interpolation that Dx (uφe,b) . 2 2 LT Lx 1¡α 2+ 2 Analogously can be bounded Dx (uφe,b) . 2 Lx §.3 Finally, after apply integration by parts » 1¡α 2 B2 A3,7(t) = A3,7(t)Dx xu χe,b dx »R 1¡α 1¡α 2 2 B3 2 B2 = uχe,bDx xu Dx xu dx R » 2! 1 1¡α 2 B 2 B = uχe,b x Dx xu dx 2 R » » ¡ 2 ¡ 2 1 1 α 1 1 α ¡ B 2 2 B2 ¡ 2 B2 = xu χe,b Dx xu dx uχe,b χe,b Dx xu dx 2 R R

= A3,7,1(t) + A3,7,2(t).

First, » 1¡α 2 | ( )| }B ( )} 8 2 B2 2 A3,7,1 t . xu t Lx Dx xu χe,b dx, R 7. PROPAGATION OF REGULARITY 56 where the last integral is the quantity that will be estimated using Gronwall’s inequality, and the other factor will be controlled after integration in time. After integration in time and Sobolev’s embedding it follows that » » » T T ¡ 2 1 1 α | | 2 2 B2 A3,7,2(t) dt . u (χe,b) Dx xu dx dt R 0 0 » » T ¡ 2 1 1 α ¤ } ( )} 8 2 2 B2 u t Lx χe,b Dx xu dx dt 0 R ! » » T 1¡α 2 2 2 2 1 sup }u(t)} s(α)+ D B u (χ ) dx dt, . H x x e,b 0¤t¤T x 0 R where the last term was already estimated in (2.62). Thus, after collecting all the information in this step and applying Gronwall’s inequality together with hypothesis (0.7), we obtain » » » » » ¡ 2 T T 1 α 2 1 2 1 ¦ 2 B2 2 B3 2 B3 2 ¤ sup Dx xu χe,b dx + xu (χe,b) dx dt + H xu χe,b dx dt c2,2 0¤t¤T R 0 R 0 R ¡ ! ¦ ¦ 1 α } } ¡ 2 B2 ¡ ¥ ¥ where c2,2 = c2,2 α; e; T; v; u0 3 α ; Dx xu0χe,b for any e 0, b 5e and v 0. H 2 2 x Lx According to the induction argument we shall assume that (0.8) holds for j ¤ m with j P Z and j ¥ 2, i.e. (2.68) » » » » » T 2 T 2 2 1+α 1 1+α 1 ¦ Bj 2 2 Bj 2 2 Bj 2 ¤ sup xu χe,b dx + Dx xu χe,b dx dt + HDx xu χe,b dx dt cj,1 0¤t¤T R 0 R 0 R for j = 1, 2, . . . , m with m ¥ 1, for any e ¡ 0, b ¥ 5e v ¥ 0.

STEP 2:

We will assume j an even integer. The case where j is odd follows by an argument similar to the case j = 1. By an analogous reasoning to one employed in the case j = 2 it follows that

1¡α 1¡α 1¡α 1¡α 1¡α 1¡α 2 Bj B 2 Bj 2 ¡ 2 Bj 1+αB 2 Bj 2 2 Bj B 2 Bj 2 Dx x tuDx xuχe,b Dx xDx xuDx xuχe,b + Dx x(u xu)Dx xuχe,b = 0 which after integration in time yields the identity » » ¡ 2 ¡ 2 1 d 1 α v 1 α 1 2 Bj 2 ¡ 2 Bj 2 Dx xu χe,b dx Dx xu (χe,b) dx 2 dt R loooooooooooooooooomoooooooooooooooooon2 R

A (t) » 1 » 1¡α 1¡α 1¡α 1¡α ¡ 2 Bj 1+αB 2 Bj 2 2 Bj B 2 Bj 2 Dx xDx xu Dx xuχe,b dx + Dx x(u xu) Dx xuχe,b dx = 0. looooooooooooooooooooooooooooomooooooooooooooooooooooooooooonR looooooooooooooooooooooooomooooooooooooooooooooooooonR

A2(t) A3(t) §.1 We claim that j+ 1+α 2 8 Dx (uηe,b) . 2 2 LT Lx 7. PROPAGATION OF REGULARITY 57

We proceed as in the case j = 2. A combination of the commutator estimate (1.5), (2.61), Holder’s¨ inequality and (2.68) yields 1+α j+ 1+α j+ 1+α 2 Bj ¤ 2 2 Dx x(uηe,b) Dx uηe,b + Dx ; ηe,b (uχe,b + uφe,b + uψe) L2 L2 L2 L2 L2 L2 T x T x T x ¡ 1+α ¡ 1+α ¦ 2 j 1+ 2 j 1+ 2 (c ) + D (uχ ) +}u } 2 + D (uφ ) . j,1 x e,b 0 Lx x e,b 2 2 2 2 looooooooooooomooooooooooooonLT Lx looooooooooooomooooooooooooonLT Lx

B1 B2 1+α j+ 2 + ηe,bDx (uψe) . 2 2 looooooooooooomooooooooooooonLT Lx

Since χe/5,e = 1 on the support of χe,b then @ P χe,b(x)χe/5,e(x) = χe,b(x) x R.

Combining Lemma 1.9 and Young’s inequality 2j+α¡1 ¡ α¡1 1 α j+ 2j 2 Bj } } 2j Dx (uχe,b) x(uχe,b) u0 2 . 2 Lx 2 Lx Lx j B (uχ ) + }u } 2 . x e,b 2 0 Lx Lx ¸ k (j¡k) γ B uχ χ + }u } 2 . j,k x e/5,e e,b 0 Lx ¤ ¤ (2.69) 0 k j 2 Lx 2(1¡α) ¤ B2 2 B 1 2 } }2 } } 5¡α c xuχe,b 2 + c xuχ b 2 + c u0 2 + c u0 2 Lx e, Lx Lx Lx ¸ 2 ¡ Bj + (j k) Bk . xuχe,b γk,j χ 8 xuχe/5,e L2 e,b L L2 x 2¤k¤j¡1 x x

+ }u} 3¡α + }u0} 2 . 8 2 Lx LT Hx Hence, taking (e, b) = (e/5, e) in (2.68) yields

¸ ¦ ¦ } } } } B1 . cj,1 + γk,jck,1 + u 3¡α + u0 L2 . (2.70) L8 H 2 x 2¤k¤j¡1 T x

B2 can be estimated as in the step 2 of the case j ¡ 1, so is bounded by the induction hypothesis. Next, since e dist (supp (η ) , supp (ψ )) ¥ e,b e 2 we have by Lemma 1.10 1+α j+ 2 η D (uψ ) η e }u } 2 . e,b x e . 8 ,b+e 8 0 Lx 2 Lx Lx Gathering the estimates above follows the claim 1. 7. PROPAGATION OF REGULARITY 58

α+1 We have proved that locally in the interval [e, b] there exists j + 2 derivatives. So, by Lemma 1.9 we obtain 2j¡α+1 ¡ + j+ + 2α + 1 α 1 α 2 α 1 ¡ 1 α j+ 2 j+ 2 2j α+1 j+ 2 D (uη ) D (uη ) }uη } 8 D (uη ) + }u } 2 , x e,b . x e,b e,b L L2 x e,b 0 Lx 2 2 2 2 T x 2 2 LT Lx LT Lx LT Lx then, as before ¡ ¡ ¡ j+ 1 α j+ 1 α j+ 1 α 2 2 ¡ 2 Dx uηe,b = cjDx (uηe,b) cj Dx ; ηe,b (uχe,b + uφe,b + uψe) , where cj is a constant depending only on j. Hence, if we proceed as in the proof of claim 1 we have ¡ j+ 1 α 2 8 (2.71) Dx uηe,b . 2 2 LT Lx Therefore » T ¡ 2 j+ 1 α | | 2 8 A1(t) dt = v Dx uηe,b . 2 2 0 LT Lx

§.2 To handle the term A2 we use the same procedure as in the previous steps. First, » » 1¡α h i 1¡α 1¡α 1¡α 2 Bj 1+αB 2 2 Bj 1+αB 2 Bj 2 2 Bj A2(t) = Dx xu Dx x; χe,b Dx xu dx + Dx xDx xu χe,b Dx xu dx »R R 1¡α h i 1¡α 2 Bj 1+αB 2 2 Bj ¡ = Dx xu Dx x; χe,b Dx xu dx A2(t) R and » 1 1¡α h i 1¡α 2 Bj 1+αB 2 2 Bj A2(t) = Dx xu Dx x; χe,b Dx xu dx 2 »R ¡ ¡ 1 j+ 1 α h i j+ 1 α 2 1+αB 2 2 (2.72) = Dx u Dx x; χe,b Dx u dx 2 R» 1 2j+1¡α 2j+1¡α ¡ 2 2+α 2 2 = Dx u HDx ; χe,b Dx u dx. 2 R Since 1 1 (2.73) HDα+2; χ2 + P (α + 2) + R (α + 2) = HP (α + 2)H, x e,b 2 n n 2 n for some positive integer n. Replacing (2.73) into (2.72) produces » » 1 2j+1¡α 2j+1¡α 1 2j+1¡α 2j+1¡α A (t) = D 2 u R (α + 2)D 2 u dx + D 2 u P (α + 2)D 2 u dx 2 2 x n x 4 x n x R» R 2j+1¡α 2j+1¡α 1 2 2 ¡ Dx u HPn(α + 2)HDx u dx 4 R

= A2,1(t) + A2,2(t) + A2,3(t).

As above we deal ﬁrst with the crucial term in the decomposition associated to A2,that is A2,1. Applying Plancherel’s identity yields » 2j+1¡α 2j+1¡α 1 2 2 A2,1(t) = u Dx Rn(α + 2)Dx u dx. 2 R 7. PROPAGATION OF REGULARITY 59

2j+1¡α We ﬁx n such that (1.13) is satisﬁed. In this case we have to take a = α + 2 and σ = 2 , to get n = j. As occurs in the previous cases it is possible for n = j + 1. 2 Thus, by construction Rj(α + 2) is bounded in Lx (see Proposition 1.11). Then { 2j+3 2 |A (t)| }u } 2 D (χ ) , 2,1 . 0 Lx x e,b 1 Lξ and » T { 2j+3 2 |A (t)| dt }u } 2 sup D (χ ) . 2,1 . 0 Lx x e,b 0 ¤ ¤ 1 0 t T Lξ

Replacing Pj(α + 2) into A2,2 » ¸j ¡ ¡ + 2j+1 α α+1 ¡ ( + ) α+1 ¡ 2j+1 α α 2 l ¡l 2 2 l 2 2l 1 2 l 2 A (t) = c + (¡1) 4 D u D χ D D u dx 2,2 4 2l 1 x x e,b x x l=0 R ¸j » + ¡ 2 ( + ) α 2 ¡ l ¡l j l+1 2 2l 1 = c2l+1( 1) 4 Dx u χe,b dx 4 R »l=0 + 2 α 2 Bj+1 2 1 = H x u (χe,b) dx 4 R j » ¸ 2 α + 2 l ¡l j¡l+1 2 (2l+1) + c + (¡1) 4 D u (χ ) dx 2 2l 1 x e,b l=1 R j¸¡1 = A2,2,1(t) + A2,2,l(t) + A2,2,j(t). l=1

Note that A2,2,1 is positive and it gives the smoothing effect after integration in time, and A2,2,j is bounded by using the local theory. To handle the remainder terms we recall that by construction § § § § § (j) § 1 1 (2.74) (χe,b) (x) . χe/3,b+e(x) . χe/9,b+10e/9(x)χe/9,b+10e/9(x) for x P R, j P Z+. So that, for j ¡ 2 » » » T T ¡ 2 | | j l+1 1 A2,2,l(t) dt . Dx u χe/3,b+e dx dt 0 »0 »R T j¡l+1 2 1 . Dx u χe/9,b+10e/9χe/9,b+10e/9 dx dt, 0 R thus if we apply (2.68) with (e/9, b + 4e/3) instead of (e, b) we obtain » » T j¡l+1 2 1 ¦ D u (χ e χ e ) dx dt ¤ c x 9 ,b+10e/9 ,b+10e/9 l,2 0 R 9 for l = 1, 2, . . . j ¡ 1. 7. PROPAGATION OF REGULARITY 60

Meanwhile, (2.75)

A2,3(t) » ¸j ¡ ¡ + 2j+1 α α+1 ¡ ( + ) α+1 ¡ 2j+1 α α 2 l+1 ¡l 2 2 l 2 2l 1 2 l 2 = ¡ c + (¡1) 4 D u HD χ D HD u dx 4 2l 1 x x e,b x x l=0 R » ¸j ¡ + 2j+1 α α+1 ¡ ( + ) ¡ α 2 l ¡l 2 2 l 2 2l 1 j l+1 = c + (¡1) 4 HD u D χ HD u dx 4 2l 1 x x e,b x l=0 R j » ¸ 2 α + 2 l ¡l j¡l+1 2 (2l+1) = c + (¡1) 4 HD u χ dx 4 2l 1 x e,b l=0 R » j » 2 ¸ 2 α + 2 j+1 2 1 α + 2 l ¡l j¡l+1 2 (2l+1) = B u (χ ) dx + c + (¡1) 4 HD u (χ ) dx 4 x e,b 4 2l 1 x e,b R l=1 R j¸¡1 = A2,3,1(t) + A2,3,l(t) + A2,3,j(t) l=1 ¥ As we can see A2,3,1 0 and it represents the smoothing effect. Besides, applying a similar argument to the employed in (2.74)-(2.75) is possible to bound the remainders terms in (2.75). Anyway, » T | | ¦ ¤ ¤ ¡ A2,3,l(t) dt . cl,2 1 l j 1. 0

§.3 Only remains to estimate A3 to ﬁnish this step.

