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.   Google Scholar

[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.   Google Scholar

[3]

E. BustamanteP. Isaza and J. Mejia, On the support of solutions to the Zakharov-Kuznetsov equation, J. Diff. Eqs, 251 (2011), 2728-2736.   Google Scholar

[4]

E. BustamanteP. Isaza and J. Mejia, On uniqueness properties of solutions of the Zakharov-Kuznetsov equation, J. Funct. Anal., 264 (2013), 2529-2549.   Google Scholar

[5]

A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension, Proc. R. Soc. Edinburgh, 126 (1996), 89-112.   Google Scholar

[6]

R. CȏteC. MuñozD. Pilod and G. Simpson, Asymptotic stability of high-dimensional Zakharov-Kuznetsov solitons, Arch. Rat. Mech. Anal., 220 (2016), 639-710.   Google Scholar

[7]

A.V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Diff. Eqs, 31 (1995), 1002-1012.   Google Scholar

[8]

L.G. FarahF. 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.   Google Scholar

[9]

A. Grünrock, A remark on the modified Zakharov-Kuznetsov equation in three space dimensions, Math. Res. Lett., 21 (2014), 127-131.   Google Scholar

[10]

A. Grünrock, On the generalized Zakharov-Kuznetsov equation at critical regularity, preprint, arXiv: 1509.09146. Google Scholar

[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.   Google Scholar

[12]

D. Han-Kwan, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Comm. Math. Phys., 324 (2013), 961-993.   Google Scholar

[13]

P. IsazaF. Linares and G. Ponce, On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., 48 (2016), 1006-1024.   Google Scholar

[14]

P. IsazaF. Linares and G. Ponce, On the propagation of regularities in solutions of the Benjamin-Ono equation, J. Funct. Anal., 270 (2016), 976-1000.   Google Scholar

[15]

P. IsazaF. 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.   Google Scholar

[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.   Google Scholar

[17]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.   Google Scholar

[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.   Google Scholar

[19]

C.E. KenigG. 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.   Google Scholar

[20]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana U. Math. J. , 40 (1991), 33-69. Google Scholar

[21]

D. LannesF. 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.   Google Scholar

[22]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.   Google Scholar

[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.   Google Scholar

[24]

F. LinaresA. 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.   Google Scholar

[25]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Disc. Contin. Dyn. Syst., 24 (2009), 547-565.   Google Scholar

[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.   Google Scholar

[27]

M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlinear Anal., 59 (2004), 425-438.   Google Scholar

[28]

F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.   Google Scholar

[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.   Google Scholar

[30]

V.E. Zakharov and E.A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286.   Google Scholar

show all references

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.   Google Scholar

[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.   Google Scholar

[3]

E. BustamanteP. Isaza and J. Mejia, On the support of solutions to the Zakharov-Kuznetsov equation, J. Diff. Eqs, 251 (2011), 2728-2736.   Google Scholar

[4]

E. BustamanteP. Isaza and J. Mejia, On uniqueness properties of solutions of the Zakharov-Kuznetsov equation, J. Funct. Anal., 264 (2013), 2529-2549.   Google Scholar

[5]

A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension, Proc. R. Soc. Edinburgh, 126 (1996), 89-112.   Google Scholar

[6]

R. CȏteC. MuñozD. Pilod and G. Simpson, Asymptotic stability of high-dimensional Zakharov-Kuznetsov solitons, Arch. Rat. Mech. Anal., 220 (2016), 639-710.   Google Scholar

[7]

A.V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Diff. Eqs, 31 (1995), 1002-1012.   Google Scholar

[8]

L.G. FarahF. 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.   Google Scholar

[9]

A. Grünrock, A remark on the modified Zakharov-Kuznetsov equation in three space dimensions, Math. Res. Lett., 21 (2014), 127-131.   Google Scholar

[10]

A. Grünrock, On the generalized Zakharov-Kuznetsov equation at critical regularity, preprint, arXiv: 1509.09146. Google Scholar

[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.   Google Scholar

[12]

D. Han-Kwan, From Vlasov-Poisson to Korteweg-de Vries and Zakharov-Kuznetsov, Comm. Math. Phys., 324 (2013), 961-993.   Google Scholar

[13]

P. IsazaF. Linares and G. Ponce, On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., 48 (2016), 1006-1024.   Google Scholar

[14]

P. IsazaF. Linares and G. Ponce, On the propagation of regularities in solutions of the Benjamin-Ono equation, J. Funct. Anal., 270 (2016), 976-1000.   Google Scholar

[15]

P. IsazaF. 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.   Google Scholar

[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.   Google Scholar

[17]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.   Google Scholar

[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.   Google Scholar

[19]

C.E. KenigG. 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.   Google Scholar

[20]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana U. Math. J. , 40 (1991), 33-69. Google Scholar

[21]

D. LannesF. 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.   Google Scholar

[22]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.   Google Scholar

[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.   Google Scholar

[24]

F. LinaresA. 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.   Google Scholar

[25]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Disc. Contin. Dyn. Syst., 24 (2009), 547-565.   Google Scholar

[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.   Google Scholar

[27]

M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlinear Anal., 59 (2004), 425-438.   Google Scholar

[28]

F. Ribaud and S. Vento, Well-posedness results for the 3D Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 44 (2012), 2289-2304.   Google Scholar

[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.   Google Scholar

[30]

V.E. Zakharov and E.A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP., 39 (1974), 285-286.   Google Scholar

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