On special regularity properties of solutions of the Zakharov-Kuznetsov equation

Felipe Linares; Gustavo Ponce

doi:10.3934/cpaa.2018074

Article Contents

2018, Volume 17, Issue 4: 1561-1572. Doi: 10.3934/cpaa.2018074

This issue Previous Article Dynamical behavior for the solutions of the Navier-Stokes equation Next Article On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system

On special regularity properties of solutions of the Zakharov-Kuznetsov equation

Felipe Linares^1, and
Gustavo Ponce^2,

1.
IMPA, Estrada Dona Castorina 110, Rio de Janeiro 22460-320, Brazil
2.
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

Received: December 2016

Revised: April 2017

Published: April 2018

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Abstract

We study special regularity properties of solutions to the initial value problem associated to the Zakharov-Kuznetsov equation in three dimensions. We show that the initial regularity of the data in a family of half-spaces propagates with infinite speed. By dealing with the finite envelope of a class of these half-spaces we extend the result to the complement of a family of cones in $\mathbb{R}^3$.

Keywords:

Nonlinear dispersive equations,
propagation of regularity.

Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B05.

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References

[1]	H. Biagioni and F. Linares, Well-posedness results for the modified Zakharov-Kuznetsov equation, Progr. Nonlinear Diff. Eqs Appl., 54 (2003), 181-189.
[2]	J.L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Roy. Soc. London Ser A, 278 (1978), 555-601.
[3]	E. Bustamante, P. Isaza and J. Mejia, On the support of solutions to the Zakharov-Kuznetsov equation, J. Diff. Eqs, 251 (2011), 2728-2736.
[4]	E. Bustamante, P. Isaza and J. Mejia, On uniqueness properties of solutions of the Zakharov-Kuznetsov equation, J. Funct. Anal., 264 (2013), 2529-2549.
[5]	A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension, Proc. R. Soc. Edinburgh, 126 (1996), 89-112.
[6]	R. Cȏte, C. Muñoz, D. Pilod and G. Simpson, Asymptotic stability of high-dimensional Zakharov-Kuznetsov solitons, Arch. Rat. Mech. Anal., 220 (2016), 639-710.
[7]	A.V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Diff. Eqs, 31 (1995), 1002-1012.
[8]	L.G. Farah, F. Linares and A. Pastor, A note on the 2D generalized Zakharov-Kuznetsov equation: local, global and scattering results, J. Diff. Eqs, 253 (2011), 2558-2571.
[9]	A. Grünrock, A remark on the modified Zakharov-Kuznetsov equation in three space dimensions, Math. Res. Lett., 21 (2014), 127-131.
[10]	A. Grünrock, On the generalized Zakharov-Kuznetsov equation at critical regularity, preprint, arXiv: 1509.09146.
[11]	A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Disc. Contin. Dyn. Syst. Ser. A, 34 (2014), 2061-2068.
[12]	D. Han-Kwan, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Comm. Math. Phys., 324 (2013), 961-993.
[13]	P. Isaza, F. Linares and G. Ponce, On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., 48 (2016), 1006-1024.
[14]	P. Isaza, F. Linares and G. Ponce, On the propagation of regularities in solutions of the Benjamin-Ono equation, J. Funct. Anal., 270 (2016), 976-1000.
[15]	P. Isaza, F. Linares and G. Ponce, On the propagation of regularity and decay of solutions to the k-generalized Korteweg-de Vries equation, Comm. Partial Diff. Eqs., 40 (2015), 1336-1364.
[16]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8 (1983), 93-128.
[17]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
[18]	C.E. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 827-838.
[19]	C.E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.
[20]	C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana U. Math. J. , 40 (1991), 33-69.
[21]	D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Prog. Nonlinear Diff. Eqs Appl., 84 (2013), 181-213.
[22]	F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.
[23]	F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov Equation, J. Funct. Anal., 260 (2011), 1060-1085.
[24]	F. Linares, A. Pastor and J.-C. Saut, Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton, Comm. Partial Diff. Eqs, 35 (2010), 1674-1689.
[25]	F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Disc. Contin. Dyn. Syst., 24 (2009), 547-565.
[26]	L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.
[27]	M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlinear Anal., 59 (2004), 425-438.
[28]	F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.
[29]	F. Ribaud and S. Vento, A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations, C. R. Acad. Sci. Paris, 350 (2012), 499-503.
[30]	V.E. Zakharov and E.A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286.