1¡α 1¡α 1¡α 2 Bj B ¡ 2 Bj B 2 Bj B Dx x(u xu)χe,b = Dx x; χe,b (u xu) + Dx x (χe,bu xu)

¡ ¡ ¡ 1 1 α j 1 α j 1 α j = ¡ D 2 B ; χ B (u2) + D 2 B ; uχ B u + uχ D 2 B (B u) 2 x x e,b x x x e,b x e,b x x x ¡ 1 1 α j (2.76) = ¡ D 2 B ; χ B ((uχ )2 + (uφ)2 + (ψ u2)) 2 x x e,b x e,b e,b e

1¡α 1¡α 2 Bj B 2 Bj B + Dx x; uχe,b x((uχe,b) + (uφe,b) + (uψe)) + uχe,bDx x( xu) = A3,1(t) + A3,2(t) + A3,3(t) + A3,4(t) + A3,5(t) + A3,6(t) + A3,7(t).

Replacing (2.76) into A3 and apply Holder’s¨ inequality » » ¸ 1¡α 1¡α 2 Bj 2 Bj A3(t) = A3,k(t) Dx xu χe,b dx + A3,7(t) Dx xu χe,b dx ¤ ¤ R R 1 k 6 » ¸ 1¡α 1¡α j+ 2 2 j ¤ }A (t)} 2 D u(t)χ (¤ + vt) + A (t)D B uχ dx. 3,k Lx x e,b 3,7 x x e,b ¤ ¤ L2 R 1 k 6 x » 1¡α ¸ 1¡α j+ 2 2 j = D u(t)χ (¤ + vt) }A (t)} 2 + A (t)D B uχ dx x e,b 3,k Lx 3,7 x x e,b L2 R x 1¤k¤6 ¡ ¸ j+ 1 α 2 ¤ = Dx u(t)χe,b( + vt) A3,k(t) + A3,7(t). 2 Lx 1¤m¤6 7. PROPAGATION OF REGULARITY 61

The ﬁrst factor on the right hand side is the quantity to be estimated. We will start by estimating the easiest term.

» » ¡ ¡ 1 1 α 1 1 α ¡ B 2 2 Bj 2 ¡ 2 Bj 2 A3,7(t) = xu χe,b (Dx xu) dx uχe,b χe,b(Dx xu) dx 2 R R

= A3,7,1(t) + A3,7,2(t).

We have that

» ¡ 2 1 α j | ( )| }B ( )} 8 2 B 2 A3,7,1 t . xu t Lx Dx xu χe,b dx, R where the last integral is the quantity that we want to estimate, and the another factor will be controlled after integration in time. After integration in time and Sobolev’s embedding

» » » T T ¡ 1 1 α | | 2 2 Bj A3,7,2(t) dt . u (χe,b) Dx xu dx dt 0 0 R ! » » T 1¡α 2 2 j 2 1 sup }u(t)} s(α)+ D B u (χ ) dx dt . H x x e,b 0¤t¤T x 0 R

where the integral expression on the right hand side was already estimated in (2.71). To handle the contribution coming from A3,1 and A3,2 we apply a combination of (1.2), (1.5) and Lemma 1.9 to obtain

1¡α j+ 2 2 }A (t)} 2 = D ; χ B ((uχ ) ) 3,1 Lx x e,b x e,b L2 x ¡ j+ 1 α j¡ 1+α ¤ }B } 2 2 2 B B 2 c xχ b 8 Dx (uχ b) + Dx xχ b x (uχ b) e, Lx e, e, e, 2 2 8 Lx Lx Lx ¡ j+ 1 α ¤ 2 } } } } } } 8 c Dx (uχ b) u 8 + c Dx (uχ b) 2 u L (2.77) e, Lx e, Lx x L2 x 2 ¡ ¡ 1¡α 1¡α 2j¡α+1 2j α 1 j+ j+ 2j¡α+1 D 2 (uχ ) }u} 8 + D 2 (uχ ) }u } }u} 8 . x e,b Lx x e,b 0 L2 Lx 2 2 x Lx Lx 1¡α j+ 2 D (uχ ) }u} 8 + }u } 2 }u} 8 . x e,b Lx 0 Lx Lx 2 Lx 7. PROPAGATION OF REGULARITY 62

and 1¡α j+ 2 2 }A (t)} 2 = D ; χ B ((uφ ) ) 3,2 Lx x e,b x e,b 2 Lx ¡ j+ 1 α 2 j¡ 1+α 2 ¤ }B } 2 2 B B c xχ b 8 Dx uφ b + Dx xχ b x uφ b e, Lx e, e, e, L2 L8 L2 x x x 1¡α j+ 2 ¤ c D uφ }u} 8 + c Dx uφ }u}L8 x e,b Lx e,b 2 x L2 Lx x 2 ¡ ¡ 1¡α 1¡α 2j¡α+1 2j α 1 j+ j+ 2j¡α+1 D 2 uφ }u} 8 + D 2 uφ }u } }u} 8 . x e,b Lx x e,b 0 L2 Lx 2 2 x Lx Lx 1¡α j+ 2 D uφ }u} 8 1 + }u } 2 . x e,b Lx 0 Lx 2 Lx 1¡α j+ 2 D (uφ ) }u} 8 + }u } 2 }u} 8 . . x e,b Lx 0 Lx Lx 2 Lx

The condition on the supports of χe,b and ψe combined with Lemma 1.10 implies 1¡α j+ 2 2 }A (t)} 2 = D ; χ B ψ u 3,3 Lx x e x e 2 Lx ¡ j+ 1 α 2 B 2 = χe,bDx x ψeu 2 Lx

}u } 2 }u} 8 . . 0 Lx Lx By using (1.1) and (1.5) 1¡α j+ 2 }A (t)} 2 = D ; uχ B (uχ ) 3,4 Lx x e,b x e,b 2 Lx 1¡α j+ 2 }B (uχ )} 8 }D (uχ )} 2 . x e,b Lx x e,b Lx and 1¡α 1¡α j+ 2 j+ 2 }A (t)} 2 }B (uχ )} 8 D (uφ ) + }B (uφ )} 8 D (uχ ) . 3,5 Lx . x e,b Lx x e,b x e,b Lx x e,b 2 2 Lx Lx An application of Lemma 1.10 leads to 1¡α j+ 2 }A (t)} 2 = uχ B D (uψ ) 3,6 Lx e,b x x e,b 2 (2.78) Lx

}u } 2 }u} 8 . . 0 Lx Lx To complete the estimate in (2.77)-(2.78) we write @ P χe,b(x) + φe,b(x) + ψe(x) = 1 x R; then

1¡α 1¡α 1¡α j+ 2 j+ 2 j+ 2 Dx (uχe,b) = Dx uχe,b + Dx ; χe,b (uχe,b + uφe,b + uψe)

= I1 + I2 + I3 + I4. 7. PROPAGATION OF REGULARITY 63

Notice that }I } 2 is the quantity to be estimated. In contrast, I is handled by using Lemma 1 Lx 4 1.10. In regards to }I } 2 and }I } 2 the Lemma 1.7 combined with the local theory, and the step 2 2 Lx 3 Lx corresponding to the case j ¡ 1 produce the required bounds. By Theorem 1.4 and Holder’s¨ inequality 1+α 1+α ¸ j+ 2 j+ 2 1 β s,β D (uφ ) }u} 4 D φ + B φ D u x e,b . Lx x e,b x e,b x 2 4 β! Lx Lx β¤j L2 x ¡ ¸ 2+ 1 α } }1/2} }1/2 2 Bβ s,β u0 2 u 8 Dx φe,b + cs,β x φe,bDx u . Lx Lx 2 L4 ¤ Lx (2.79) x β 2 ¸ 1 j¡β+ α+1 } }1/2} }1/2 Bβ 2 u0 2 u 8 + x φe,bDx u . Lx Lx P β! L2 β Q1(j) x ¸ 1 j¡β+ α+1 Bβ 2 + x φe,bHDx u P β! L2 β Q2(j) x t u where Q1(j), Q2(j) denotes odd integers and even integers in 0, 1, . . . , j respectively. Bβ To estimate the second term in (2.79), note that x φe,b is supported in [e/4, b] then ¸ ¸ 1 j¡β+ α+1 1 j¡β+ α+1 Bβ 2 2 x φe,bDx u . 1[e/8,b]Dx u P β! L2 P β! L2 β Q1(j) x β Q1(j) x ¸ ¡ α+1 1 1 1/2 j β+ 2 . χe/8,b+e/4 Dx u P β! L2 β Q1(j) x ¸ ¡ α+1 1 j β+ 2 . ηe/24,b+7e/24Dx u . P β! L2 β Q1(j) x Hence, after integrate in time and apply (2.68) with (e, b) = (e/24, b + 7e/24) we obtain ¸ ¸ 1 j¡β+ α+1 ¦ 2 1/2 8 ηe/24,b+7e/24Dx u . (cj¡β,1) P β! L2 L2 P β Q1(j) T x β Q1(j) by the induction hypothesis. Analogously, we can handle the third term in (2.79) ¸ ¸ ¡ α+1 1 β j β+ 2 ¦ 1/2 B HD u (c ¡ ) + }u} 3¡α x φe,b x . j β,1 8 2 2 P β! L L2 P LT Hx β Q2(j),β j T x β Q2(j),β j 8.

Therefore, after integrate in time and apply Holder’s¨ inequality we have j+ 1+α 2 8 Dx (uφe,b) . 2 2 LT Lx Next, by interpolation and Young’s inequality 1¡α 1+α j+ 2 j+ 2 (2.80) D (uφ ) D (uφ ) + }u } 2 8. x e,b . x e,b 0 Lx 2 2 2 2 LT Lx LT Lx 7. PROPAGATION OF REGULARITY 64

If we apply (2.79)-(2.80) then ¡ j+ 1 α 2 8 Dx (uφe,b) . 2 2 LT Lx Finally, after collecting all information and apply Gronwall’s inequality we obtain » » » » » ¡ 2 T T 1 α + 2 1 + 2 1 ¦ 2 Bj 2 Bj 1 2 Bj 1 2 ¤ sup Dx xu χe,b dx + x u χe,b dx dt + H x u χe,b dx dt cj,2 0¤t¤T R 0 R 0 R ¡ ! ¦ ¦ 1 α j } } ¡ 2 B ¡ ¥ ¡ where cj,2 = cj,2 α; e; T; v; u0 3 α ; Dx xu0χe,b for any e 0, b 5e and v 0. H 2 2 x Lx This finishes the induction process. To justify the previous estimates we shall follow the following argument of regularization. For P s ¡ 3¡α µ ¦ arbitrary initial data u0 H (R) s 2 , we consider the regularized initial data u0 = ρµ u0 with P 8 ¡ ¥ } } ρ C0 (R), supp ρ ( 1, 1), ρ 0, ρ L1 = 1 and ¡1 ρµ(x) = µ ρ(x/µ), for µ ¡ 0. µ µ ¦ The solution u of the IVP (0.1) corresponding to the smoothed data u0 = ρµ u0, satisfies 8 uµ P C([0, T] : H (R)), we shall remark that the time of existence is independent of µ. Therefore, the smoothness of uµ allows us to conclude that » » » T 2 m+ 1+α 1 Bm µ 2 2 2 µ 2 sup ( x u ) χe,b dx + Dx u (χe,b) dx dt 0¤t¤T R 0 R » » T 2 m+ 1+α 1 ¦ 2 µ 2 ¤ + HDx u (χe,b) dx dt c 0 R ¦ ¦ µ m µ = } } 3¡α B where c c α; e; T; v; u0 ; x u0 χe,b L2 . In fact our next task is to prove that the constant H 2 x ¦ x c is independent of the parameter µ. The independence of the parameter µ ¡ 0 can be reached first noticing that µ } } ¤ } } }x} 8 } } } } } } u 3¡α u0 3¡α ρµ L = u0 3¡α ρµ 1 = u0 3¡α . 0 2 2 ξ 2 Lx 2 Hx Hx Hx Hx ¤ P Next, since χe,b(x) = 0 for x e, then restricting µ (0, e) it follows by Young’s inequality » 8 » 8 2 Bm µ 2 ¦ Bm ( x u0 ) dx = ρµ x u01[0,8) dx e e ¤ } } }Bm } ρµ 1 u0 2 (( 8)) Lξ x Lx 0, m = }B u } 2 8 . x 0 Lx((0, )) Using the continuous dependence of the solution upon the data we have that

µ sup }u (t) ¡ u(t)} 3¡α ÝÑ 0. 2 µÑ0 tP[0,T] Hx ¦ Combining this fact with the independence of the constant c from the parameter µ , weak com- P s ¡ 3¡α pactness and Fatou’s Lemma, the theorem holds for all u0 H (R), s 2 . 7. PROPAGATION OF REGULARITY 65

REMARK 2.20. The proof of TheoremB remains valid for the defocussing dispersive generalized Benjamin-Ono equation # B ¡ α+1B ¡ B P tu Dx xu u xu = 0, x, t R, 0 α 1, u(x, 0) = u0(x). In this direction, the propagation of regularity holds for u(¡x, ¡t), being u(x, t) a solution of (0.1). In other words, this means that for initial data satisfying the conditions (0.6) and (0.7) on the left hand side of the real line, the TheoremB remains valid backward in time.

