ON the REGULARITY of SOLUTIONS to the K-GENERALIZED KORTEWEG-DE VRIES EQUATION

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 146, Number 9, September 2018, Pages 3759–3766 http://dx.doi.org/10.1090/proc/13506 Article electronically published on June 11, 2018

ON THE REGULARITY OF SOLUTIONS TO THE k-GENERALIZED KORTEWEG-DE VRIES EQUATION

C. E. KENIG, F. LINARES, G. PONCE, AND L. VEGA

(Communicated by Catherine Sulem)

Abstract. This work is concerned with special regularity properties of solutions to the k-generalized Korteweg-de Vries equation. In [Comm. Partial Dif- ferential Equations 40 (2015), 1336–1364] it was established that if the initial l + datum is u0 ∈ H ((b, ∞)) for some l ∈ Z and b ∈ R, then the corresponding solution u(·,t) belongs to Hl((β,∞)) for any β ∈ R and any t ∈ (0,T). Our goal here is to extend this result to the case where l>3/4.

1. Introduction In this note we study the regularity of solutions to the initial value problem (IVP) associated to the k-generalized Korteweg-de Vries equation ∂ u + ∂3u + uk∂ u =0,x,t∈ R,k∈ Z+, (1.1) t x x u(x, 0) = u0(x). The starting point is a property found by Isaza, Linares, and Ponce [4] concerning the propagation of smoothness in solutions of the IVP (1.1). To state it we ﬁrst recall the following well-posedness (WP) result for the IVP (1.1): 3/4+ R Theorem A1. If u0 ∈ H ( ), then there exist T = T (u0 + ; k) > 0 and a 3 2 4 , unique solution u = u(x, t) of the IVP (1.1) such that + (i) u ∈ C([−T,T]:H3/4 (R)), 4 ∞ (ii) ∂xu ∈ L ([−T,T]:L (R)) (Strichartz), T (1.2) r 2 + (iii) sup |J ∂xu(x, t)| dt < ∞ for r ∈ [0, 3/4 ], x −T ∞ (iv) sup |u(x, t)|2 dx < ∞, −∞ −T ≤t≤T 2 1/2 with J =(1− ∂x) . Moreover, the map data-solution u0 → u(x, t) is locally 3/4+ R 3/4+ continuous (smooth) from H ( ) into the class XT deﬁned in (1.2).

Received by the editors April 9, 2016, and, in revised form, April 29, 2016 and September 25, 2016. 2010 Mathematics Subject Classiﬁcation. Primary 35Q53; Secondary 35B05. Key words and phrases. Nonlinear dispersive equation, propagation of regularity. The ﬁrst author was supported by the NSF grant DMS-1265249. The second author was supported by CNPq and FAPERJ/Brazil. The fourth author was supported by ERCEA Advanced Grant 2014 669689 - HADE and by the MINECO project MTM2014-53850-P.

c 2018 American Mathematical Society 3759

Licensed to Univ of Chicago. Prepared on Mon Aug 5 17:38:22 EDT 2019 for download from IP 205.208.116.24. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3760 C. E. KENIG, F. LINARES, G. PONCE, AND L. VEGA

If k ≥ 2, then the result holds in H3/4(R).Ifk =1, 2, 3,thenT can be taken arbitrarily large.

For the proof of Theorem A1 we refer to [6], [1], and [3]. The proof of our main result, Theorem 1.1, is based on an energy estimate argument for which the estimate (ii) in (1.2) (i.e., the time integrability of ∂xu(·,t)∞) is essential. However, we remark that from the WP point of view it is not optimal. For a detailed discussion on the WP of the IVP (1.1) we refer to [7, Chapters 7-8]. Now we enunciate the result obtained in [4] regarding propagation of regularities which motivates our study here:

