# American Institute of Mathematical Sciences

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

## The Fourier restriction norm method for the Zakharov-Kuznetsov equation

 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.
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.

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##### 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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