A question that arise from the proof of the TheoremB is to determine the behavior of the partial derivatives in the left hand side of the real line as time evolves for the function » 8 j 2 (2.81) Bxu (x, t) dx. e¡vt A consequence of the TheoremB is the following corollary, that describe the asymptotic behavior of the function in (2.81).

3¡α COROLLARY 2.21. Let u P C [¡T, T] : H 2 (R) be a solution of the equation in (0.1) described by TheoremB. Then, for any t P (0, T] and δ ¡ 0 » 8 1 j 2 c Bxu (x, t) dx ¤ , ¡8 xx¡yj+δ t ? where x¡ = maxt0, ¡xu, c is a positive constant and xxy := 1 + x2.

For the proof of this corollary we use the following lemma provided by Segata and Smith [41].

LEMMA 2.22. Let f : [0, 8) ÝÑ [0, 8) be a continuous function. If for a ¡ 0 » a f (x) dx ¤ cap, 0 then for every δ ¡ 0, there » 8 f (x) dx ¤ c(p). x yp+δ 0 x t u + PROOF. Let ψk k¥0 be a smooth partition of unity of R such that k¡1 k+1 ¤ ¤ ¤ supp(ψk) [2 , 2 ] for k = 1, 2, , and supp(ψ0) [¡1, 1]. So that, we split the integral of the function f as » 8 ¸8 » 8 ψ (x) ψ (x) k f (x) dx = k f (x) dx xxyp+δ xxyp+δ 0 k=0 0 » 8 ¸8 » 8 ψ (x) ψ (x) = 0 f (x) dx + k f (x) dx xxyp+δ xxyp+δ 0 k=1 0 » » ¸8 k+1 1 ψ (x) 2 ψ (x) = 0 ( ) + k ( ) p+δ f x dx p+δ f x dx. xxy k¡1 xxy 0 k=1 2 7. PROPAGATION OF REGULARITY 66

Notice that,the ﬁrst term can be easily bounded by using the hypothesis, that is » 1 ψ (x) 0 f (x) dx ¤ c. x yp+δ 0 x

Also, a similar argument applies when we deal with each function ψk in the sum above. So that,

» k+1 » k+1 2 ψ (x) 1 2 k f (x) dx ¤ f (x)dx ¡ x yp+δ (k¡1)(p+δ) 2k 1 x 2 0 ¡ ¤ c2 kδ+2p+δ, for k ¥ 1.

Finally, we gather the estimates above to obtain » 8 f (x) dx ¤ 2c 1 + 22p+1 , for every δ ¡ 0, x yp+δ loooooooomoooooooon 0 x :=c(p) as was required.

REMARK 2.23. Observe that the lemma also applies when integrating a non-negative function on the interval [¡(a + e), ¡e], implying decay on the left half-line.

PROOFOF COROLLARY 2.21. We shall recall that TheoremB with x0 = 0 asserts that any e ¡ 0 » 8 j 2 ¦ sup Bxu (x, t) dx ¤ c . tP[0,T] e¡vt

For fixed t P [0, T] we split the integral term as follows » » » 8 e 8 j 2 j 2 j 2 Bxu (x, t) dx = Bxu (x, t) dx + Bxu (x, t) dx. e¡vt e¡vt e The second term in the right hand side is easily bounded by using TheoremB with v = 0. So that, we just need to estimate the first integral in the right hand side. Notice that after making a change of variables, » » e ¡e j 2 j 2 ¦ Bxu (x, t) dx = Bxu (x + 2e, t) dx ¤ c . e¡vt ¡(e¡vt) So that, by using the lemma 2.22 and the remark 2.23 we find » » ¡e e ¦ 1 j 2 1 j 2 c Bxu (x + 2e, t) dx = Bxu (x, t) dx ¤ . ¡8 xx + 2eyj+δ ¡8 xxyj+δ tj In summary, we have proved that for all j P Z+, j ¥ 2 and any δ ¡ 0, » e ¦ 1 j 2 c (2.82) Bxu (x, t) dx ¤ , ¡8 xxyj+δ t and » 8 j 2 ¦ (2.83) Bxu (x, t) dx ¤ c . e 7. PROPAGATION OF REGULARITY 67

If we apply the lemma 2.22 to (2.83) we obtain a more extra decay in the right hand side, this allow us to obtain a uniform expression that combines (2.82) and (2.83), that is, there exist a constant c such that for any t P (0, T] and δ ¡ 0 » 8 1 j 2 c Bxu (x, t) dx ¤ . ¡8 xx¡yj+δ t Chapter 3 Fractional Korteweg-de Vries equation: propagation of regularity

In this chapter we establish Theorem C, the third main result of this work. Roughly,we will show that the solutions of the IVP # B ¡ αB ¡ B P tu Dx xu u xu = 0, x, t R, 0 α 1, u(x, 0) = u0(x), also satisfy the propagation of regularity principle showed in the previous chapter. We shall emphasize the difference that in the case of the fKdV is weak in comparaisson with that of the Benjamin-Ono equation Also, a direct application of the commutator decomposition (1.9) is the smoothing effect associated to solutions of the IVP (0.9).

1 PROPOSITION 3.1. Let ϕ denote a nondecreasing smooth function such that supp ϕ (¡1, 2) and | P ¡ P 8 ϕ [0,1] 1. For j Z, we deﬁne ϕj(x) = ϕ(x j). Let u C ([0, T] : H (R)) be a smooth solution of ¥ ¡ 1 (0.9) with 0 α 1. Assume also that s 0 and r 2 . Then,

» » 1/2 T § § § § ! § s+ α §2 § s+ α §2 1 § 2 § § 2 § Dx u(x, t) + HDx u(x, t) ϕj(x) dx dt 0 R 1/2 }B } } } 8 } } 8 1 + T + xu 1 8 + T u L Hr u L Hs . . LT Lx T x T x

PROOF. As was mentioned above the proof use the decomposition (1.9) and a application of Mikhlin’s Theorem. For a detailed description of the proof see [33, Proposition 2.12].

sα ¡ α REMARK 3.2. In the particular case that u0 is in the Sobolev space H (R), sα = 2 2 , the gain of local derivatives is summarized in the following inequality

» » 1/2 T r § § § § ! § 2 §2 § 2 §2 B ( ) + HB ( ) ¤ } } sα xu x, t xu x, t dx dt C r; T; u0 Hx 0 ¡r

68 1. PROOF OF THEOREMC 69 for any r ¡ 0. The proof follows combining the ideas used in the proof of Proposition 3.1 and the inequality }B u} 8 x L1 L }u} 8 sα }u0}Hs e T x . LT Hx . x The last inequality above is widely know in the literature and its proof is based on energy estimate combined with commutator estimates Kato-Ponce type. For a detailed proof see for instance [31, Chapter 9, p. 221]

1. Proof of TheoremC

We shall use two induction arguments. One for m P Z+ with m ¥ 2 and the other one for m + αk, where k is a positive integer to be speciﬁed later in the proof. Since the solutions of the IVP (0.9) are translation invariant, then without loss of generality we will take x0 = 0. Also, it will be assumed that solutions of the IVP (0.9) are real valued functions with as much regularity as required.

CASE m = 2 :

STEP 1: B2 B2 2 Formally, we apply x to the equation in (0.9) followed by a multiplication by xuχe,b to obtain

B2B B2 2 ¡B2 αB B2 2 B2 B B2 2 x tu xuχe,b xDx xu xuχe,b + x (u xu) xuχe,b = 0 after integrating in the position variable, we obtain the energy identity » » » 1 1 d B2 2 2 ¡ v B2 2 2 ¡ B2 αB B2 2 xu χe,b dx xu χe,b dx xDx xu xuχe,b dx 2 dt R looooooooooooomooooooooooooon2 R looooooooooooooomooooooooooooooonR

» B1(t) B2(t) B2 B B2 2 + x (u xu) xuχe,b dx = 0. loooooooooooooomoooooooooooooonR

B3(t) ¡ ¡ §.1 To handle B1 ﬁrst notice that there exists c 0 and r 0 such that 1 2 1 ¤ (3.1) χe,b (x + vt) = 2χe,b(x + vt)χe,b(x + vt) c1[¡r,r](x) for all (x, t) P R ¢ [0, T]. Thus, by Remark 3.2, it follows that » » » T T 1 | | B2 2 2 B1(t) dt . xu χe,b dxdt 0 »0 »R T r (3.2) B2 2 . xu dxdt 0 ¡r ¤ } } sα c u0 Hx ; r; T . §.2 Now, we extract information from the term handling the dispersive part of the equation in (0.9). 1. PROOF OF THEOREMC 70

Combining integration by parts and Plancherel’s identity it follows that » h i ¡1 B2 α+1 B2 B2(t) = xu HDx ; χe,b xu dx 2 »R h i ¡1 2 α+1 2 = Dxu HDx ; χe,b Dxu dx. 2 R Then using (1.9) h i 1 1 (3.3) HDα+1; χ = ¡R (α + 1) ¡ P (α + 1) + HP (α + 1)H x e,b n 2 n 2 n for some nonnegative integer n to be ﬁxed. This will help to obtain the smoothing effect.

Indeed, replacing this decomposition into B2 yields

» » 1 2 2 1 2 2 B2(t) = DxuRn(α + 1)Dxu dx + DxuPn(α + 1)Dxu dx 2 R» 4 R ¡ 1 2 2 DxuHPn(α + 1)HDxu dx 4 R

= B2,1(t) + B2,2(t) + B2,3(t).

The quantity of terms n, will be chosen according to the following rule (see (1.13))

2n + 1 ¤ α + 5 ¤ 2n + 3, which clearly implies n = 2.

In view of (1.14) the remainder operator R2(α + 1) satisﬁes { 2 2 α+5 2 (3.4) D R (α + 1)D f } f } 2 D χ , x 2 x L2 . Lx x e,b x 1 Lξ for f in a suitable class of functions. For our proposes a combination of (3.4) with Holder’s¨ inequality and Plancherel’s identity is sufﬁcient to obtain » § § { | | § 2 2 § } }2 α+5 2 B2,1(t) . uDxR2(α + 1)Dxu dx . u0 L2 Dx χe,b . x 1 R Lx Therefore, integrating in time yields » T | | ¤ B2,1(t) dt c. 0

Replacing P2(α + 1) into B2,2 and B2,3 produce » » α + 1 2+ α 2 1 α + 1 1+ α 2 (3) B (t) = D 2 u χ2 dx ¡ c D 2 u χ2 dx 2,2 4 x e,b 3 16 x e,b R » R + α 2 ( ) α 1 2 2 5 + c5 Dx u χe,b dx 64 R

= B2,2,1(t) + B2,2,2(t) + B2,2,3(t). 1. PROOF OF THEOREMC 71 and » α + 1 2+ α 2 1 B (t) = HD 2 u χ2 dx 2,3 4 x e,b R » » α + 1 1+ α 2 (3) α + 1 α 2 (5) ¡ 2 2 2 2 c3 HDx u χe,b dx + c5 HDx u χe,b dx 16 R 64 R

= B2,3,1(t) + B2,3,2(t) + B2,3,3(t).

Notice that B2,2,1 and B2,3,1 are positive and represent the smoothing effect. Then we need to bound the terms B2,2,2, B2,2,3, B2,3,2, and B2,3,3. After integrating in time the terms B2,2,2, B2,3,2, B2,2,3, and B2,3,3 can be handled by using the local theory i.e. » T | ( )| } ( )} sα P t u B2,2+l,2 t dt . sup u t Hx for l 0, 1 , 0 0¤t¤T and » T | ( )| } ( )} sα P t u B2,2+l,3 t dt . sup u t Hx for l 0, 1 . 0 0¤t¤T

§.3 Finally, we handle B3 ﬁrstly applying integration by parts as follows » » 1 5 B B2 2 2 ¡ 1 B2 2 2 B3(t) = xu xu χe,b dx u xu χe,b dx 2 R 2 R

= B3,1(t) + B3,2(t).

1 8 Since u satisfies the Strichartz estimate i.e. Bxu P L ([0, T] : L (R)) in Theorem 0.4, then » 2 | ( )| }B ( )} 8 B2 2 (3.5) B3,1 t . xu t Lx xu χe,b dx. R We shall also point out that the integral expression on the right hand side of (3.5) will be estimated by using Gronwall’s inequality. An application of Sobolev’s embedding produces » 2 1 | ( )| } ( )} 8 B2 2 B3,2 t . u t Lx xu χe,b dx R » 1 2 2 2 } ( )} sα B . sup u t Hx xu χe,b dx. 0¤t¤T R We finish this step gathering the estimates above, this together with an application of Gronwall’s inequality yield 2 2+ α 2 2+ α 2 ¦ B2 ¤ 2 ¤ 2 ¤ ¤ (3.6) sup uχe,b( + vt) 2 + Dx uηe,b( + vt) + HDx uηe,b( + vt) c x Lx 2 2 2 2 2,1 0¤t¤T LT Lx LT Lx for any e ¡ 0, b ¥ 5e and v ¥ 0. At this point several issues shall be clarified and fixed. First, we indicate the dependence on ¦ the parameters involved in the constant c2,1, this will be crucial later when we will consider the ¦ ¦ } } B2 limit process. More precisely, c = c α; e; T; v; u0 sα ; u0χe,b 2 . 2,1 2,1 Hx x Lx 1. PROOF OF THEOREMC 72

These families of constants are distinguished in our argument, so that we will differentiate P + ¦ them by ﬁxing the following notation: for m, n Z , the number cm,n corresponds to the case of m¡derivatives in the nth step of the induction process. Without more delays we proceed to the case 2 in our inductive process. This step is summarized in the following diagram B2 2 α/2B2 2 1 xuχe,b Dx xu(χe,b)

α/2 Dx α/2B2 2 αB2 2 1 Dx xuχe,b Dx xu(χe,b) .

