In this note we study the generalized 2D Zakharov-Kuznetsov equations
$\partial_tu+\Delta\partial_xu+u^k\partial_xu=0$ for $k\ge 2$. By an iterative
method we prove the local well-posedness of these equations in the Sobolev
spaces $H^s(\mathbb{R}^2)$ for $s>1/4$ if $k=2$, $s>5/12$ if $k=3$ and
$s>1-2/k$ if $k\ge 4$

Ribaud, Francis

Vento, Stéphane

English

arXiv

In this note we study the generalized 2D Zakharov-Kuznetsov equations $\partial_tu+\Delta\partial_xu+u^k\partial_xu=0$ for $k\ge 2$. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $s>1/4$ if $k=2$, $s>5/12$ if $k=3$ and $s>1-2/k$ if $k\ge 4$

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A note on the Cauchy problem for the 2D generalizedZakharov-Kuznetsov equationsFrancis Ribaud, Ste´phane VentoTo cite this version:Francis Ribaud, Ste´phane Vento. A note on the Cauchy problem for the 2D generalizedZakharov-Kuznetsov equations. 2011. <hal-00642572>HAL Id: hal-00642572https://hal.archives-ouvertes.fr/hal-00642572Submitted on 18 Nov 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A note on the Cauchy problem for the 2Dgeneralized Zakharov-Kuznetsov equationsFrancis Ribaud∗ Ste´phane Vento†Abstract. In this note we study the generalized 2D Zakharov-Kuznetsovequations ∂tu + ∆∂xu + uk∂xu = 0 for k ≥ 2. By an iterative methodwe prove the local well-posedness of these equations in the Sobolev spacesHs(R2) for s > 1/4 if k = 2, s > 5/12 if k = 3 and s > 1− 2/k if k ≥ 4.Keywords: KdV-like equations, Cauchy problemAMS Classification: 35Q53, 35B65, 35Q601 IntroductionIn this short note, we are interested with the Cauchy problem associated tothe generalized Zakharov-Kuznetsov (gZK) equationsut +∆ux + ukux = 0, (1.1)in two-dimensional space and for k = 2, 3, 4, .... These equations are naturalmulti-dimensional generalizations of the well-known generalized Korteweg-de Vries equations and have been derived in [8] when k = 1 to model thepropagation of nonlinear ionic-sonic waves in a magnetized plasma.We give sharp results concerning the well-posedness issue in standardSobolev spaces Hs(R2) for suitable s ∈ R. This work follows and use similartechnics as in the paper [6] where we proved that the 3D associated problemfor k = 1 is locally well-posed in H1+(R3).Remark that the Sobolev spaces H˙s(R2) are invariant by the naturalrescaling of the equation as soon as s = sk := 1 − 2/k. Thus a naturalquestion is whether (gZK) is well-posed in Hs(R2) for s > sk.∗Laboratoire d’Analyse et de Mathe´matiques Applique´es - Universite´ Paris-Est - 5 Bd.Descartes - Champs-Sur-Marne, 77454 Marne-La-Valle´e Cedex 2 (FRANCE)†Laboratoire Analyse, Ge´ome´trie et Applications - Universite´ Paris 13 - Institut Galile´e,99 avenue J.B. Cle´ment,93430 Villetaneuse (FRANCE)1Theorem 1.1. For any u0 ∈ Hs(R2) withs > 1/4 if k = 2,s > 5/12 if k = 3,s > 1− 2/k if k ≥ 4,there exist T > 0, a Banach space XsT and a unique solution u of theCauchy problem associated to (1.1) with u(0) = u0 such that u ∈ XsT ∩Cb([0, T ],Hs(R2)). Moreover, the flow-map u0 7→ u is Lipschitz on everybounded set of Hs(R2).This theorem improves the recent works of Farah, Linares and Pastor in[2]-[4]-[5] where local well-posedness was obtained in Hs(R2) for s > 3/4 if2 ≤ k ≤ 8 and s > sk if k > 8.In view of the