3/4+ + Theorem A2 ([4]). Let u0 ∈ H (R).Ifforsomel ∈ Z and for some x0 ∈ R, ∞ l 2 l 2 2 (1.3) ∂xu0L ((x0,∞)) = |∂xu0(x)| dx < ∞, x0 then the solution u = u(x, t) of the IVP (1.1) provided by Theorem A1 satisﬁes that for any v>0 and ǫ>0, ∞ j 2 (1.4) sup (∂xu) (x, t) dx < c, 0≤t≤T x0+ǫ−vt l + 2 for j =0, 1,...,l with c = c(l; u03/4 ,2; ∂xu0L ((x0,∞)); v; ǫ; T ). In particular, for all t ∈ (0,T],therestrictionofu(·,t) to any interval of the form (a, ∞) belongs to Hl((a, ∞)). Moreover, for any v ≥ 0, ǫ>0,andR>0,

T x0+R−vt l+1 2 (1.5) (∂x u) (x, t) dxdt < c, 0 x0+ǫ−vt l with c = c(l; u ; ∂ u 2 ; v; ǫ; R; T ). 0 3/4+,2 x 0 L ((x0,∞))

Theorem A2 tells us that the Hl-regularity (l ∈ Z+) on the right hand side of the data travels forward in time with inﬁnite speed. Notice that since the equation is reversible in time a gain of regularity in Hs(R) cannot occur at t>0, so u(·,t) fails to be in Hl(R) due to its decay at −∞. In this regard, it was also shown in [4] that for any δ>0andt ∈ (0,T)andj =1,...,l, ∞ 1 c (∂j u)2(x, t) dx ≤ , x j+δ x t −∞ − j + 2 with c = c(u03/4 ,2; ∂xu0L ((x0,∞)); x0; δ), x− =max{0; −x},andx = (1 + x2)1/2. The aim of this note is to extend Theorem A2 to the case where the local regularity of the datum u0 in (1.3) is measure with a fractional exponent. Thus, our main result is:

3/4+ Theorem 1.1. Let u0 ∈ H (R).Ifforsomes ∈ R,s>3/4, and for some x0 ∈ R, ∞ s 2 s 2 2 (1.6) J u0L ((x0,∞)) = |J u0(x)| dx < ∞, x0

then the solution u = u(x, t) of the IVP (1.1) provided by Theorem A1 satisﬁes that for any v>0,andǫ>0, ∞ (1.7) sup (J ru)2(x, t) dx < c, 0≤t≤T x0+ǫ−vt r + 2 for r ∈ (3/4,s] with c = c(l; u03/4 ,2; J u0L ((x0,∞)); v; ǫ; T ). Moreover, for any v ≥ 0, ǫ>0,andR>0,

T x0+R−vt (1.8) (J s+1u)2(x, t) dxdt < c, 0 x0+ǫ−vt s with c = c(l; u ; J u 2 ; v; ǫ; R; T ). 0 3/4+,2 0 L ((x0,∞)) From the results in section 2 it will be clear that we need only consider the case s ∈ (3/4, ∞) − Z+. The rest of this paper is organized as follows: section 2 contains some preliminary estimates required for Theorem 1.1, whose proof will be given in section 3.

2. Preliminary estimates

Let Ta be a pseudo-diﬀerential operator with the symbol r (2.1) σ(Ta)=a(x, ξ) ∈ S ,r∈ R, so that

2πix·ξ (2.2) Taf(x)= a(x, ξ)f(ξ) e dξ. Rn The following result is the singular integral realization of a pseudo-diﬀerential operator, whose proof can be found in [8, Chapter 4]. Theorem A3. Using the above notation (2.1)-(2.2) one has that

