no. 2
Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 347-371

This article is concerned with the Zakharov–Kuznetsov equation

∂ t u+∂ x Δu+u∂ x u=0.(0.1)
We prove that the associated initial value problem is locally well-posed in H s (ℝ 2 ) for s>1 2 and globally well-posed in H 1 (ℝ×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.

DOI : 10.1016/j.anihpc.2013.12.003
Classification : 35A01, 35Q53, 35Q60
Keywords: Zakharov–Kuznetsov equation, Initial value problem, Well-posedness, Bilinear Strichartz estimates, Bourgain's spaces 
@article{AIHPC_2015__32_2_347_0,
     author = {Molinet, Luc and Pilod, Didier},
     title = {Bilinear {Strichartz} estimates for the {Zakharov{\textendash}Kuznetsov} equation and applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {347--371},
     year = {2015},
     publisher = {Elsevier},
     volume = {32},
     number = {2},
     doi = {10.1016/j.anihpc.2013.12.003},
     mrnumber = {3325241},
     zbl = {1320.35106},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2013.12.003/}
}
TY  - JOUR
AU  - Molinet, Luc
AU  - Pilod, Didier
TI  - Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 347
EP  - 371
VL  - 32
IS  - 2
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2013.12.003/
DO  - 10.1016/j.anihpc.2013.12.003
LA  - en
ID  - AIHPC_2015__32_2_347_0
ER  - 
%0 Journal Article
%A Molinet, Luc
%A Pilod, Didier
%T Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 347-371
%V 32
%N 2
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2013.12.003/
%R 10.1016/j.anihpc.2013.12.003
%G en
%F 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) no. 2, pp. 347-371. doi: 10.1016/j.anihpc.2013.12.003

[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 | Zbl | MR

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

[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 | Zbl

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

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

[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 | Zbl | MR

[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 | EuDML | Zbl | Numdam

[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 | Zbl | MR

[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 | Zbl | MR

[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 | Zbl | MR

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

[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 | Zbl | MR

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

[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 | Zbl | MR

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

[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 | EuDML | Zbl | MR

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

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

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

Cité par Sources :