ill-posedness result obtained in [5], Theorem 1.1 is optimal(up to the end point) for k ≥ 4 whereas in the particular cases k = 2 and 3,we still have a gap (respectively 1/4 and 1/12) compared with the scalingindex. Concerning the end point s = sk, local well-posedness could perhapsbe reached by following the strategy developed in [7], but with a flow-maponly continuous.In a standard way our proof is based on a fixed point scheme applied tothe Duhamel formulation of (1.1):u(t) = U(t)u0 −1k + 1∫ t0U(t− t′)∂x(uk+1)(t′)dt′. (1.2)where U(t) = e−t∆∂x denotes the propagator associated with the linearpart of (1.1). Following the works of Kenig, Ponce and Vega [3] on theKdV equation, we use in a crucial way some sharp dispersive estimates forfree solutions. More precisely, these estimates are the well known Katosmoothing effect‖∇U(t)ϕ‖L∞x L2yT. ‖ϕ‖L2 , (1.3)which allows to gain one derivative in each spatial direction, and the maximalin time estimate‖U(t)ϕ‖L4xL∞yT . ‖ϕ‖Hs , s > 3/4, (1.4)proved in [4]. On the other hand, similarly to the generalized KdV equations,the previous bound is no more sufficient to deal with low non-linearities(k = 2, 3) and we need the following L2x based maximal in time estimate(see [1])‖U(t)ϕ‖L2xL∞yT . ‖ϕ‖Hs , s > 3/4. (1.5)22 Proof of the main result2.1 The case k ≥ 4As mentioned in the introduction, we want to take advantage of the L4xL∞yTlinear estimate (1.4). This motivates the choice of our resolution space:XsT = {u ∈ Cb([0, T ],Hs(R2)) : ‖u‖XsT <∞},where‖u‖XsT = ‖u‖L∞T Hsxy+ ‖〈∇〉s+1u‖L∞x L2yT+ ‖〈∇〉s−3/4+u‖L4xL∞yT .Combining estimates (1.3)-(1.4) as well as the straightforward bound‖U(t)ϕ‖L∞T L2 . ‖ϕ‖L2 , (2.1)we get‖U(t)ϕ‖XsT . ‖ϕ‖Hs . (2.2)Note that the bound for the second term can be handled by using a low-highfrequencies decomposition and next Bernstein inequality and estimate (1.3).Having the linear part under control, it remains to deal with the integralterm. It is not too hard to adapt the proofs of Propositions 3.5-3.6-3.7 in[6] to deduce∥∥∥∥∫ t0U(t− t′)∂xuk+1(t′)dt′∥∥∥∥XsT. ‖〈∇〉s−1∂xuk+1‖L1xL2yT. (2.3)The multi-dimensional version of Theorem A.13 in [3] applies and leads to‖〈∇〉s−1∂xuk+1‖L1xL2yT. ‖〈∇〉suk+1‖L1xL2yT. ‖〈∇〉su‖L7+x L14/3−yT‖u‖kL7k/6−x L7k/2+yT. (2.4)We claim that the first product in the right hand side of (2.4) is controlledby the XsT norm of u. Indeed, an interpolation argument shows that‖〈∇〉αu‖LpxLqyT. ‖u‖XsT (2.5)as soon as there exists θ ∈ [0, 1] such that1p=1− θ4,1q=θ2, α =(s+7θ − 34)−.3Taking α = s, i.e. θ = 3/7+, it follows that‖〈∇〉su‖L7+x L14/3−yT. ‖u‖XsT .If we choose now θ = 4/7k in (2.5), we infer‖〈∇〉(s−3/4+1/k)−u‖L( 14−17k)−1x L7k/2yT. ‖u‖XsT .In order to get the desired contraction factor, we will interpolate this in-equality with the bound‖〈∇〉(s−1)+u‖L∞−xyT. T 0+‖〈∇〉su‖L∞T L2xy . T0+‖u‖XsT . (2.6)This leads to‖〈∇〉(s−3/4+1/k)−u‖L(( 14−17k)−1)+x L7k/2+yT. T 0+‖u‖XsT . (2.7)By virtue of the Sobolev inequalities, we get‖u‖L7k/6−x L7k/2+yT. ‖〈∇〉(1/4−1/k)+u‖L(( 14−17k)−1)+x L7k/2+yT. ‖〈∇〉(s−3/4+1/k)−u‖L(( 14−17k)−1)+x L7k/2+yT. T 0+‖u‖XsT (2.8)for s − 3/4 + 1/k > 1/4 − 1/k, that is s > sk. Gathering together (2.2)-(2.3)-(2.4)-(2.7) and (2.8) we infer that‖F (u)‖XsT . ‖u0‖Hs + T0+‖u‖k+1XsT,where F (u) denote the right hand side of (1.2). The well-posedness