Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications

Molinet, Luc; Pilod, Didier

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 347-371 / Harvested from Numdam

Résumé

This article is concerned with the Zakharov–Kuznetsov equation $\begin{matrix} \partial_{t} u + \partial_{x} Δ u + u \partial_{x} u = 0 . & (0.1) \end{matrix}$ We prove that the associated initial value problem is locally well-posed in $H^{s} (ℝ^{2})$ for $s > \frac{1}{2}$ and globally well-posed in $H^{1} (ℝ \times 𝕋)$ and in $H^{s} (ℝ^{3})$ for $s > 1$ . Our main new ingredient is a bilinear Strichartz estimate in the context of Bourgain's spaces which allows to control the high-low frequency interactions appearing in the nonlinearity of (0.1). In the $ℝ^{2}$ case, we also need to use a recent result by Carbery, Kenig and Ziesler on sharp Strichartz estimates for homogeneous dispersive operators. Finally, to prove the global well-posedness result in $ℝ^{3}$ , we need to use the atomic spaces introduced by Koch and Tataru.

Publié le : 2015-01-01

MR 3325241 | Zbl 1320.35106

DOI : https://doi.org/10.1016/j.anihpc.2013.12.003
Classification: 35A01, 35Q53, 35Q60

@article{AIHPC_2015__32_2_347_0,
     author = {Molinet, Luc and Pilod, Didier},
     title = {Bilinear Strichartz estimates for the Zakharov--Kuznetsov equation and applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {32},
     year = {2015},
     pages = {347-371},
     doi = {10.1016/j.anihpc.2013.12.003},
     mrnumber = {3325241},
     zbl = {1320.35106},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2015__32_2_347_0}
}

Molinet, Luc; Pilod, Didier. Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) pp. 347-371. doi : 10.1016/j.anihpc.2013.12.003. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_2_347_0/

Bibliographie

[1] M. Ben-Artzi, H. Koch, J.-C. Saut, Dispersion estimates for third order equations in two dimensions, Commun. Partial Differ. Equ. 28 (2003), 1943 -1974 | MR 2015408 | Zbl 1060.35122

[2] J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal. 3 (1993), 157 -178 | MR 1209300 | Zbl 0787.35096

[3] A. Carbery, C.E. Kenig, S. Ziesler, Restriction for homogeneous polynomial surfaces in $ℝ^{3}$ , Trans. Am. Math. Soc. 365 (2013), 2367 -2407 | MR 3020102 | Zbl 1273.42008

[4] A.V. Faminskii, The Cauchy problem for the Zakharov–Kuznetsov equation, Differ. Equ. 31 no. 6 (1995), 1002 -1012 | MR 1383936 | Zbl 0863.35097

[5] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables déspace (d'après Bourgain), Astérisque 237 (1996), 163 -187 | MR 1423623 | Zbl 0870.35096

[6] A. Grünrock, S. Herr, The Fourier restriction norm method for the Zakharov–Kuznetsov equation, Discrete Contin. Dyn. Syst., Ser. A 34 (2014), 2061 -2068 | MR 3124726 | Zbl 1280.35124

[7] M. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009), 917 -941 , Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27 (2010), 971 -972 | Numdam | Numdam | Zbl 1188.35162 | Zbl 1169.35372

[8] C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation, Commun. Pure Appl. Math. 46 (1993), 527 -620 | MR 1211741 | Zbl 0808.35128

[9] E.A. Kuznetsov, V.E. Zakharov, On three dimensional solitons, Sov. Phys. JETP 39 (1974), 285 -286

[10] H. Koch, D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Commun. Pure Appl. Math. 58 (2005), 217 -284 | MR 2094851 | Zbl 1078.35143

[11] D. Lannes, F. Linares, J.-C. Saut, The Cauchy problem for the Euler–Poisson system and derivation of the Zakharov–Kuznetsov equation, Prog. Nonlinear Differ. Equ. Appl. 84 (2013), 181 -213 | MR 3185896 | Zbl 1273.35263

[12] F. Linares, A. Pastor, Well-posedness for the two-dimensional modified Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 41 (2009), 1323 -1339 | MR 2540268 | Zbl 1197.35242

[13] F. Linares, A. Pastor, J.-C. Saut, Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton, Commun. Partial Differ. Equ. 35 (2010), 1674 -1689 | MR 2754059 | Zbl 1198.35228

[14] F. Linares, J.-C. Saut, The Cauchy problem for the 3D Zakharov–Kuznetsov equation, Discrete Contin. Dyn. Syst. 24 (2009), 547 -565 | MR 2486590 | Zbl 1170.35086

[15] L. Molinet, J.-C. Saut, N. Tzvetkov, Global well-posedness for the KPII equation on the background of non localized solution, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28 no. 5 (2011), 653 -676 | Numdam | MR 2838395 | Zbl 1279.35079

[16] F. Ribaud, S. Vento, Well-posedness results for the 3D Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 44 (2012), 2289 -2304 | MR 3023376 | Zbl 1251.35135

[17] F. Rousset, N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009), 477 -496 | Numdam | MR 2504040 | Zbl 1169.35374

[18] F. Rousset, N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures Appl. 90 (2008), 550 -590 | MR 2472893 | Zbl 1159.35063

[19] F. Planchon, On the cubic NLS on 3D compact domains, J. Inst. Math. Jussieu 13 (2014), 1 -18 | MR 3134013 | Zbl 1290.35249

[20] D. Tataru, On global existence and scattering for the wave maps equation, Am. J. Math. 123 (2001), 37 -77 | MR 1827277 | Zbl 0979.35100