The columns on the left hand side indicates the propagation of regularity, which is carried out by steps of length α/2, except for the ﬁnal step that will be exempliﬁed later. Instead, the columns on the right hand side give us the smoothing effect obtained at the corresponding level of propagation. Finally, the diagonal lines are to indicate the dependence of the smoothing effect in the next level of propagation. Now that the strategy has been explained we proceed to estimate. STEP 2: α 2 B2 After applying the operator Dx x to the equation in (0.9) and multiply the resulting equation by α 2 B2 2 Dx xuχe,b(x + vt) one gets » » 1 d α 2 v α 2 1 B2 2 2 ¡ B2 2 2 xDx u χe,b dx xDx u χe,b dx 2 dt R loooooooooooooooomoooooooooooooooon2 R

» » B1(t) α α α α ¡ B2 2 αB B2 2 2 B2 2 B B2 2 2 xDx Dx xu xDx uχe,b dx + xDx (u xu) xDx uχe,b dx = 0. loooooooooooooooooooomoooooooooooooooooooonR loooooooooooooooooooomoooooooooooooooooooonR

B2(t) B3(t) } } §.1 As we indicated in the diagram above, to handle B1 1 it is only required to use (3.6). Then, » » » T T |v| α 2 1 ¦ | | ¤ B2 2 2 ¤ (3.7) B1(t) dt xDx u χe,b dx dt c2,1. 0 2 0 R

§.2 The term B2 shall be rewritten as was done before in order to obtain the corresponding smoothing effect in this step. Thus, we apply integration by parts and Plancherel’s identity to get » 1 2+ α h i 2+ α ¡ 2 α+1 2 2 (3.8) B2(t) = Dx u HDx ; χe,b Dx u dx. 2 R By using the commutator decomposition (3.3) into (3.8)we have » » α α α α 1 2+ 2 2+ 2 1 2+ 2 2+ 2 B2(t) = Dx uRn(α + 1)Dx u dx + Dx uPn(α + 1)Dx u dx 4 R» 4 R α α 1 2+ 2 2+ 2 ¡ Dx uHPn(α + 1)HDx u dx 4 R

= B2,1(t) + B2,2(t) + B2,3(t), 1. PROOF OF THEOREMC 73 for some positive integer n to be fixed. From this term we obtain the smoothing effect by using a similar analysis as the used in the previous step. This argument allow us to fix n = 2, and for this particular value, the operator 2 2 R2(α + 1) is bounded from L (R) into L (R). Hence replacing P2(α + 1) into B2,2 and B2,3 lead us to, in first place that » » α + 1 2 1 α + 1 2 B (t) = D2+αu (χ2 ) dx ¡ c D1+αu (χ2 )(3)dx 2,2 4 x e,b 3 16 x e,b R » R α + 1 α 2 2 (5) + c5 (Dx u) (χe,b) dx 64 R

= B2,2,1(t) + B2,2,2(t) + B2,2,3(t), and in second place » » α + 1 2 1 α + 1 2 B (t) = HD2+αu (χ2 ) dx ¡ c HD1+αu (χ2 )(3)dx 2,3 4 x e,b 3 16 x e,b R » R α + 1 α 2 2 (5) + c5 (HDx u) χe,b dx 64 R

= B2,3,1(t) + B2,3,2(t) + B2,3,3(t).

The terms B2,2,1 and B2,3,1 are positives and give us the smoothing effect. We estimate next B2,2,2 and B2,3,2. To avoid repetitions, we only show the procedures how to bound B2,3,2. A similar analysis is applied to B2,2,2. Before carry on, we shall remember that for any e ¡ 0 and b ¥ 5e the function b 1 ϕe,b = χe,b is smooth. ¡ ¥ So that, for χe/5,e 0 and b 5e, we consider the following decomposition h i 1+α 1+α ¡ 1+α ϕe/3,b+eDx u = Dx (uϕe/3,b+e) Dx ; ϕe/3,b+e u, then a combination of Lemma 1.8 and interpolation produce 1+α 1+α } } } } ϕ + D u D (uϕ + ) + ϕ + l u Hα e/3,b e x 2 . x e/3,b e 2 e/3,b e Hx x Lx Lx B2 } } } } . x (uϕe/3,b+e) 2 + ϕe/3,b+e Hl u Hα Lx x x B2 } } } } } } uϕe/3,b+e 2 + u sα + ϕe/3,b+e l u Hα . . x Lx Hx Hx x Thus, » T 2 | | 1+α B2,2,2(t) dt = c ϕe/3,b+eDx u L2 L2 0 T x B2 2 } }2 . xuϕe/3,b+e 2 2 + u 8 sα LT Lx LT Hx ¤ } } sα c u0 Hx ; e; v; T , where the last inequality is a consequence of the local theory, interpolation and (3.2). 1. PROOF OF THEOREMC 74

Similarly, » » » T T 2 | | 1+α 1 B2,3,2(t) dt . HDx u χe/3,b+edx dt 0 0 R ¤ } } sα c u0 Hx ; e; v; T .

After integrate in time » T | | 8 P t u B2,2+l,3(t) dt for l 0, 1 . 0 §.3 Finally, we deal with the term B3, which corresponds to the nonlinear part of the equation in (0.9). First, we decompose the nonlinearity as follows α 2+ α B2 2 B ¡ 2 B xDx (u xu) χe,b = Dx (u xu)χe,b

1 h 2+ α i = D 2 ; χ B (uχ )2 + (uφ)2 + u2ψ 2 x e,b x e,b e,b e h 2+ α i α ¡ 2 B B2 2 B Dx ; uχe,b x (uχe,b + uφe,b + uψe) + uχe,b xDx xu = B3,1(t) + B3,2(t) + B3,3(t) + B3,4(t) + B3,5(t) + B3,6(t) + B3,7(t).

2 ¡ For that, it is sufﬁcient to estimate the L (R) norm of the terms B3,l for l = 1, 2, . . . , 7. Combining Lemma 1.2 and Lemma 1.8 it is obtained }B } 2 B3,1 xχe,b l (uχe,b) 2 . Hx ˙ 2+α/2 Lx Hx 2+ α 2 2 2 . (uχe,b) + Dx (uχe,b) L2 L2 x x α 2+ 2 }u}L8 }u } 2 + D (uχ ) . x 0 Lx x e,b 2 Lx and }} }B } 2 B3,2 L2 . xχe,b Hl (uφe,b) + x x H˙ 2 α/2 x α 2+ 2 }u}L8 }u } 2 + D (uφ ) . . x 0 Lx x e,b 2 Lx

Since the weighted functions χ and ψe satisfy hypothesis of Lemma 1.10, then e,b α 2+ 2 2 }B } 2 = χ D (u ψe) }u}L8 }u } 2 , 3,3 Lx e,b x 2 . x 0 Lx Lx and α 2+ 2 }B } 2 = uχ D (uψe) }u } 2 }u}L8 . 3,6 Lx e,b x 2 . 0 Lx x Lx To handle the terms B and B , we use the commutator estimate (1.5) to yield 3,4 3,1 2+ α }} 2 }B } B3,4 2 Dx (uχ b) x(uχ b) 8 Lx . e, 2 e, Lx Lx and 2+ α 2+ α }} 2 }B } 2 }B } B3,5 2 Dx (uφ b) x(uχ b) 8 + Dx (uχ b) x(uφ b) 8 . Lx . e, 2 e, Lx e, 2 e, Lx Lx Lx 1. PROOF OF THEOREMC 75

s R The nonlocal character of the operator Dx, for s 2N, implies that several terms above shall be estimated. First, an immediate application of Theorem 1.4 yield α α α α 2+ 2 2+ 2 2+ 2 1+ 2 D (uφ ) D φ }u} 2 + φ D u + Bxφ HD u x e,b 2 . x e,b Lx e,b x 2 e,b x 2 Lx BMO Lx Lx α + B2φ D 2 u x e,b x 2 Lx α α 2 2 1 2+ 2 D B φ }u } 2 + χ D u + }u} sα . x x e,b 8 0 Lx e/8,b+e/4 x 2 Hx Lx Lx α 1 2+ 2 χ D u + }u} sα . . e/8,b+e/4 x 2 Hx Lx Notice that the second term on the right hand side is bounded by local theory. By using the weighted functions properties combined with (3.6) it follows » T 2+ α 2+ α 1 2 ¤ 1/2 2 Dx uχe/8,b+e/4 dt T Dx uχe/8,b+e/4 L2 L2 L2 0 x T x α 1/2 2+ 2 T D uη + . x e/24,b 7e/24 2 2 LT Lx ¦ 1/2 . c2,1 .

Similarly 2+ α ¦ 1/2 D 2 uφ c . x e,b 1 2 . 2,1 L Lx T 2+ α Finally, to estimate D 2 (uχ ) , we write x e,b 2 Lx

2+ α α h 2+ α i 2 ¡ B2 2 2 Dx (uχe,b) = χe,b xDx u + Dx ; χe,b (uχe,b + uφe,b + uψe) .

Thus, a combination of Lemma 1.8 and interpolation lead us to α α 2+ 2 2 2 D (uχ ) χ B D u + }u} sα . x e,b 2 . e,b x x 2 Hx Lx Lx Notice that the ﬁrst term on the right hand side is the quantity to be estimated by Gronwall’s inequality.

Concerning to B3,7 we obtain after applying integration by parts » » 1 α 2 1 1 α 2 ¡ B 2 B2 2 ¡ 2 B2 2 B3,7(t) = xuχe,b xDx u dx u χe,b xDx u dx 2 R 2 R

= B3,7,1(t) + B3,7,2(t).

On one hand, we have » α 2 | ( )| }B ( )} 8 B2 2 2 B3,7,1 t . xu t Lx xDx u χe,bdx, R the integral expression on the right hand side will be estimate by Gronwall’s inequality and by B P 1 8 Theorem 0.4 we have xu LT Lx . 1. PROOF OF THEOREMC 76

On the other hand, by Sobolev’s embedding it follows that

» 1 2+ α | ( )| } ( )} 8 2 ( 2 )2 B3,7,2 t . u t Lx χe,b Dx u dx R » 1 α 2 2 2 2 } ( )} sα B . sup u t Hx χe,b xDx u dx. 0¤t¤T R

Integrating in time yields

» » » T T 1 α 2 2 2 2 | ( )| } ( )} sα B B3,7,2 t dt . sup u t Hx χe,b xDx u dx dt, 0 0¤t¤T 0 R

where the integral expression corresponds to the B1 term already estimated in (3.7). Gathering the estimates corresponding to B1, B2 and B3 combined with Gronwall’s inequality and integration in time yields that for any e ¡ 0, b ¥ 5e and v ¥ 0

α 2 2 2 ¦ B2 2 ¤ 2+α 2+α ¤ sup Dx uχe,b( + vt) + D uηe,b 2 2 + HD uηe,b 2 2 c , x 2 x L Lx x L Lx 2,2 0¤t¤T Lx T T

α ¦ ¦ 2 2 where c = c α; e; T; v; }u } sα ; D B u χ ¡ 0. 2,2 2,2 0 Hx x x 0 e,b 2 Lx At this point, we shall determine the number of steps in the second inductive process that allow us to reach the next integer. For this reason we will consider the following cases:

2 ¤ 1 (a) If 2k+1 α k for some positive integer k, then are required 2k + 1 steps in the second inductive process. 1 ¤ 2 (b) Instead, if k+1 α 2k+1 for some positive integer k, then are required 2k + 2 steps in the second inductive process.

A detailed description of the number of steps can be obtained directly from (a)-(b). More precisely, r 2 s r¤s there are required α steps in both cases, where denotes the greatest integer. The different considerations are made to differentiate when the number of steps are even or odd. A graphical description of the process described here is presented below. Henceforth, for comfort in the notation we will consider α satisfying the condition (a). As part of the second inductive process we shall assume that

2 2 2 j+1 j+1 αj 2+α 2+α ¦ B2 2 ¤ 2 2 ¤ (3.9) sup xDx uχe,b( + vt) + Dx uηe,b + HDx uηe,b c2,j+1 ¤ ¤ 2 0 t T Lx 2 2 2 2 LT Lx LT Lx for every e ¡ 0, b ¥ 5e, v ¥ 0, and j = 0, 1, ¤ ¤ ¤ , 2k ¡ 1. In fact, the previous cases in the induction process are summarized in the following diagram: 1. PROOF OF THEOREMC 77

B2 2 α/2B2 2 1 xuχe,b Dx xu(χe,b)

α/2 Dx α/2B2 2 αB2 2 1 Dx xuχe,b Dx xu(χe,b)

α/2 Dx αB2 2 3α/2B2 2 1 Dx xuχe,b Dx xu(χe,b)

α/2 Dx 3α/2B2 2 2αB2 2 1 Dx xuχe,b Dx xu(χe,b)

α/2 Dx . . . . α/2 Dx ¡ (2k 1)α/2B2 2 αkB2 2 1 Dx xuχe,b Dx xu(χe,b)

1¡α/2 Dx 1¡α/2B2 2 B3 2 1 Dx xuχe,b xu(χe,b)

The last before the last case in the diagram is the one that we will carry out in the next step, we will present all the issues corresponding to this case.