resultfollows then from standard arguments.2.2 The case k = 2The proof in this case follows the same lines as in the case k ≥ 4, but withthe L4x norm replaced with a L2x maximal in time norm. So let us endow theXsT space with the norm‖u‖XsT = ‖u‖L∞T Hsxy+ ‖〈∇〉s+1u‖L∞x L2yT+ ‖〈∇〉s−3/4+u‖L2xL∞yT ,4for any s > 1/4. Using now (1.5), we easily see that‖U(t)u0‖XsT . ‖u0‖Hs , (2.9)and ∥∥∥∥∫ t0U(t− t′)∂xu3(t′)dt′∥∥∥∥XsT. ‖〈∇〉s−1∂xu3‖L1xL2yT. (2.10)Again, the fractional Leibniz rule yields the bound‖〈∇〉s−1∂xu3‖L1xL2yT. ‖〈∇〉su‖L7/2+x L14/3−yT‖u‖2L14/5−x L7+yT.By interpolation, we get‖〈∇〉αu‖LpxLqyT. ‖u‖XsT (2.11)for α, p and q satisfying1p=1− θ2,1q=θ2, α =(s+7θ − 34)−for 0 ≤ θ ≤ 1. On one hand, we deduce that‖〈∇〉s−1∂xu3‖L1xL2yT. ‖u‖XsT ,were we took θ = 3/7+ in (2.11). On the other hand, for θ = 2/7−, we infer‖〈∇〉(s−1/4)−u‖L14/5−x L7+yT. ‖u‖XsT ,which interpolated with (2.6) gives‖〈∇〉(s−1/4)−u‖L14/5−x L7+yT. T 0+‖u‖XsT .This yields the desired result for s > 1/4.2.3 The case k = 3Now we consider the intermediate case k = 3. To prove our result, wedefine the resolution space as the intersection of the two previous spaces,i.e. equipped with the norm‖u‖XsT = ‖u‖L∞T Hsxy+ ‖〈∇〉s+1u‖L∞x L2yT+ ‖〈∇〉s−3/4+u‖(L2x∩L4x)L∞yT .5Actually we don’t require the full range L2x ∩ L4x for the maximal in timenorm, but only‖〈∇〉s−3/4+u‖L3xL∞yT . ‖u‖XsT. (2.12)According to (2.2)-(2.3)-(2.9)-(2.10) we have‖F (u)‖XsT . ‖u0‖Hs + ‖〈∇〉su4‖L1xL2yT.Using again the Leibniz rule for fractional derivatives, we infer‖〈∇〉su4‖L1xL2yT. ‖〈∇〉su‖L21/4+x L14/3−yT‖u‖3L63/17−x L21/2+yT.From an interpolation argument with (2.12), we easily check that both thesenorms are acceptable as soon as s > 5/12.References[1] A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsovequation. Differential Equations 31 (1995), no. 6, 1002–1012[2] L.G. Farah, F. Linares and A. Pastor, A note on the 2D generalizedZakharov-Kuznetsov equation: local, global, and scattering results.2011, arXiv:1108.3714[3] C E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering re-sults for the generalized Korteweg-de Vries equation via the contractionprinciple. Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620.[4] F. Linares and A. Pastor, Well-posedness for the two-dimensional mod-ified Zakharov-Kuznetsov equation. SIAM J. Math. Anal. 41 (2009),no. 4, 1323–1339.[5] F. Linares and A. Pastor, Local and global well-posedness for the 2Dgeneralized Zakharov-Kuznetsov equation. J. Funct. Anal. 260 (2011),no. 4, 1060–1085.[6] F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation. Preprint, arXiv:1111.2850[7] S. Vento, Well-posedness for the generalized Benjamin-Ono equationswith arbitrary large initial data in the critical space. Int. Math. Res.Not. IMRN 2010, no. 2, 297–319.[8] V.E. Zakharov and E.A. Kuznetsov, On three dimensional solitons. Sov.Phys. JETP., 39 (1974), 285–286.6

A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations

