# American Institute of Mathematical Sciences

July 2018, 17(4): 1561-1572. doi: 10.3934/cpaa.2018074

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

 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

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 $\mathbb{R}^3$.

Citation: Felipe Linares, Gustavo Ponce. On special regularity properties of solutions of the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1561-1572. doi: 10.3934/cpaa.2018074
##### 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.

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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.
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