STEP 2k + 1: A standard argument lead us to the energy identity » » 1 d 1¡ α 2 v 1¡ α 2 1 B2 2 2 ¡ B2 2 2 xDx u χe,bdx xDx u (χe,b) dx 2 dt R loooooooooooooooooomoooooooooooooooooon2 R

B1(t) (3.10) » » 1¡ α 1¡ α 1¡ α 1¡ α ¡ B2 2 αB B2 2 2 B2 2 B B2 2 2 xDx Dx xu xDx uχe,b dx + xDx (u xu) xDx uχe,b dx = 0. looooooooooooooooooooooooomooooooooooooooooooooooooonR loooooooooooooooooooooooomoooooooooooooooooooooooonR

B2(t) B3(t)

} } Notice that estimating the term B1 2 is straightforward in the previous cases, because it is a direct consequence of the former cases. However, this cannot be done in this step, instead a new approach is required. Since the smoothing effect obtained in the case j = 2k ¡ 1 is 2 + αk, then by hypothesis

α 2 1 2 + αk ¥ 3 ¡ for α P , . 2 2k + 1 k } } Therefore, the regularity obtained in (3.9) is enough to provide a bound for B1 1. To do this, we B2 αk ﬁrst decompose the term xDx u as follows: h i B2 αk B2 αk ¡ 2+αk (3.11) xDx (uηe,b) = ηe,b xDx u Dx ; ηe,b (uχe,b + uφe,b + uψe) . 1. PROOF OF THEOREMC 78

Then, by Lemma 1.8 2 αk 2 αk h 2+αk i B D (uη ) ¤ B D uη + D ; η (uχ + uφ + uψe) x x e,b 2 x x e,b 2 x e,b e,b e,b 2 Lx Lx Lx 2 αk B D uη + }η } }uχ } 1+αk + }uφ } 1+αk . x x e,b 2 e,b l,2 e,b H˙ x e,b H˙ x (3.12) Lx 2+αk + η D (uψe) e,b x 2 Lx

= I1 + I2 + I3 + I4. Notice that after integrating in time, the inequality (3.9) implies that » » !1/2 T 2 } } 2+αk 2 ¦ 1/2 I1 L2 = Dx u ηe,b dx dt . (c2,j+1) . T 0 R The terms I and I can be bounded via Lemma 1.9 and Young’s inequality 2 3 1+αk B2 } } D (uχe,b) (uχe,b) 2 + u0 L2 x 2 . x Lx x Lx B2 } } uχe,b 2 + u sα , . x Lx Hx and 1+αk B2 } } D (uφe,b) (uφe,b) 2 + u0 L2 x 2 . x Lx x Lx B2 } } uφe,b 2 + u sα . . x Lx Hx Moreover, } } 1+αk B2 } } I2 2 = D (uχe,b) sup uχe,b + u 8 sα . LT x 2 2 . x 2 LT Hx LT Lx 0¤t¤T On the other hand, there exists τ ¡ 0 such that P ¢ φe,b(x + vt) . 1[¡τ,τ](x) for all (x, t) R [0, T], Thus, in view of Remark 3.2 » » 1/2 T ! B2 B2 2 2 xuφe,b L2 L2 . xu φe,bdx dt T x 0 R » » 1/2 T τ ! B2 2 . xu dx dt 0 ¡τ ¤ } } sα c u0 Hx ; τ . Next, by Lemma 1.10, it follows that 2+αk (3.13) η D (uψe) }u } 2 . e,b x 2 . 0 Lx Lx Gathering together the estimates above lead us to D2+αk(uη ) 8. x e,b 2 2 LT Lx This inequality combined with Lemma 1.9 implies that

6¡α ( + )¡ α α 1 2k 2 3¡ 2(2+αk) 2 2+αk } } 2(2+αk) Dx (uηe,b) Dx (uηe,b) uηe,b 2 . 2 . 2 Lx Lx Lx 1. PROOF OF THEOREMC 79

1¡ α B2 2 Since we need to estimate xDx uηe,b, a decomposition as that in (3.11) is sufﬁcient to our proposes, i.e.

1¡ α 1¡ α h 1¡ α i B2 2 B2 2 ¡ B2 2 xDx uηe,b = xDx (uηe,b) xDx ; ηe,b (uχe,b + uφe,b + uψe) .

For the sake of brevity, we omit the computations behind these commutators estimates. Never- theless, similar arguments as those used in (3.12)-(3.13) lead to 1¡ α B2D 2 uη 8. x x e,b 2 2 LT Lx } } The estimates above are in fact a way to bound the term B1 1 in (3.10). More precisely, » T 1¡ α 2 |B (t)| dt = v B2D 2 uη 8. 1 x x e,b 2 2 0 LT Lx §.2 A previous argument (see step 1) implies that » » ¡ α ¡ α ¡ α ¡ α 1 3 2 3 2 1 3 2 3 2 B2(t) = Dx uRn(α + 1)Dx u dx + Dx uPn(α + 1)Dx u dx 2 R» 4 R ¡ α ¡ α (3.14) 1 3 2 3 2 ¡ Dx uHPn(α + 1)HDx u dx 4 R

= B2,1(t) + B2,2(t) + B2,3(t).

A similar argument to the used in step 1 allow us to ﬁx n = 2 above. 2 According to Proposition 1.11, the remainder term R2(α + 1) is a bounded operator in L (R). Therefore » ¡ α ¡ α 1 3 2 3 2 B2,1(t) = uDx R2(α + 1)Dx u dx, 2 R and by Holder’s¨ inequality 3¡ α 3¡ α { | | } } 2 2 } }2 7 2 B2,1(t) u0 2 Dx R2(α + 1)Dx u u0 2 D χ . . Lx 2 . Lx x e,b Lx 1 Lξ

Thus, » T { | | } }2 7 2 B2,1(t) dt . T u0 2 Dx χe,b . 0 1 If we decompose the terms in (3.14) it follows that » » α + 1 2 1 α + 1 2 (3) B (t) = HB3u χ2 dx ¡ c B2u χ2 dx 2,2 4 x e,b 3 16 x e,b R » R + ( ) α 1 B 2 2 5 + c5 (H xu) χe,b dx 64 R

= B2,2,1(t) + B2,2,2(t) + B2,2,3(t), 1. PROOF OF THEOREMC 80

and » » α + 1 2 1 α + 1 2 (3) B (t) = B3u χ2 dx ¡ c HB2u χ2 dx 2,3 4 x e,b 3 16 x e,b R » R + ( ) α 1 B 2 2 5 + c5 ( xu) χe,b dx 64 R

= B2,3,1(t) + B2,3,2(t) + B2,3,3(t).

As before B2,2,1 and B2,3,1 provide the smoothing effect after integration in time. The terms B2,2,2 and B2,3,2 are easily handled by using Proposition 3.2 and the arguments used in (3.1)-(3.2). Meanwhile, the low regularity in the terms B2,2,3 and B2,3,3 is handled by using local theory. §.3 To ﬁnish this step, we turn our attention to the term involving the nonlinear part of the equation 1¡ α 3¡ α B2 2 B ¡ 2 B xDx (u xu) χe,b = Dx (u xu)χe,b

1 h 3¡ α i = D 2 ; χ B (uχ )2 + (uφ)2 + u2ψ 2 x e,b x e,b e,b e h 3¡ α i 1¡ α ¡ 2 B B2 2 B Dx ; uχe,b x (uχe,b + uφe,b + uψe) + uχe,b xDx xu = B3,1(t) + B3,2(t) + B3,3(t) + B3,4(t) + B3,5(t) + B3,6(t) + B3,7(t). 2¡ In the decomposition above is sufﬁcient to estimate the Lx norm of the terms B3,l, for l = 1, 2, ¤ ¤ ¤ , 6. Combining (1.2) and Lemma 1.8 it follows that }} 1 2 B3,1 L2 . χe,b l (uχe,b) 3¡ α x Hx H˙ 2 x 3¡ α 2 2 2 . u 2 + Dx (uχe,b) Lx L2 x ¡ α 3 2 }u}L8 }u } 2 + D (uχ ) . x 0 Lx x e,b 2 Lx and ¡ α 3 2 }B } 2 }u}L8 }u } 2 + D (uφ ) . 3,2 Lx . x 0 Lx x e,b 2 Lx

Since χe,b and ψe satisfy e dist (supp (χ ) , supp (ψ )) ¥ ¡ 0, e,b e 2 then by Lemma 1.10, it follows that ¡ α 3 2 2 }B } 2 = χ D (u ψe) }u}L8 }u } 2 , 3,3 Lx e,b x 2 . x 0 Lx Lx and ¡ α 3 2 }B } 2 = uχ D (uψe) }u } 2 }u}L8 . 3,6 Lx e,b x 2 . 0 Lx x Lx Instead, the terms B and B are handled by using (1.5) 3,4 3,1 3¡ α }} 2 }B } B3,4 2 Dx (uχ b) x(uχ b) 8 Lx . e, 2 e, Lx Lx 1. PROOF OF THEOREMC 81 and 3¡ α 3¡ α }} 2 }B } 2 }B } B3,5 2 Dx (uφ b) x(uχ b) 8 + Dx (uχ b) x(uφ b) 8 . Lx . e, 2 e, Lx e, 2 e, Lx Lx Lx Notice that

1¡ α 1¡ α h 3¡ α i B2 2 B2 2 ¡ 2 xDx (uχe,b) = χe,b xDx u Dx ; χe,b (uχe,b + uφe,b + uψe) , so that a combination of interpolation, local theory and Lemma 1.8 yield 1¡ α 1¡ α B2 2 B2 2 B2 B2 } } Dx (uχe,b) χe,b Dx u + uχe,b 2 + uφe,b 2 + u sα . x 2 . x 2 x Lx x Lx Hx Lx Lx An immediate application of Theorem 1.4 implies that D2+αk(uφ ) x e,b 2 Lx 2+αk 2+αk 1+αk 2 αk D φ }u} 2 + φ D u + Bxφ HD u + B φ D u . x e,b Lx e,b x 2 e,b x 2 x e,b x 2 BMO Lx Lx Lx 1 2+αk 1 1+αk 1 αk }u } 2 + χ D u + χ HD u + χ D u . 0 Lx e/8,b+e/4 x 2 e/8,b+e/4 x 2 e/8,b+e/4 x 2 Lx Lx Lx

= A1 + A2 + A3 + A4. The ﬁrst and fourth term on the right hand side above are bounded after integrate in time. The second and third term require extra arguments to control them after integrate in time. Since α satisﬁes 2 1 ¤ α for k P Z+, 2k + 1 k it is clear that αk is positive and strictly less than one for all positive integer k, then is straightfor- } } 8 ward to see that A4 1 . Interpolation combined with Calderon’s commutator estimate (1.1) implies that 1+αk A = Bxφ HD u 3 e,b x 2 Lx 1+αk 1+αk = H Bxφ D u ¡ [H; Bxφ ] D u e,b x e,b x 2 Lx 1+αk Bxφ D u + }u} sα , . e,b x 2 Hx Lx and B 1+αk B2 B } } xφe,bD u (u xφe,b) 2 + u sα x 2 . x Lx Hx Lx 1 B2 } } s xuχ b+ 2 + u H α . . e/8, e/4 Lx x The last inequality above is obtained combining interpolation and the properties of the weighted functions. Hence, } } } } } } A3 1 c u0 sα ; e; T; v + u 8 sα . . Hx LT Hx Next, the properties of weighted functions and (3.9) imply } } 1/2 ¦ 1/2 A2 1 . T c k . 2,2 Applying similar arguments allow us to estimate D2+αk uφ . x e,b 2 Lx 1. PROOF OF THEOREMC 82

Finally, integration by parts give us » » 1 1¡ α 2 1 1 1¡ α 2 ¡ B 2 B2 2 ¡ 2 B2 2 B3,7(t) = xuχe,b xDx u dx u χe,b xDx u dx 2 R 2 R

= B3,7,1(t) + B3,7,2(t).

Observe that » 1¡ α 2 | ( )| }B ( )} 8 B2 2 2 B3,7,1 t . xu t Lx xDx u χe,bdx, R

B P 1 8 By the local theory xu LT Lx , and the integral expression will be estimated by means of Gron- wall’s inequality. By Sobolev’s embedding it follows that » 1 1¡ α 2 | ( )| } ( )} 8 2 B2 2 B3,7,2 t . u t Lx χe,b xDx u dx R ! » 1 α 2 2 2 2 } ( )} sα B . sup u t Hx χe,b xDx u dx 0¤t¤T R and noticing that after integrating in time we obtain » ! » » T T 1 α 2 2 2 2 | ( )| } ( )} sα B B3,7,2 t dt . sup u t Hx χe,b xDx u dx dt, 0 0¤t¤T 0 R where the integral expression corresponds to the B1 term, which was already estimated in (3.7). We conclude this step gathering the estimates corresponding to B1, B2 and B3 combined with Gronwall’s inequality and integration in time to obtain for any e ¡ 0, b ¥ 5e and v ¥ 0, 1¡ α 2 2 2 ¦ B2 2 ¤ B3 B3 ¤ sup Dx uχe,b( + vt) + uηe,b 2 2 + H uηe,b 2 2 c , x 2 x L Lx x L Lx 2,2k 0¤t¤T Lx T T ¡ α ¦ ¦ 1 2 2 where c = c α; e; T; v; }u } sα ; D B u χ ¡ 0. 2,2k+1 2,2k+1 0 Hx x x 0 e,b 2 Lx The inequality above ﬁnishes the ﬁrst induction argument.