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C. R. Acad. Sci. Paris, Ser. I 350 (2012) 499–503Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial Differential EquationsA Note on the Cauchy problem for the 2D generalizedZakharov–Kuznetsov equationsUne Note sur le problème de Cauchy pour les équations de Zakharov–Kuznetsov 2DgénéraliséesFrancis Ribaud a, Stéphane Vento ba Laboratoire d’analyse et de mathématiques appliquées, Université Paris-est, 5 boulevard Descartes, Champs-Sur-Marne, 77454 Marne-La-Vallée cedex 2, Franceb Laboratoire analyse, géométrie et applications, Université Paris 13, Institut Galilée, 99, avenue J.B. Clément, 93430 Villetaneuse, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 18 November 2011Accepted 16 May 2012Available online 30 May 2012Presented by the Editorial BoardIn this Note we study the generalized 2D Zakharov–Kuznetsov equations ∂t u + ∂xu +uk∂xu = 0 for k  2. By an iterative method we prove the local well-posedness of theseequations in the Sobolev spaces Hs(R2) for s > 1/4 if k = 2, s > 5/12 if k = 3 ands > 1 − 2/k if k  4.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous étudions dans cette Note les équations de Zakharov–Kuznetsov 2D généralisées ∂t u +∂xu + uk∂xu = 0 pour k  2. Il est établi que le problème de Cauchy peut être résolu parune méthode itérative dans les espaces de Sobolev Hs(R2) pour s > 1/4 si k = 2, s > 5/12si k = 3 et s > 1 − 2/k si k  4.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeNous nous intéressons au problème de Cauchy associé aux équations de Zakharov–Kuznetsov généralisées (gZK)ut + ux + ukux = 0, (1)en dimension deux d’espace et pour des non-linéarités k  2. Ces équations sont des extensions multi-dimensionnellesnaturelles de l’équation de Korteweg–de Vries (KdV) et ont été utilisées (lorsque k = 1) par Zakharov et Kuznetsov [8] pourdécrire la propagation d’ondes non-linéaires ioniques soniques dans un plasma magnétique.Ces équations jouissent de l’invariance par changement d’échelle suivante : si u est solution de (1), alors uλ(t, x, y) =λ2/ku(λ3t, λx, λy) est encore solution de (1). On en déduit que la norme Sobolev Ḣ s(R2) est invariante par cette transfor-mation si et seulement si s = sk := 1 − 2/k. Ce type d’invariance nous suggère d’étudier cette équation dans les espacesHs(R2) pour s > sk . Notre but ici est de montrer que les solutions de (gZK) ont effectivement un bon comportement enrégularité sous-critique (s > sk), au moins pour k assez grand. Plus précisément, nous montrons le résultat suivant.E-mail addresses: francis.ribaud@univ-mlv.fr (F. Ribaud), vento@math.univ-paris13.fr (S. Vento).1631-073X/$ – see front matter © 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2012.05.007500 F. Ribaud, S. Vento / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 499–503Théorème 0.1. Pour tout u0 ∈ Hs(R2) avec s > 1/4 si k = 2, s > 5/12 si k = 3 et s > sk si k  4, il existe T > 0, un espace deBanach X sT et une unique solution u du problème de Cauchy associé (gZK) avec u(0) = u0 telle que u ∈ X sT ∩ Cb([0, T ], Hs(R2)). Enoutre, le flot solution u0 → u est Lipschitz sur toute partie bornée de Hs(R2).Ce théorème améliore les travaux récents de Farah, Linares et Pastor [2,4,5] dans lesquels le caractère bien posé duproblème est prouvé dans les espaces Hs(R2) pour s > 3/4 si 2  k  7 et s > sk si k  8.Notons d’autre part que notre résultat est optimal au moins pour k  4. En effet, il a été établi dans [5] que le flotsolution dans l’espace critique Hsk (R2), s’il existe, ne peut être uniformément continu. Dans les cas particuliers k = 2 et 3,une différence de régularité