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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. V. Faminskiĭ, The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Equations, 31 (1995), 1002-1012.  Google Scholar

[5]

A. V. Faminskiĭ, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Differential Equations, 2008, 23 pp.  Google Scholar

[6]

J.-M. Ghidaglia and J.-C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195. doi: 10.1007/BF02429863.  Google Scholar

[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.  Google Scholar

[8]

A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Differential Equations, 15 (2010), 1001-1031.  Google Scholar

[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.  Google Scholar

[24]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. V. Faminskiĭ, The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Equations, 31 (1995), 1002-1012.  Google Scholar

[5]

A. V. Faminskiĭ, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Differential Equations, 2008, 23 pp.  Google Scholar

[6]

J.-M. Ghidaglia and J.-C. Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci., 3 (1993), 169-195. doi: 10.1007/BF02429863.  Google Scholar

[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.  Google Scholar

[8]

A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Differential Equations, 15 (2010), 1001-1031.  Google Scholar

[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.  Google Scholar

[24]

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

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