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

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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 con- tinuous 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 propagac¸ao˜ de regularidade de soluc¸oes˜ associadas a modelos dispersivos nao˜ locais. O primeiro a ser considerado corresponde a equac¸ao˜ dispersiva generalizada Benjamin-Ono (DGBO), que pode ser observada como a interpolac¸ao˜ dis- persiva entre a equac¸ao˜ de Benjamin-Ono (BO) e a equac¸ao˜ de Korteweg-de Vries (KdV). Para a segunda analise,´ trataremos da equac¸ao˜ fracionaria´ de KdV, a qual, diferente da DGBO, pode ser interpretado como uma perturbac¸ao˜ dispersiva da equac¸ao˜ de Burger. Recentemente, foi mostrado que as soluoes˜ das equac¸oes˜ 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 equac¸oes,˜ a DGBO e a fKdV, sao˜ mais desafiadoras que as apresentadas para as equac¸oes˜ BO e KdV. No caso da DGBO, nos,´ primeiramente, provamos um resultado de boa colocac¸ao˜ local associado ao problema de valor inicial, utilizando um argumento de compacidade. Na segunda parte da tese, nos´ estudamos a propagac¸ao˜ de regularidade nas soluc¸oes˜ da DGBO. A fim de lidar com esse problema, introduzimos uma nova abordagem. De fato, a abordagem utilizada para lidar com estas equac¸oes˜ 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 propagac¸ao˜ desejadas. Finalmente, a tecnica´ usada na propagac¸ao˜ de regularidade nas soluc¸oes˜ da DGBO permite-nos verificar que as soluc¸oes˜ da fKdV tambem´ satisfazem tal fenomenoˆ de propagac¸ao˜ da regularidade. Nessa situac¸ao,˜ 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 ¡ Da+1B u + uB u = 0, x, t P R, 0 a 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 x f (x) , 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(x) f (x)) (x), Ñ + p 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.
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