CASE m Next, our argument will combine two induction process at the same time to reach our goal. The ﬁrst induction process consists in to assume that m derivatives are propagated to then prove that m + 1 derivatives are propagated; this is summarized in the diagram below.

B2 B3 B4 xuχe,b xuχe,b xuχe,b

Bx Bm¡1 Bm Bm+1 x uχe,b x uχe,b x uχe,b More precisely, it is described in the following ﬁgure 1. PROOF OF THEOREMC 83

Bm 2 α/2Bm 2 1 x uχe,b Dx x u(χe,b)

α/2 Dx α/2Bm 2 αBm 2 1 Dx x uχe,b Dx x u(χe,b)

α/2 Dx αBm 2 3α/2Bm 2 1 Dx x uχe,b Dx x u(χe,b)

α/2 Dx 3α/2Bm 2 2αBm 2 1 Dx x uχe,b Dx x u(χe,b)

α/2 Dx . . . . α/2 Dx ¡ (2k 1)α/2Bm 2 αkBm 2 1 Dx x uχe,b Dx x u(χe,b)

1¡α/2 Dx 1¡α/2Bm 2 Bm+1 2 1 Dx x uχe,b x u(χe,b)

As part of this induction process, we shall assume that for any e ¡ 0, b ¥ 5e, v ¥ 0, the following holds:

(I) for l = 2, 3, ¤ ¤ ¤ , m and j = 0, 1, ¤ ¤ ¤ , 2k ¡ 1

2 2 2 j+1 j+1 αj α α ¦ Bl 2 ¤ 2 Bl 2 Bl ¤ (3.15) sup xDx uχe,b( + vt) + Dx xuηe,b + HDx xuηe,b cn,j ¤ ¤ 2 0 t T Lx 2 2 2 2 LT Lx LT Lx

! αj ¦ ¦ l 2 = } } sα B ¡ where cl,j cl,j α; e; T; v; j; u0 Hx ; xDx u0χe,b 0. 2 Lx (II) For l = 2, 3, ¤ ¤ ¤ , m ¡ 1

1¡ α 2 2 2 ¦ sup Bl D 2 uχ (¤ + vt) + Bl+1uη + HBl uη ¤ c , (3.16) x x e,b 2 x e,b 2 2 x e,b 2 2 l,2k 0¤t¤T Lx LT Lx LT Lx

as usual we indicate the dependence of the parameters involved behind the constant in the right hand side above. More precisely,

¡ α ¦ ¦ l 1 2 c = c α; e; T; v; j; }u } sα ; B D u χ ¡ 0. l,j l,j 0 Hx x x 0 e,b 2 Lx

STEP 2: 1. PROOF OF THEOREMC 84

Once our induction assumptions are ﬁxed, we proceed as before, that is, we obtain the weighted energy estimates » » 1 d α 2 v α 2 1 2 Bm 2 ¡ 2 Bm 2 Dx x u χe,b dx Dx x u χe,b dx 2 dt R looooooooooooooooomooooooooooooooooon2 R

B1(t) (3.17) » » α α α α ¡ 2 Bm αB 2 Bm 2 2 Bm B 2 Bm 2 Dx x Dx xu Dx x uχe,b dx + Dx x (u xu) Dx x uχe,b dx = 0. loooooooooooooooooooooomoooooooooooooooooooooonR looooooooooooooooooooooooomooooooooooooooooooooooooonR

B2(t) B3(t) ¡ } } §.1 First, by the induction hypothesis (3.15), we take l = m and j = 2k 1, then B1 1 is LT bounded. More precisely, » » » T T α 2 1 ¦ | | 2 Bm 2 B1(t) dt . Dx x u χe,b dx . cm,1. 0 0 R

§.2 Next, as usual in our argument to handle B2, we ﬁrst rewrite it as follows » 1 α α 2 Bm αB 2 2 Bm B2(t) = Dx x u Dx x; χe,b Dx x u dx 2 R» 1 α h i α ¡ 2 Bm α+1 2 2 Bm = Dx x u HDx ; χe,b Dx x u dx. 2 R

At this point several remarks shall be made. More precisely, B2 has different representations according the number m be even or odd. The full description of these issues is presented below. P (I) If m Q1, then » 1 m+ α h i m+ α ¡ 2 α+1 2 2 B2(t) = Dx u HDx ; χe,b Dx u dx. 2 R

(II) If m P Q2, then » 1 m+ α h i m+ α ¡ 2 α+1 2 2 B2(t) = HDx u HDx ; χe,b HDx u dx. 2 R We will only focus our attention on (II). Nevertheless, the way to proceed when (I) holds was described in the case m = 2. As was done in the previous cases, we incorporate the commutator decomposition (1.9) to obtain » » α α α α 1 m+ 2 m+ 2 1 m+ 2 m+ 2 B2(t) = HDx uRn(α + 1)HDx u dx + HDx uPn(α + 1)HDx u dx 2 R» 4 R α α (3.18) 1 m+ 2 m+ 2 ¡ Dx HuHPn(α + 1)HDx Hu dx 4 R

= B2,1(t) + B2,2(t) + B2,3(t), for some positive integer n. We ﬁx n satisfying α 2n + 1 ¤ α + 1 + 2 m + ¤ 2n + 3, 2 1. PROOF OF THEOREMC 85 from this we obtain n = m. Thus, in view of Plancherel’s identity, Holder’s¨ inequality and Propo- sition 1.11 α α m+ 2 m+ 2 |B (t)| }Hu(t)} 2 D Rm(α + 1)D Hu(t) 2,1 . Lx x x 2 Lx { 2m+2α+1 2 }u } 2 D χ . . 0 Lx x e,b 1 Lξ

Therefore, » T |B (t)| dt T}u } 2 . 2,1 . 0 Lx 0 In addition to this, the terms in the decomposition (3.18) can be written as » 1 α + 1 m+α 2 2 B2,2(t) = HDx u χe,b dx 4 R » m¸¡1 d α + 1 c (¡1) ¡ 2 (2d+1) + 2d+1 H m d+α 2 d Dx u χe,b dx 4 4 R d=1 » α + 1 c (2m+1) ¡ 2m+1 (H α )2 2 m Dx u χe,b dx 4 4¸ R ¸ = B2,2,1(t) + B2,2,d(t) + B2,2,d(t) + B2,2,m(t) P ¡ P ¡ d Q1(m 1) d Q2(m 1) and » ¸m d α + 1 c (¡1) m¡d+ α 2 ( ) = 2d+1 2 ( 2 )(2d+1) B2,3 t d Dx u χe,b dx 4 4 R »d=0 1 α + 1 m+α 2 2 = Dx u χe,b dx 4 R » m¸¡1 d α + 1 c (¡1) ¡ 2 (2d+1) + 2d+1 m d+α 2 d Dx u χe,b dx 4 4 R d=1 » α + 1 c (2m+1) ¡ 2m+1 ( α )2 2 m Dx u χe,b dx 4 4¸ R ¸ = B2,3,1(t) + B2,3,d(t) + B2,3,d(t) + B2,3,m(t). P ¡ P ¡ d Q1(m 1) d Q2(m 1)

The terms B2,2,1 and B2,3,1 are positive and represent the smoothing effect after integrate in time. } } } } Besides, B2,2,m 1 and B2,3,m 1 are easily controlled by means of the local theory. The remainder terms require extra arguments besides local theory to be estimated, we proceed to describe how to handle these terms. P ¡ First, for d Q1(m 1) » » » T T 2 | | m¡d+α 1 B2,2,d(t) dt . HDx u χe/3,b+e dx dt 0 »0 »R T 2 1 (3.19) m¡d+α 2 . HDx u χe/9,b+10e/9 dx dt 0 R ¤ ¦ cm¡d,2, 1. PROOF OF THEOREMC 86

while for d P Q2(m ¡ 1) » » » T T 2 | | Bm¡d α 1 B2,2,d(t) dt . x Dx u χe/3,b+e dx dt 0 »0 »R T 2 1 (3.20) Bm¡d α 2 . x Dx u χe/9,b+10e/9 dx dt 0 R ¤ ¦ cm¡d,2. The inequalities in (3.19) and (3.20) are obtained combining the properties of the weighted functions and (3.15). Similar arguments can be applied to obtain » T | | ¦ P ¡ Y ¡ B2,3,d(t) dt . cm¡d,2 for any d Q1(m 1) Q2(m 1). 0

§.3 The term B3 is handled as in the previous steps, this is, ﬁrst we decompose it according to the case. This is: (I) if m is even, then α m+ α Bm 2 B 2 B x Dx (u xu) χe,b = cmDx (u xu)χe,b c h m+ α i 2 = m D 2 ; χ B (uχ )2 + uφ + u2ψ 2 x e,b x e,b e,b e h m+ α i α ¡ 2 B Bm 2 B cm Dx ; uχe,b x (uχe,b + uφe,b + uψe) + uχe,b x Dx xu = B3,1(t) + B3,2(t) + B3,3(t) + B3,4(t) + B3,5(t) + B3,6(t) + B3,7(t); (II) instead for m odd, α m+ α Bm 2 B ¡ 2 B x Dx (u xu) χe,b = cmHDx (u xu)χe,b c h m+ α i 2 = m HD 2 ; χ B (uχ )2 + uφ + u2ψ 2 x e,b x e,b e,b e h m+ α i α ¡ 2 B Bm 2 B cm HDx ; uχe,b x (uχe,b + uφe,b + uψe) + uχe,b x Dx xu c h m+ α i 2 = m H D 2 ; χ B (uχ )2 + uφ + u2ψ 2 x e,b x e,b e,b e c m+ α 2 + m [H; χ ] D 2 B (uχ )2 + uφ + u2ψ 2 e,b x x e,b e,b e h m+ α i ¡ 2 B cmH Dx ; uχe,b x (uχe,b + uφe,b + uψe) m+ α α ¡ 2 B Bm 2 B cm [H; uχe,b] Dx x (uχe,b + uφe,b + uψe) + uχe,b x Dx xu = B3,1(t) + B3,2(t) + B3,3(t) + B3,4(t) + B3,5(t) + B3,6(t) + B3,7(t) + B3,8(t) + B3,9(t) + B3,10(t) + B3,11(t) + B3,12(t) + B3,13(t). We will focus on working the harder case to estimate i.e. when m is odd. The case m even, is simpler. For the sake of simplicity we will gather the terms according to the tools used for its estimation without matter the order. 1. PROOF OF THEOREMC 87

First, by Lemma 1.10 it is clear that }B } 2 }u(t)} 8 }u } 2 for l = 1, 2, 3, 4. 3,3l Lx . Lx 0 Lx A combination of Lemma 1.2, Lemma 1.8, interpolation and the Calderon’s commutator estimate (1.1), give us α m+ 2 }B } 2 }u} 8 }u } 2 + D (uχ ) , 3,1 Lx . Lx 0 Lx x e,b 2 Lx α m+ 2 }B } 2 }u} 8 }u } 2 + D (uφ ) , 3,2 Lx . Lx 0 Lx x e,b 2 Lx and }B + } 2 }u} 8 }u} sα for l P t0, 1u. 3,4 l Lx . Lx Hx Next, applying commutator’s estimate (1.5) leads to ¡ α m+1 2 }B } 2 [H; uχ ] D Bx ((uχ )) 3,7 Lx . e,b x e,b 2 Lx α m+ 2 }Bx(uχ )}L8 D (uχ ) . e,b x x e,b 2 Lx and " + # m+1¡α j 1 2 }B } 2 = |c | H D ; uχ B (uφ ) 3,8 Lx m x e,b x e,b 2 Lx " + # m+1¡α j 1 2 B . Dx ; uχe,b x(uφe,b) 2 Lx α α m+ 2 m+ 2 D (uφ ) }Bx(uχ )}L8 + D (uχ ) }Bx(uφ )}L8 . . x e,b 2 e,b x x e,b 2 e,b x Lx Lx In order to bound the remainder terms we use (1.5) from where we get that αj m+ 2 }B } 2 = |c | H D ; uχ B (uχ ) 3,10 Lx m x e,b x e,b 2 Lx h ¡ α i m+1 2 D ; uχ Bx(uχ ) . x e,b e,b 2 Lx α m+ 2 }Bx(uχ )}L8 D (uχ ) , . e,b x x e,b 2 Lx and α m+ 2 }B } 2 }Bx(uχ )}L8 D (uφ ) . 3,11 Lx . e,b x x e,b 2 Lx To ﬁnish with the estimates above, we replace B3,13 into (3.17), from that it is obtained a term which can be estimated by using integration by parts, Gronwall’s inequality and the Strichartz’s estimate in Theorem 0.4. Notwithstanding all the terms above have been estimated, several issues have to be clariﬁed. s R First, the non-local behavior of operator Dx for any s 2N do not allow us to know where force us to localize the function u and its derivatives in the places where the regularity is available. 1. PROOF OF THEOREMC 88

m+ α For this reason, the term D 2 (uχ ) appearing in the estimates above must be bounded, x e,b 2 Lx we do this by using the decomposition trick, this is

α α h α i m+ 2 m+ 2 m+ 2 (3.21) Dx (uχe,b) = χe,bDx u + Dx ; χe,b (uχe,b + uφe,b + uψe) , and if we assume that m is odd the decomposition (3.21) have to be rewritten as follows

m+ α α α 2 Bm 2 Bm 2 Dx (uχe,b) = cm [H; χe,b] x Dx u + cmH χe,b x Dx u h α i m+ 2 + Dx ; χe,b (uχe,b + uφe,b + uψe) , where cm is a non-null constant. A direct application of Calderon’s commutator estimate (1.1), Lemma 1.8 and interpolation imply that α α ¡ α ¡ α m+ 2 m 2 m 1+ 2 m 1+ 2 D (uχ ) }u} sα + χ B D u + D (uχ ) + D (uφ ) x e,b 2 . Hx e,b x x 2 x e,b 2 x e,b 2 Lx Lx Lx Lx α } } Bm 2 }Bm } }Bm } u sα + χ b Dx u + (uχ b) 2 + (uφ b) 2 . Hx e, x 2 x e, Lx x e, Lx Lx

= A1 + A2 + A3 + A4.