par rapport à l’indice de changement d’échelle (respectivement 1/4 et 1/12) subsiste toujours.Notre preuve est basée sur un argument de point fixe appliqué à la formulation intégrale (3). Inspirés par la stratégiede Kenig, Ponce et Vega [3] sur KdV, nous utilisons des estimations dispersives optimales sur la solution linéaire associée.Plus précisément, nous tirons avantage de l’estimation régularisante (4) qui permet de regagner une dérivée dans chaquedirection spatiale, ainsi que de l’estimation maximale en temps (5). Ceci nous permet de traiter le cas des hautes non-linéarités (k  4). Par ailleurs, à l’instar de ce qui se produit pour les équations de KdV généralisées, cette dernière inégalitén’est plus suffisante pour traiter les basses non-linéarités (k = 2,3) et nous avons alors besoin de l’estimation maximale (6)basée sur l’espace L2x .Ces estimations permettent de contrôler la partie linéaire de (3) et il reste ensuite à estimer le terme intégral. Dansce but, nous utilisons les versions non-homogènes retardées de (4)–(6)–(5) qui permettent de récupérer la dérivée perduedans le terme non-linéaire de l’équation. En combinant celles ci avec une décomposition en paraproduit appliquée au termenon-linéaire, nous obtenons le contrôle désiré sur le terme intégral. Enfin, travaillant dans le cas sous-critique, il n’est pasdifficile de sortir de ces estimations une puissance de T (pour des solutions définies sur [0, T ]), ce qui permet d’achever leprocessus d’itération pour T suffisamment petit.1. IntroductionIn this short Note, we are interested with the Cauchy problem associated to the generalized Zakharov–Kuznetsov (gZK)equationsut + ux + ukux = 0, (2)in two-dimensional space and for k = 2,3,4, . . . . These equations are natural multi-dimensional generalizations of the well-known generalized Korteweg–de Vries equations and have been derived in [8] when k = 1 to model the propagation ofnonlinear ionic-sonic waves in a magnetized plasma.We give sharp results concerning the well-posedness issue in standard Sobolev spaces Hs(R2) for suitable s ∈ R. Thiswork follows and use similar technics as in the paper [6] where we proved that the 3D associated problem for k = 1 islocally well-posed in H1+(R3).Remark that the Sobolev spaces Ḣ s(R2) are invariant by the natural rescaling of the equation as soon as s = sk := 1−2/k.Thus a natural question is whether (gZK) is well-posed in Hs(R2) for s > sk .Theorem 1.1. For any u0 ∈ Hs(R2) with⎧⎨⎩s > 1/4 if k = 2,s > 5/12 if k = 3,s > 1 − 2/k if k  4,there exist T > 0, a Banach space X sT and a unique solution u of the Cauchy problem associated to (2) with u(0) = u0 such thatu ∈ X sT ∩ Cb([0, T ], Hs(R2)). Moreover, the flow-map u0 → u is Lipschitz on every bounded set of Hs(R2).This theorem improves the recent works of Farah, Linares and Pastor in [2,4,5] where local well-posedness was obtainedin Hs(R2) for s > 3/4 if 2  k  8 and s > sk if k > 8.In view of the ill-posedness result obtained in [5], Theorem 1.1 is optimal (up to the end point) for k  4 whereas in theparticular cases k = 2 and 3, we still have a gap (respectively 1/4 and 1/12) compared with the scaling index. Concerningthe end point s = sk , local well-posedness could perhaps be reached by following the strategy