(2.3) Taf(x)= k(x, x − y)f(y)dy, if x/∈ supp(f), Rn where k ∈ C∞(Rn × Rn −{0}) satisﬁes : ∀ α, β ∈ (Z+)n ∀ N ≥ 0, α β −(n+m+|β|+N) (2.4) |∂x ∂z k(x, z)|≤Aα,β,N,δ |z| , |z|≥δ, if n + m + |β| + N>0 uniformly in x ∈ Rn. To simplify the exposition we restrict ourselves to the one-dimensional case x ∈ R, where in the next section these results will be applied. As a direct consequence of Theorem A3 one has Corollary 2.1. Let m ∈ Z+ and l ∈ R.Ifg ∈ L2(R) and f ∈ Lp(R), p ∈ [2, ∞], with distance(supp(f); supp(g)) ≥ δ>0, then m l (2.5) f∂x J g2 ≤ cfpg2. ∞ R ′ ∞ R Next, let θj ∈ C ( ) −{0} with θj ∈ C0 ( )for j =1, 2and

(2.6) distance(supp(1 − θ1); supp(θ2)) ≥ δ>0.

s Lemma 2.2. Let f ∈ H (R),s<0, and Ta be a pseudo-diﬀerential operator of 0 2 order zero (a ∈ S ).Ifθ1f ∈ L (R),then 2 (2.7) θ2Taf ∈ L (R). Proof of Lemma 2.2. Since

θ2 Taf = θ2Ta(θ1f)+θ2Ta((1 − θ1)f), 2 combining the hypothesis and the continuity of Ta in L (R) it follows that θ2Ta(θ1f) ∈ L2(R). Also

θ2(x) Ta((1 − θ1)f)(x) ∞ 2πixξ = θ2(x)a(x, ξ)((1 − θ1)f)(ξ)e dξ −∞ 2πixξ1 2πixξ2 = ( θ2(x)a(x, ξ1 + ξ2)(1 − θ1)(ξ1)e dξ1)f(ξ2)e dξ2 (2.8) =:b(x,ξ) = Tbf(x)= θ2(x)k(x, x − z)(1 − θ1(z))f(z)dz 2m −2m = θ2(x)k(x, x − z)(1 − θ1(z))J J f(z)dz with −2ma}) s>0. Then for any ǫ>0 and any r ∈ (0,s], (2.9) J rf ∈ L2({x>a+ ǫ}). Proof of Proposition 2.3. Deﬁne g = J sf ∈ L2({x>a}); s −s ∞ thus J f ∈ H (R). Let θj ∈ C (R),j=1, 2, with θ1(x)=1forx ≥ a + ǫ/4, supp θ1 ⊂{x>a},and θ2(x)=1forx ≥ a + ǫ and supp θ2 ⊂{x>a+ ǫ/2}; 2 iβ therefore θ1g ∈ L (R). Let T = J ,β∈ R. By Lemma 2.2 s+iβ 2 θ2Tg = θ2J f ∈ L (R), and since f ∈ L2(R), iβ 2 θ2J f ∈ L (R). Hence, by the Three Lines Theorem it follows that z 2 θ2J f ∈ L (R),z= r + iβ, r ∈ [0,s],β∈ R, which completes the proof.

∞ Remark 2.4. In a similar manner one has: for ǫ>0letϕǫ ∈ C (R) with ϕǫ(x)= ′ 1,x≥ ǫ, supp ϕǫ ⊂{x>ǫ/2},and ϕǫ(x) ≥ 0. Then: + m 2 ′ (I) If m ∈ Z and ϕǫJ f ∈ L (R), then ∀ ǫ > 2ǫ,

′ j 2 R ϕǫ ∂xf ∈ L ( ),j=0, 1,...,m.

Z+ j 2 R ′ (II) If m ∈ and ϕǫ∂xf ∈ L ( ),j=0, 1,...,m,then ∀ ǫ > 2ǫ, m 2 ϕǫ′ J f ∈ L (R). s 2 ′ (III) If s>0andJ (ϕǫ f), f ∈ L (R), then ∀ ǫ > 2ǫ, s 2 ϕǫ′ J f ∈ L (R). s 2 ′ (IV) If s>0andϕǫJ f, f ∈ L (R), then ∀ ǫ > 2ǫ, s 2 J (ϕǫ′ f) ∈ L (R).