It is straightforward to see that A1 is bounded, this is a consequence of the local theory. The term, A3 is handled by ﬁrst noticing that

P (3.22) χe/5,e(x)χe,b(x) = χe,b(x) for all x R, then combining Lemma 1.9 and Young’s inequality

¡ m¸1 }Bm } }Bm } Bd } } x (uχe,b) 2 . x uχe,b 2 + γm,d xuχe/5,e + u Hsα . Lx Lx L2 x d=2 x

Considering a slightly modiﬁcation of the weights involved in (3.22) it is possible to follow the same arguments as above to estimate A4, however, in order to avoid repetitions we will omit the calculations. Further, after integrate in time and use the inductive hypothesis, this is (3.15), then

¸m } } ¦ 1/2 } } A3 1 γm,d c + u 8 sα . LT . d,0 LT Hx d=2

Similarly, combining properties of the weighted functions and the induction hypothesis (3.16) produces

¸m } } ¦ 1/2 } } A4 1 λm,d c + u 8 sα . LT . d,2k LT Hx d=2 1. PROOF OF THEOREMC 89

m+ α Finally, to estimate D 2 (uφ ) we decouple it by using Theorem 1.4 as follows x e,b 2 Lx ¸ m+ α m+ α 1 m¡d+ α 2 2 } } Bd 2 Dx (uφe,b) . Dx φe,b 8 u0 L2 + xφe,bDx u L2 L2 L L4 x L2 L2 T x T x P d! T x d Q1(m) ¸ 1 m¡d+ α Bd 2 + xφe,bHDx u L2 L2 P d! T x d Q2(m) ¸ 1 m¡d+ α } } 2 . u0 L2 + ϕe/8,b+e/4Dx u x L2 L2 P d! T x d Q1(m) ¸ ¡ α 1 m d+ 2 + ϕe/8,b+e/4HDx u L2 L2 P d! T x d Q2(m) ¸ 1 m¡d+ α } } 2 . u0 L2 + ηe/24,b+7e/24Dx u x L2 L2 P d! T x d Q1(m) ¸ ¡ α 1 m d+ 2 + ηe/24,b+7e/24HDx u L2 L2 P d! T x d Q2(m) ¸m } } 1 ¦ 1/2 . u0 L2 + cm¡d , x d! ,1 d=0 where the last inequality is a consequence of inductive hypothesis (3.15).

This step ﬁnishes gathering together the estimates corresponding to B1, B2 and B3 that combined with Gronwall’s inequality and integration in time α 2 ¦ Bm 2 ¤ } αBm }2 } αBm }2 sup x Dx uχe,b( + vt) + Dx x uηe,b 2 2 + HDx x uηe,b 2 2 cm , 2 LT Lx LT Lx . ,2 0¤t¤T Lx where as usual we indicate the full dependence of the parameters behind the constant i.e. α ¦ ¦ 2 m c = c α; e; T; v; }u } sα ; D B u χ ¡ 0. m,2 m,2 0 Hx x x 0 e,b 2 Lx

STEP 2k + 1 : The corresponding energy weighted estimate for this step is » » 1 d 1¡ α 2 v 1¡ α 2 1 Bm 2 2 ¡ Bm 2 2 x Dx u χe,b dx x Dx u χe,b dx 2 dt R looooooooooooooooooomooooooooooooooooooon2 R

B1(t) (3.23) » » 1¡ α 1¡ α 1¡ α 1¡ α ¡ Bm 2 αB Bm 2 2 Bm 2 B Bm 2 2 x Dx Dx xu x Dx uχe,b dx + x Dx (u xu) x Dx uχe,b dx = 0. looooooooooooooooooooooooomooooooooooooooooooooooooonR loooooooooooooooooooooooooomoooooooooooooooooooooooooonR

B2(t) B3(t)

§.1 We shall point out that the representation of B1 may change according to the case. More precisely, P (I) If m Q1 then » m+1¡ α 2 m+1¡ α 2 B (t) = ¡v D 2 u η2 dx = ¡v D 2 uη . 1 x e,b x e,b 2 R Lx 1. PROOF OF THEOREMC 90

(II) If m P Q2 then » m+1¡ α 2 m+1¡ α 2 B (t) = ¡v HD 2 u η2 dx = ¡v HD 2 uη . 1 x e,b x e,b 2 R Lx For the sake of simplicity we will study the case (I), it will be clear from the context how to proceed in the case (II). Since m+αk m+αk h m+αk i Dx (uηe,b) = Dx uηe,b + Dx ; ηe,b (uχe,b + uφe,b + uψe),

j+1 " j+1 # m+α 2 m+α 2 = HDx uηe,b + H Dx ; ηe,b (uχe,b + uφe,b + uψe)

j+1 m+α 2 + [H; ηe,b] Dx (uχe,b + uφe,b + uψe) , then combining (1.5), Lemma 1.10 and interpolation m+αk m+αk h m+αk i D (uη ) ¤ D uη + D ; η (uχ + uφ + uψe) x e,b 2 x e,b 2 x e,b e,b e,b 2 Lx Lx Lx m+1¡ α m¡ α m¡ α D 2 uη + D 2 (uχ ) + D 2 (uφ ) . x e,b 2 x e,b 2 x e,b 2 Lx Lx Lx ¡ α m+1 2 + η D (uψe) e,b x 2 Lx m+αk }Bm } }Bm } } } D uη b + (uχ b) 2 + (uφ b) 2 + u0 2 . . x e, 2 x e, Lx x e, Lx Lx Lx Since P (3.24) χe/5,e(x)χe,b(x) = χe,b(x) for all x R, then combining Lemma 1.9 and Young’s inequality give us ¸m }Bm } ¤ Bd Bm¡d x (uχe,b) 2 cm,d xuχe/5,e x χe,b Lx L2 d=0 x ¡ m¸1 }Bm } Bd Bm¡d = x uχe,b 2 + cm,d xuχe/5,e x χe,b Lx L2 d=0 x ¡ m¸1 ¤ }Bm } Bm¡d Bd x uχe,b 2 + cm,d x χe,b xuχe/5,e Lx L8 L2 d=0 x x ¡ m¸1 }Bm } Bd } } . x uχe,b 2 + γm,d xuχe/5,e + u Hsα . Lx L2 x d=2 x Hence, by (3.15)

m¸¡1 ¦ 1/2 ¦ 1/2 }Bm } } } 8 x uχe,b 8 2 cm + γm,d c + u L Hsα . LT Lx . ,1 d,1 T x d=0 and ¸m }Bm } Bd } } x (uφe,b) 2 . γm,d xuχe/20,e/4 + u Hsα . Lx L2 x d=2 x 1. PROOF OF THEOREMC 91

Analogously ¸m ¦ 1/2 }Bm } } } 8 (3.25) x (uφe,b) 8 2 γm,d c + u L Hsα . LT Lx . d,1 T x d=2 Gathering the estimated terms above allow us to obtain Dm+αk(uη ) 8. x e,b 2 2 LT Lx Therefore, by interpolation 1¡ α BmD 2 (uη ) }u } + Dm+αk(uη ) . x x e,b 2 2 . 0 2 x e,b 2 2 LT Lx LT Lx With this information at hand, we can estimate the term with the regularity required, this is achieved localizing the commutator expression as follows

1¡ α 1¡ α h 1¡ α i Bm 2 Bm 2 ¡ Bm 2 x Dx uηe,b = x Dx (uηe,b) x Dx ; ηe,b (uχe,b + uφe,b + uψe) , that by using a similar argument as before we obtain 1¡ α BmD 2 uη 8. x x e,b 2 2 LT Lx So that, » » T T 1¡ α 2 |B (t)|dt = v BmD 2 uη dt 8. 1 x x e,b 2 0 0 Lx §.2 As was evidenced at the beginning of this step, several case shall be considered. In fact, the term B2 is represented in different ways according be the case P (I) If m Q1, then » 1 m+1¡ α h i m+1¡ α ¡ 2 α+1 2 2 B2(t) = Dx u HDx ; χe,b Dx u dx. 2 R

(II) If m P Q2 then » 1 m+1¡ α h i m+1¡ α ¡ 2 α+1 2 2 B2(t) = HDx u HDx ; χe,b HDx u dx. 2 R To show how to proceed in the case that m is odd, we will use the expression above and the commutator decomposition as we have previously described to obtain » » ¡ α ¡ α ¡ α ¡ α 1 m+1 2 m+1 2 1 m+1 2 m+1 2 B2(t) = HDx uRn(α + 1)HDx u x + HDx uPn(α + 1)HDx u dx 2 R» 4 R ¡ α ¡ α 1 m+1 2 m+1 2 ¡ HDx uHPn(α + 1)HDx Hu 4 R

= B2,1(t) + B2,2(t) + B2,3(t). By choosing n according to the rule α 2n + 1 ¤ α + 1 + 2 m + 1 ¡ ¤ 2n + 3, 2 which clearly implies n = m. 1. PROOF OF THEOREMC 92

Next, an application of Proposition 1.11 implies that the remainder term Rm(α + 1) is bounded 2¡ in the L norm; moreover it allows us to handle B2,1 as follows x » ¡ α ¡ α 1 m+1 2 m+1 2 B2,1(t) = HDx uRm(α + 1)Dx Hu dx 2 »R ¡ α ¡ α 1 m+1 2 m+1 2 = HuDx Rm(α + 1)Dx Hu dx. 2 R Combining Holder’s¨ inequality and ¡ α ¡ α m+1 2 m+1 2 |B (t)| }Hu(t)} 2 D Rm(α + 1)D Hu 2,1 . Lx x x 2 Lx { } }2 2m+3 2 u0 2 Dx (χ ) . . Lx e,b 1 Lξ From this it is easy to obtain that » T | | ¤ B2,1(t) dt c. 0 The control in B2,1 allow us to ﬁx the number of terms in B2,2 and B2,3. » ¸m d α + 1 c (¡1) ¡ 2 ( ) = 2d+1 m+1 d ( 2 )(2d+1) B2,2 t d Dx u χe,b dx 4 4 R »d=0 2 1 α + 1 m+1 2 = Dx u χe,b dx 4 R » m¸¡1 d α + 1 c (¡1) ¡ 2 (2d+1) + 2d+1 m+1 d 2 (3.26) d Dx u χe,b dx 4 4 R d=1 » α + 1 c (2m+1) ¡ 2m+1 ( )2 2 m Dxu χe,b dx 4 4 R m¸¡1 = B2,2,1(t) + B2,2,d(t) + B2,2,m+1(t), d=2 and » ¸m d α + 1 c (¡1) ¡ 2 ( ) = 2d+1 H m+1 d ( 2 )(2d+1) B2,3 t d Dx u χe,b dx 4 4 R »d=0 2 1 α + 1 m+1 2 = HDx u χe,b dx 4 R » m¸¡1 d α + 1 c (¡1) ¡ 2 (2d+1) + 2d+1 H m+1 d 2 (3.27) d Dx u χe,b dx 4 4 R d=1 » α + 1 c (2m+1) ¡ 2m+1 (H )2 2 m Dxu χe,b dx 4 4 R m¸¡1 = B2,3,1(t) + B2,3,d(t) + B2,3,m+1(t). d=2

Concerning the terms B2,2,1 and B2,3,1, after integrate in time they grant the smoothing effect. The remainders terms in (3.26) and (3.27) have to be estimated separately. 1. PROOF OF THEOREMC 93

In this sense, for d P t1, 2, ¤ ¤ ¤ , m ¡ 1u » » » T T 2 | | l m+1¡d 1 P t u B2,l+1,d(t) dt . H Dx u χe/3,b+e dx dt for l 0, 1 . 0 0 R Then a combination of (3.15) and properties of the weighted functions imply that » T | | ¦ P t u ¤ ¤ ¤ ¡ B2,l+1,d(t) dt . cm+1¡d,2k+1 for any l 0, 1 , d = 2, 3, , m 2. 0 The cases not considered in the estimation above are handled by means of the local theory.