developed in [7], but with aflow-map only continuous.In a standard way our proof is based on a fixed point scheme applied to the Duhamel formulation of (2):u(t) = U (t)u0 − 1k + 1t∫U(t − t′)∂x(uk+1)(t′)dt′, (3)0F. Ribaud, S. Vento / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 499–503 501where U (t) = e−t∂x denotes the propagator associated with the linear part of (2). Following the works of Kenig, Ponce andVega [3] on the KdV equation, we use in a crucial way some sharp dispersive estimates for free solutions. More precisely,these estimates are the well-known Kato smoothing effect∥∥∇U (t)ϕ∥∥L∞x L2yT  ‖ϕ‖L2 , (4)which allows to gain one derivative in each spatial direction, and the maximal in time estimate∥∥U (t)ϕ∥∥L4x L∞yT  ‖ϕ‖Hs , s > 3/4, (5)proved in [4]. On the other hand, similarly to the generalized KdV equations, the previous bound is no more sufficient todeal with low non-linearities (k = 2,3) and we need the following L2x based maximal in time estimate (see [1])∥∥U (t)ϕ∥∥L2x L∞yT  ‖ϕ‖Hs , s > 3/4. (6)2. Proof of the main result2.1. The case k  4As mentioned in the introduction, we want to take advantage of the L4x L∞yT linear estimate (5). This motivates the choiceof our resolution space:Y sT ={u ∈ Cb([0, T ], Hs(R2)): ‖u‖Y sT < ∞},where‖u‖Y sT = ‖u‖L∞T Hsxy +∥∥〈∇〉s+1u∥∥L∞x L2yT + ∥∥〈∇〉s−3/4+ u∥∥L4x L∞yT .Since we do not have any available fractional Leibniz rule in R2, we will rather work in the Besov version X sT defined bythe norm ‖u‖X sT = ‖‖ ju‖Y sT ‖2(N) where ( j) j0 is a Littlewood–Paley analysis such that 0 localizes spatial frequenciesin a ball {|ξ |  1} and  j , j > 0 in an annulus {|ξ | ∼ 2 j}.Combining estimates (4)–(5) as well as the straightforward bound ‖U (t)ϕ‖L∞T L2  ‖ϕ‖L2 , we get∥∥U (t)u0∥∥XsT  ‖u0‖Hs . (7)Having the linear part under control, it remains to deal with the integral term. It is not too hard to adapt the proofs ofPropositions 3.5–3.7 in [6] to deduce∥∥∥∥∥t∫0U(t − t′)∂xuk+1(t′)dt′∥∥∥∥∥XsT∥∥2sj∥∥ j(uk+1)∥∥L1x L2yT ∥∥2(N). (8)To estimate the nonlinear term, we use the following paraproduct decomposition j(uk+1) =  j( ∑r j−Cr+1uk∑p=0(Pr+1u)p(Pru)k−p), (9)for some C  0 and where Pr = ∑rp=0 p . From this we get∥∥ j(uk+1)∥∥L1x L2yT ∑r j∥∥ru(Pru)k∥∥L1x L2yT ∑r j‖ru‖L7+x L14/3−yT ‖Pru‖kL7k/6−x L7k/2+yT. (10)Next, an interpolation argument shows that for any r  0,2αr‖ru‖L px LqyT  ‖ru‖XsT , (11)as soon as there exists θ ∈ [0,1] such that1p= 1 − θ4,1q= θ2, α =(s + 7θ − 34)−.Taking α = s, i.e. θ = 3/7+ , it follows that2sr‖ru‖L7+ L14/3−  ‖ru‖XsT . (12)x yT502 F. Ribaud, S. Vento / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 499–503If we choose now θ = 4/7k in (11), we infer2(s−3/4+1/k)−r‖ru‖L( 14 − 17k)−1x L7k/2yT ‖ru‖XsT .In order to get the desired contraction factor, we will interpolate this inequality with the bound2(s−1)+r‖ru‖L∞−xyT  T0+2s+r‖ru‖L∞T L2xy . (13)This leads to2(s−3/4+1/k)−r‖ru‖L(( 14 − 17k)−1)+x L7k/2+yT T 0+‖ru‖XsT .By virtue of the Bernstein inequality, we get‖Pru‖L7k/6−x L7k/2+yT r∑p=02(s−3/4+1/k)− p‖pu‖L(( 14 − 17k)−1)+x L7k/2+yT T 0+‖u‖XsT , (14)for s − 3/4 + 1/k > 1/4 − 1/k, that is s > sk . The discrete Young inequality and (10)–(12)–(14) yield∥∥2sj∥∥ j(uk+1)∥∥L1x L2yT ∥∥2(N)  T 0+‖u‖kXsT ∥∥(1r02−sr) ∗r ‖ru‖XsT ∥∥2(N)  T 0+‖u‖k+1XsT . (15)Gathering