The same results hold with θ1,θ2, as in (2.6), instead of χǫ,χǫ′ . Next, we recall some inequalities obtained in [5]. Theorem A4 ([5]). If s>0 and p ∈ (1, ∞),then s s s (2.10) J (fg)p ≤ c(f∞J gp + J fpg∞) and s s s [J ; f]gp = J (fg) − fJ gp (2.11) s−1 s ≤ c(∂f∞J gp + J fpg∞). Also we shall use the following elementary estimate whose proof is similar to that found in [2, Chapter 6]. ∞ R ′ ∞ R R Lemma 2.5. Let φ ∈ C ( ) with φ ∈ C0 ( ).Ifs ∈ , then for any l> |s − 1| +1/2, s s−1 l ′ s−1 (2.12) [J ; φ]f2 + [J ; φ]∂xf2 ≤ c J φ 2 J f2.

3. Proof of Theorem 1.1

Without loss of generality x0 =0.Forǫ>0andb ≥ 5ǫ deﬁne the families of functions ∞ R χǫ,b ,φǫ,b, φǫ,b,ψǫ ∈ C ( ), ′ with χǫ,b ≥ 0, χǫ,b (x)=0,x≤ ǫ, χǫ,b (x)=1,x≥ b :

1 χ′ (x) ≥ 1 (x), ǫ,b 10(b − ǫ) [2ǫ,b−2ǫ]

supp(φǫ,b), supp(φǫ,b) ⊂ [ǫ/4,b],

φǫ,b(x)=φǫ,b(x)=1,x∈ [ǫ/2,ǫ], (3.1)

supp(ψǫ) ⊂ (−∞,ǫ/2],

χǫ,b(x)+φǫ,b(x)+ψǫ(x)=1,x∈ R,

2 2 R χǫ,b(x)+φǫ,b (x)+ψǫ(x)=1,x∈ . Hence,

distance(supp(χǫ,b ); supp(ψǫ)) ≥ ǫ/2.

Formally, we apply the operator J s to the equation in (1.1) and multiply by s 2 J uχǫ (x + vt) to obtain after integration by parts the identity 1 d (J su)2(x, t)χ2(x + vt) dx 2 dt

− v (J su)2(x, t)χχ′(x + vt) dx A1

3 s 2 ′ + (∂x J u) (x, t)χχ(x + vt) dx (3.2) 2

1 − (J su)2(x, t)∂3(χ2(x + vt)) dx 2 x A2

s s 2 + J (u∂xu) J u(x, t)χ (x + vt) dx =0, A3 where in χ the indexes ǫ, b were omitted. We shall do that from now on. Case. s ∈ (3/4, 1). First, we observe that combining (1.2) and the results in section 2 yields that for any R>0, T R (3.3) |J ru(x, t)|2 dxdt < ∞∀r ∈ [0, 7/4+]. 0 −R Thus, after integration in time the terms A1 and A2 in (3.2) are bounded. So it only remains to handle A3. Thus, s s s J (u∂xu)χ = J (u∂xuχ) − [J ; χ](u∂xu) s s s = uχJ ∂xu +[J ; uχ]∂xu − [J ; χ](u∂xu) (3.4) s s s = uχJ ∂xu +[J ; uχ]∂x(u(χ + φ + ψ)) − [J ; χ](u∂xu)

= B1 + B2 + B3 + B4 + B5.

Inserting B1 in (3.2) one obtains a term which can be estimated by integration by parts, Gronwall’s inequality, and (1.2). Using (2.11) it follows that s (3.5) B22 ≤ c∂x(uχ)∞J (uχ)2 and s s (3.6) B32 ≤ c(∂x(uχ)∞J (uφ)2 + ∂x(uφ)∞J (uχ)2).