§.3 The term B2, can be rewritten according be the case: P (I) If m Q1, then there exists a non-null constant c = c(m) such that 1¡ α h m+1¡ α i Bm 2 B 2 B 2 2 2 x Dx (u xu)χe,b = cm Dx ; χe,b x (uχe,b) + (uφe,b) + u ψe h m+1¡ α i 2 B + cm Dx ; uχe,b x ((uχe,b) + (uφe,b) + (uψe))

1¡ α Bm 2 B + uχe,b x Dx xu.

(II) If m P Q2 then there exists a non-null constant c = c(m) such that 1¡ α h m+1¡ α i 2 Bm 2 B 2 B 2 2 x Dx (u xu)χe,b = cmH Dx ; χe,b x (uχe,b) + uφe,b + u ψe m+1¡ α 2 2 B 2 2 + cm [H; χe,b] Dx x (uχe,b) + uφe,b + u ψe

h m+1¡ α i 2 B + cmH Dx ; uχe,b x ((uχe,b) + (uφe,b) + (uψe))

m+1¡ α 2 B + cm [H; uχe,b] Dx x ((uχe,b) + (uφe,b) + (uψe)) 1¡ α Bm 2 B + uχe,b x Dx xu = B3,1(t) + B3,2(t) + B3,3(t) + B3,4(t) + B3,5(t) + B3,6(t) + B3,7(t) + B3,8(t) + B3,9(t) + B3,10(t) + B3,11(t) + B3,12(t) + B3,13(t). To show how to proceed in the case m odd we combine Lemma 1.2 and Lemma 1.8 to obtain h ¡ α i m+1 2 2 }B } 2 = |cm| H D ; χ Bx (uχ ) 3,1 Lx x e,b e,b 2 Lx h ¡ α i m+1 2 2 D ; χ Bx (uχ ) . x e,b e,b 2 Lx (3.28) }B } 2 . xχe,b Hl (uχe,b) m+1¡ α x H˙ 2 x ¡ α m+1 2 }u} 8 }u } 2 + D (uχ ) , . Lx 0 Lx x e,b 2 Lx and h ¡ α i 2 m+1 2 }B } 2 = |c | H D ; χ B uφ 3,2 Lx m x e,b x e,b L2 (3.29) x ¡ α m+1 2 }u} 8 }u } 2 + D (uφ ) . . Lx 0 Lx x e,b 2 Lx 1. PROOF OF THEOREMC 94

Since the weighted functions χe,b and ψe satisfy the hypothesis of Lemma 1.10 e dist (supp(χ ), supp(ψ )) ¥ ¡ 0, e,b e 2 then h ¡ α i m+1 2 2 }B } 2 = |cm| H D ; χ Bx u ψe 3,3l Lx x e,b 2 Lx h ¡ α i m+1 2 2 D ; χ Bx u ψe . x e,b 2 Lx ¡ α m+1 2 2 = c χ D u ψe e,b x 2 Lx

}u} 8 }u } 2 for l = 1, 2, 3, 4. . Lx 0 Lx On the other hand, the Calderon’s commutator estimate (1.1) and Lemma 1.2 lead to ¡ α m 1 2 2 }B } 2 = |cm| [H; χ ] B D H (uχ ) 3,4 Lx e,b x x e,b 2 Lx 1¡ α }Bm } 2 2 χ b 8 HDx (uχ b) . x e, Lx e, 2 Lx ¡ α 1 2 D (uχ ) }uχ }L8 . x e,b 2 e,b x Lx

}u} 1 }u} 8 , . Hx Lx and ¡ α 2 m 1 2 }B } 2 = |c | [H; χ ] B D H uφ 3,5 Lx m e,b x x e,b L2 x αj Bm+1 H 2 2 . x χe,b 8 Dx (uχe,b) Lx L2 x αj 2 ( ) . Dx uφe,b uφe,b 8 2 Lx Lx

}u} 1 }u} 8 . . Hx Lx Moreover, by the commutator estimate (1.5) we have ¡ α m+1 2 }B } 2 [H; uχ ] D Bx ((uχ )) 3,7 Lx . e,b x e,b 2 Lx (3.30) ¡ α m+1 2 }Bx(uχ )}L8 D (uχ ) . e,b x x e,b 2 Lx and " + # m+1¡α j 1 2 }B } 2 = |c | H D ; uχ B (uφ ) 3,8 Lx m x e,b x e,b 2 Lx " + # m+1¡α j 1 (3.31) 2 B . Dx ; uχe,b x(uφe,b) 2 Lx ¡ α ¡ α m+1 2 m+1 2 D (uφ ) }Bx(uχ )}L8 + D (uχ ) }Bx(uφ )}L8 . . x e,b 2 e,b x x e,b 2 e,b x Lx Lx 1. PROOF OF THEOREMC 95

Furthermore, an application of (1.5) yields

αj m+ 2 }B } 2 = |c | H D ; uχ B (uχ ) 3,10 Lx m x e,b x e,b 2 Lx h ¡ α i m+1 2 (3.32) D ; uχ Bx(uχ ) . x e,b e,b 2 Lx ¡ α m+1 2 }Bx(uχ )}L8 D (uχ ) , . e,b x x e,b 2 Lx and

αj m+ 2 }B } 2 = |c | H D ; uχ B (uχ ) 3,11 Lx m x e,b x e,b 2 Lx h ¡ α i m+1 2 (3.33) D ; uχ Bx(uχ ) . x e,b e,b 2 Lx ¡ α m+1 2 }Bx(uχ )}L8 D (uφ ) . . e,b x x e,b 2 Lx

Notice that in the inequalities (3.28)-(3.29) and (3.30)-(3.33) there are several terms which have been not estimated yet. Firstly,

m+1¡ α 1¡ α h m+1¡ α i 2 Bm 2 2 Dx (uχe,b) = cm x HDx uχe,b + Dx ; χe,b (uχe,b + uφe,b + uψe) 1¡ α 1¡ α h m+1¡ α i Bm 2 ¡ Bm 2 2 = H χe,b x Dx u [H; χe,b] x Dx u + Dx ; χe,b (uχe,b + uφe,b + uψe) ,

where cm is a non-null constant. Since the arguments to estimate the expression on the right have been previously used, we will only indicate the tools used. Combining the Calderon’s commutator estimate (1.1), (1.5) and interpolation imply that

¡ α ¡ α m+1 2 m 1 2 m m D (uχ ) B D uχ + }u} sα + }B (uχ ) } 2 + }B (uφ ) } 2 . x e,b 2 . x x e,b 2 Hx x e,b Lx x e,b Lx Lx Lx

Notice that the ﬁrst term on the right hand side is the quantity to be estimated. The third and fourth term are handled by using similar arguments as those described in (3.24)-(3.25). Next, the condition

2 1 α α P , ñ m + 1 ¡ ¤ m + αk 2k + 1 k 2 1. PROOF OF THEOREMC 96 is used to obtain information from the previous cases. This is achieved combining Theorem 1.4 and (3.15), that is, ¸ m+αk m+αk } } 1 Bd m+αk¡d Dx (uφe,b) . Dx φe,b 8 u0 L2 + xφe,bDx u L2 L2 L L4 x d! L2 L2 T x T x dPQ (m) T x 1 ¸ 1 Bd m+αk¡d + xφe,bHDx u d! L2 L2 dPQ (m) T x 2 ¸ } } 1 m+αk¡d . u0 L2 + ϕe/8,b+e/4Dx u x d! L2 L2 dPQ (m) T x 1 ¸ 1 m+αk¡d + ϕe/8,b+e/4HDx u d! L2 L2 dPQ (m) T x 2 ¸ } } 1 m+αk¡d . u0 L2 + ηe/24,b+7e/24Dx u x d! L2 L2 dPQ (m) T x 1 ¸ 1 m+αk¡d + ηe/24,b+7e/24HDx u L2 L2 P d! T x d Q2(m) ¸m } } 1 ¦ 1/2 . u0 L2 + cm+αk¡d k¡ . x d! ,2 1 d=0

So that interpolation provide the bound ¡ α m 1 2 m+αk B D (uφ ) }u } 2 + D (uφ ) . x x e,b 2 2 . 0 Lx x e,b 2 2 LT Lx LT Lx 1¡ α Analogously is estimated BmD 2 uφ . x x e,b 2 2 L Lx T Inserting B3,13 into (3.23) it is obtained a term which can be estimated by integration by parts, 1 8 Gronwall’s inequality and the Strichartz’s estimate in Theorem 0.4, i.e Bxu P L ([0, T] : L (R)) . Finally, gathering together the estimates corresponding to this step combined with Gronwall’s inequality and integration in time, we obtain the desired estimate, this is, for any e ¡ 0, b ¥ 5e and v ¥ 0, 1¡ α 2 2 2 ¦ sup BmD 2 uχ (¤ + vt) + Bm+1uη + HBm+1uη c , x x e,b 2 x e,b 2 2 x e,b 2 2 . m,2k+1 0¤t¤T Lx LT Lx LT Lx ¡ α ¦ ¦ 1 2 m where c = c α; k; e; T; v; }u } sα ; D B u χ ¡ 0. m,2k+1 m,2k+1 0 Hx x x 0 e,b 2 Lx This estimate ﬁnish the 2-induction process. Throughout the proof we have always assumed as much regularity as possible on the function u, nevertheless the way to proceed in the general case involves the device of regularization of Bona, Smith [4]. + µ P sα More precisely, we consider initial data u0 H (R), then we regularize the initial data u0 = ¦ P 8 ρµ u0, where ρ is a positive smooth function with compact support, more precisely ρ C0 (R) ¡ } } with supp(ρ) ( 1, 1) and ρ 1 = 1. 1. PROOF OF THEOREMC 97

For µ ¡ 0 we deﬁne the family ¡ x ρ (x) = µ 1ρ x P R. µ µ µ µ The solution u associated to the IVP (0.9) with initial data u0 satisﬁes 8 uµ P C ([0, T] : H (R)) . Applying the results obtained in the section of the 2-inductive process to the function uµ allows us conclude that, for any e ¡ 0, b ¥ 5e, v ¥ 0, the following holds: (a) for l = 2, 3, ¤ ¤ ¤ , m and j = 0, 1, ¤ ¤ ¤ , 2k ¡ 1 2 2 2 j+1 j+1 αj α α ¦ Bl 2 µ ¤ Bl 2 µ Bl 2 µ ¤ sup xDx u χe,b( + vt) + xDx u ηe,b + xHDx u ηe,b cl,j ¤ ¤ 2 0 t T Lx 2 2 2 2 LT Lx LT Lx ! αj ¦ ¦ µ l 2 µ = s B ¡ where cl,j cl,j α; k; e; T; v; l; j; u0 H α ; xDx u0 χe,b 0; x 2 Lx (b) for l = 2, 3, ¤ ¤ ¤ , m 1¡ α 2 2 ¦ sup Bl D 2 uµ(t)χ (¤ + vt) + Bl+1uµη + HBl+1uµη ¤ c , x x e,b x e,b 2 2 x e,b 2 2 l,2k+1 0¤t¤T 2 LT Lx LT Lx ¦ ¦ 1¡ α µ Bn 2 B2 µ ¡ where c = c α; k; e; T; v; n; u sα ; Dx u χ b 0. l,2k+1 l,2k+1 0 Hx x x 0 e, 2 Lx ¦ ¦ Next, we prove that the constants cl,j and cl,2k+1 are independent of the parameter µ. First notice that µ ¤ } } x ¤ } } u ˙ sα u0 ˙ sα ρµ 8 u0 sα . 0 Hx Hx Lx Hx ¤ P Next, the weighted function χe,b(x) = 0 for x e, therefore when we restrict µ (0, e) it follows by Young’s inequality that » » 8 αj 2 8 αj 2 Bl 2 µ ¦ Bl 2 xDx u0 dx = ρµ xDx u01[0,8) dx e e αj ¤ } } Bl 2 ρµ 1 xDx u0 2 8 L ((0, )) αj Bl 2 = xDx u0 , L2((0,8)) l = 2, 3, ¤ ¤ ¤ , m and j = 0, 1, ¤ ¤ ¤ , 2k ¡ 1. Similarly, it can be proved that for µ P (0, e) the following inequality holds 1¡ α 2 1¡ α Bl 2 B2 µ ¤ Bl 2 B2 ¤ ¤ ¤ Dx u χ b Dx u0 for l = 2, 3, , m. x x 0 e, 2 x x 2 8 Lx L ((0, )) Also, as part of the argument, the continuous dependence upon the initial data is used, this is

µ sup }u (t) ¡ u(t)} sα ÝÑ 0. Hx Ñ 0¤t¤T µ 0 ¦ The proof ﬁnish combining the remarks underlined above, the independence of the constants cl,j ¦ and cl,2k+1, of the parameter µ, these joint with a weak compactness argument and Fatou’s Lemma allow to conclude the proof. 1. PROOF OF THEOREMC 98

l A consequence of the TheoremC is the following corollary, that describe the asymptotic behavior of the function in (2.81).

2¡ α COROLLARY 3.3. Let u P C [¡T, T] : H 2 (R) be a solution of the equation in (0.9) described by TheoremC. Then, for any t P (0, T] and δ ¡ 0 » 8 1 j 2 c Bxu (x, t) dx ¤ , ¡8 xx¡yj+δ t where x¡ = maxt0, ¡xu and c is a positive constant.

PROOF. The proof follows the same ideas that those used in the proof of corollary 2.21. Bibliography

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