together (7)–(8) and (15) we infer that∥∥F (u)∥∥XsT  ‖u0‖Hs + T 0+‖u‖k+1XsT ,where F (u) denotes the right hand side of (3). The well-posedness result follows then from standard arguments.2.2. The case k = 2The proof in this case follows the same lines as in the case k  4, but with the L4x norm replaced with an L2x maximal intime norm. So let us endow the Y sT space with the norm‖u‖XsT = ‖u‖L∞T Hsxy +∥∥〈∇〉s+1u∥∥L∞x L2yT + ∥∥〈∇〉s−3/4+ u∥∥L2x L∞yT ,for any s > 1/4 and define ‖u‖X sT = ‖‖ ju‖Y sT ‖2(N) . Using now (6), we easily see that estimates (7) and (8) with k = 2 holdin these settings. Again, the paraproduct decomposition (9) and Hölder inequality yield the bound∥∥ j(u3)∥∥L1x L2yT ∑r j‖ru‖L7/2+x L14/3−yT ‖Pru‖2L14/5−x L7+yT.By interpolation, we get2αr‖ru‖L px LqyT  ‖ru‖XsT , (16)for α, p and q satisfying1p= 1 − θ2,1q= θ2, α =(s + 7θ − 34)−,for 0  θ  1. On one hand, we deduce that2sr‖ru‖L7/2+x L14/3−yT  ‖ru‖XsT ,were we took θ = 3/7+ in (16). On the other hand, for θ = 2/7− , we infer2(s−1/4)−r‖ru‖L14/5−x L7+yT  ‖ru‖XsT ,which interpolated with (13) gives2(s−1/4)−r‖ru‖L14/5−x L7+yT  T0+‖ru‖XsT .This yields the desired result for s > 1/4.F. Ribaud, S. Vento / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 499–503 5032.3. The case k = 3Now we consider the intermediate case k = 3. To prove our result, we define the resolution space as the intersection ofthe two previous spaces, i.e. equipped with the norm‖u‖XsT = ‖u‖L∞T Hsxy +∥∥2(s+1) j‖ ju‖L∞x L2yT ∥∥2(N) + ∥∥2(s−3/4+) j‖ ju‖(L2x∩L4x )L∞yT ∥∥2(N).Actually we do not require the full range L2x ∩ L4x for the maximal in time norm, but only2(s−3/4+) j‖ ju‖L3x L∞yT  ‖ ju‖XsT . (17)According to the previous estimates we have∥∥F (u)∥∥XsT  ‖u0‖Hs +∥∥∥∥2sj ∑r j∥∥ru(Pru)3∥∥L1x L2yT∥∥∥∥2(N).Using Hölder inequality, we infer2sr∥∥ru(Pru)3∥∥L1x L2yT  (2sr‖ru‖L21/4+x L14/3−yT)‖Pru‖3L63/17−x L21/2+yT.From an interpolation argument with (17), we easily check that both these norms are acceptable as soon as s > 5/12.AcknowledgementsThe authors thank Felipe Linares for useful comments concerning the earlier version of this work.References[1] A.V. Faminskii, The Cauchy problem for the Zakharov–Kuznetsov equation, Differential Equations 31 (6) (1995) 1002–1012.[2] L.G. Farah, F. Linares, A. Pastor, A note on the 2D generalized Zakharov–Kuznetsov equation: local, global, and scattering results, arXiv:1108.3714, 2011.[3] C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm.Pure Appl. Math. 46 (4) (1993) 527–620.[4] F. Linares, A. Pastor, Well-posedness for the two-dimensional modified Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 41 (4) (2009) 1323–1339.[5] F. Linares, A. Pastor, Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation, J. Funct. Anal. 260 (4) (2011) 1060–1085.[6] F. Ribaud, S. Vento, Well-posedness results for the 3D Zakharov–Kuznetsov equation, preprint, arXiv:1111.2850.[7] S. Vento, Well-posedness for the generalized Benjamin–Ono equations with arbitrary large initial data in the critical space, Int. Math. Res. Not. IMRN(2) (2010) 297–319.[8] V.E. Zakharov, E.A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP 39 (1974) 285–286.

A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations

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