The Fourier restriction norm method for the Zakharov-Kuznetsov equation

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May 2014, 34(5): 2061-2068. doi: 10.3934/dcds.2014.34.2061

The Fourier restriction norm method for the Zakharov-Kuznetsov equation

Axel Grünrock ^1, and Sebastian Herr ^2,

1.	Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut, Universitätsstraße 1, 40225 Düsseldorf, Germany
2.	Universität Bielefeld, Fakultät für Mathematik, Postfach 10 01 31, 33501 Bielefeld, Germany

Received February 2013 Revised June 2013 Published October 2013

Abstract
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The Cauchy problem for the Zakharov-Kuznetsov equation is shown to be locally well-posed in $H^s(\mathbb{R}^2)$ for all $s>\frac{1}{2}$ by using the Fourier restriction norm method and bilinear refinements of Strichartz type inequalities.

Keywords: Fourier restriction norm method., bilinear estimates, low regularity, Zakharov-Kuznetsov equation, well-posedness.

Mathematics Subject Classification: Primary: 35Q53; Secondary: 42B3.

Citation: Axel Grünrock, Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2061-2068. doi: 10.3934/dcds.2014.34.2061

References:

[1]	M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for third order equations in two dimensions, Comm. Partial Differential Equations, 28 (2003), 1943-1974. doi: 10.1081/PDE-120025491.
[2]	H. A. Biagioni and F. Linares, Well-Posedness Results for the Modified Zakharov-Kuznetsov Equation, In Nonlinear equations: Methods, models and applications (Bergamo, 2001), 181-189, volume 54 of Progr. Nonlinear Differential Equations Appl., pages Birkhäuser, Basel, 2003.
[3]	J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.
[4]	A. V. Faminskiĭ, The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Equations, 31 (1995), 1002-1012.
[5]	A. V. Faminskiĭ, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Differential Equations, 2008, 23 pp.
[6]	J.-M. Ghidaglia and J.-C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195. doi: 10.1007/BF02429863.
[7]	J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.
[8]	A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.
[9]	C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003.
[10]	C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[11]	C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.
[12]	H. Koch and N. Tzvetkov., On the local well-posedness of the Benjamin-Ono equation in $H^s(\mathbbR)$, Int. Math. Res. Not., 26 (2003), 1449-1464. doi: 10.1155/S1073792803211260.
[13]	E. W. Laedke and K.-H. Spatschek, Nonlinear ion-acoustic waves in weak magnetic fields, Phys. Fluids, 25 (1982), 985-989. doi: 10.1063/1.863853.
[14]	D. Lannes, F. Linares and J.-C. Saut, The Cauchy Problem for the Euler-Poisson System and Derivation of the Zakharov-Kuznetsov Equation. ArXiv e-prints, May 2012.
[15]	F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339. doi: 10.1137/080739173.
[16]	F. Linares and A. Pastor., Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., 260 (2011), 1060-1085. doi: 10.1016/j.jfa.2010.11.005.
[17]	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 Differential Equations, 35 (2010), 1674-1689. doi: 10.1080/03605302.2010.494195.
[18]	F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565. doi: 10.3934/dcds.2009.24.547.
[19]	M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlinear Anal., 59 (2004), 425-438. doi: 10.1016/j.na.2004.07.022.
[20]	F. Ribaud and S. Vento, Well-Posedness results for the three-dimensional Zakharov-Kuznetsov Equation, SIAM J. Math. Anal., 44 (2012), 2289-2304. doi: 10.1137/110850566.
[21]	F. Ribaud and S. Vento, A Note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 499-503. doi: 10.1016/j.crma.2012.05.007.
[22]	J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Differential Equations, 15 (2010), 1001-1031.
[23]	B. K. Shivamoggi, The Painlevé analysis of the Zakharov-Kuznetsov equation, Phys. Scripta, 42 (1990), 641-642. doi: 10.1088/0031-8949/42/6/001.
[24]	V. E. Zakharov and E. A. Kuznetsov, Three-dimensional solitons, Sov. Phys. JETP, 39 (1974), 285-286.

show all references

References:

[1]	M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for third order equations in two dimensions, Comm. Partial Differential Equations, 28 (2003), 1943-1974. doi: 10.1081/PDE-120025491.
[2]	H. A. Biagioni and F. Linares, Well-Posedness Results for the Modified Zakharov-Kuznetsov Equation, In Nonlinear equations: Methods, models and applications (Bergamo, 2001), 181-189, volume 54 of Progr. Nonlinear Differential Equations Appl., pages Birkhäuser, Basel, 2003.
[3]	J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.
[4]	A. V. Faminskiĭ, The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Equations, 31 (1995), 1002-1012.
[5]	A. V. Faminskiĭ, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Differential Equations, 2008, 23 pp.
[6]	J.-M. Ghidaglia and J.-C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195. doi: 10.1007/BF02429863.
[7]	J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.
[8]	A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.
[9]	C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003.
[10]	C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[11]	C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.
[12]	H. Koch and N. Tzvetkov., On the local well-posedness of the Benjamin-Ono equation in $H^s(\mathbbR)$, Int. Math. Res. Not., 26 (2003), 1449-1464. doi: 10.1155/S1073792803211260.
[13]	E. W. Laedke and K.-H. Spatschek, Nonlinear ion-acoustic waves in weak magnetic fields, Phys. Fluids, 25 (1982), 985-989. doi: 10.1063/1.863853.
[14]	D. Lannes, F. Linares and J.-C. Saut, The Cauchy Problem for the Euler-Poisson System and Derivation of the Zakharov-Kuznetsov Equation. ArXiv e-prints, May 2012.
[15]	F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339. doi: 10.1137/080739173.
[16]	F. Linares and A. Pastor., Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., 260 (2011), 1060-1085. doi: 10.1016/j.jfa.2010.11.005.
[17]	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 Differential Equations, 35 (2010), 1674-1689. doi: 10.1080/03605302.2010.494195.
[18]	F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565. doi: 10.3934/dcds.2009.24.547.
[19]	M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlinear Anal., 59 (2004), 425-438. doi: 10.1016/j.na.2004.07.022.
[20]	F. Ribaud and S. Vento, Well-Posedness results for the three-dimensional Zakharov-Kuznetsov Equation, SIAM J. Math. Anal., 44 (2012), 2289-2304. doi: 10.1137/110850566.
[21]	F. Ribaud and S. Vento, A Note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 499-503. doi: 10.1016/j.crma.2012.05.007.
[22]	J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Differential Equations, 15 (2010), 1001-1031.
[23]	B. K. Shivamoggi, The Painlevé analysis of the Zakharov-Kuznetsov equation, Phys. Scripta, 42 (1990), 641-642. doi: 10.1088/0031-8949/42/6/001.
[24]	V. E. Zakharov and E. A. Kuznetsov, Three-dimensional solitons, Sov. Phys. JETP, 39 (1974), 285-286.

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