To bound B4 and B5 we apply Corollary 2.1 and (2.12), respectively, to get s (3.7) B42 = uχJ ∂x(uψ)2 ≤ cu∞u2 and

(3.8) B52 ≤ c∂xu∞u2.

Collecting the above information (3.4)-(3.8) in (3.2) we obtain (1.7) for any r ∈ (3/4, 1),v>0, and ǫ>0, and that for any v>0,ǫ>0, T R−vt s 2 (J ∂xu) dxdt < ∞, 0 ǫ−vt from which using the results and Remark 2.4, one obtains (1.8). Case. s ∈ (m, m +1),m∈ Z+. We assume (1.7) and (1.8) with s ≤ m. Hence, from the results in section 2 it follows that for any ǫ>0,R>0, and r ∈ [0,m], T R−vt r 2 (3.9) (J ∂xu) dxdt < ∞. 0 ǫ−vt Again the starting point is the energy estimate identity (3.2). After integrating in time, the terms A1 and A2 can be easily bounded using (3.9). So it suﬃces to consider A3. Thus, using the notation introduced in (3.1) we have 1 χJ s(u∂ u)=J s(uχ∂ u) − [J s; χ]∂ (u2) x x 2 x 1 = uχJ s∂ u +[J s; uχ]∂ u − [J s; χ]∂ (u2) x x 2 x (3.10) s s = uχJ ∂xu +[J ; uχ]∂x(u(χ + φ + ψ)) 1 − [J s; χ]∂ ((u2)(χ2 +(φ)2 + ψ)) 2 x = E + E + E + E + E + E + E . 1 2 3 4 5 6 7 Inserting E1 in (3.2) one obtains a term which can be estimated by integration by parts, Gronwall’s inequality, and (1.2). From (2.11) we see that s (3.11) E22 ≤ c∂x(uχ)∞J (uχ)2 and s s (3.12) E32 ≤ c(∂x(uχ)∞J (uφ)2 + ∂x(uφ)∞J (uχ)2).

For E4 it follows from Corollary 2.1 that s (3.13) E42 = uχJ ∂x(uψ)2 ≤ cu∞u2.

For E5 and E6 a combination of (2.10) and (2.12) yields the estimates s 2 E52 ≤[J ; χ]∂x((uχ) )2 (3.14) s 2 s ≤ cJ ((uχ) )2 ≤ cu∞J (uχ)2 and s 2 s 2 E62 ≤[J ; χ]∂x((uφ) )2≤J ((uφ) )2 (3.15) ≤ cu J s(uφ) . ∞ 2 Finally, using Corollary 2.1 we write

s 2 s 2 (3.16) E72 ≤[J ; χ]∂x(u ψ)2 = χJ ∂x(u ψ)2 ≤ cu∞u2. To complete the estimates in (3.11), (3.12), (3.14), and (3.15) we observe that s s s J (uχ)=J uχ +[J ; χ](u(χ + φ + ψ)) = G1 + G2,

2 where G1 is the term whose L -norm we are estimating and G2 is of lower order 2 (hence bounded by assumption), and J (uφ)2 is bounded by (1.8) (assumption). Collecting the above information in (3.2) we obtain the desired result.

Acknowledgment The authors would like to thank an anonymous referee whose comments helped to improve the presentation of this work.

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Department of Mathematics, University of Chicago, Chicago, Illinois 60637 Email address: [email protected]

IMPA, Instituto Matematica´ Pura e Aplicada, Estrada Dona Castorina 110, 22460- 320, Rio de Janeiro, RJ, Brazil Email address: [email protected]

Department of Mathematics, University of California, Santa Barbara, California 93106 Email address: [email protected]

UPV/EHU, Departamento de Matematicas,´ Apto. 644, 48080 Bilbao, Spain – and – Basque Center for Applied Mathematics, E-48009 Bilbao, Spain Email address: [email protected]