Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes

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May 2012, 11(3): 1097-1109. doi: 10.3934/cpaa.2012.11.1097

Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes

1.	Department of Mathematics, University of Bergen, 5008 Bergen, Norway

Received November 2010 Revised February 2011 Published December 2011

Abstract
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Motivated by the work of Grujić and Kalisch, [Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations 15 (2002) 1325--1334], we prove the local well-posedness for the periodic KdV equation in spaces of periodic functions analytic on a strip around the real axis without shrinking the width of the strip in time.

Keywords: Generalized Korteweg-deVries equation, real-analytic solutions, local well-posedness..

Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A0.

Citation: Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097

References:

[1]	A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with $l_p$ initial data,, J. Differential Equations, 215 (2005), 429. doi: 10.1016/j.jde.2004.12.012.
[2]	A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $R^n$,, J. Differential Equations, 240 (2007), 145. doi: 10.1016/j.jde.2007.05.022.
[3]	M. Bjørkavåg and H. Kalisch, Exponential Convergence of a Spectral Projection of the KdV Equation,, Physics Letters A, 365 (2007), 278. doi: 10.1016/j.physleta.2006.12.085.
[4]	J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV-equation,, Ann. Inst. Henri Poincar\'e, 22 (2005), 783. doi: 10.1016/j.anihpc.2004.12.004.
[5]	A. de Bouard, N. Hayashi and K. Kato, Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations,, Ann. Inst. Henri Poincar\'e, 6 (1995), 673.
[6]	J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Parts II,, Geometric Funct. Anal., 3 (1993), 209. doi: 10.1007/BF01895688.
[7]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1.
[8]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations and applications,, J. Functional Anal., 211 (2004), 173. doi: 10.1016/S0022-1236(03)00218-0.
[9]	A. B. Ferrari and E. S. Titi, Gevrey regularity for nonlinear analytic parabolic equations,, Comm. Partial Differential Equations, 23 (1998), 1. doi: 10.1080/03605309808821336.
[10]	C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations,, J. Functional Anal., 87 (1989), 359. doi: 10.1016/0022-1236(89)90015-3.
[11]	J. Gorsky and A. A. Himonas, Construction of non-analytic solutions for the generalized KdV equation,, J. Math. Anal. Appl., 303 (2005), 522. doi: 10.1016/j.jmaa.2004.08.055.
[12]	Z. Grujić, Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation,, J. Dynam. Differential Equations, 12 (2000), 217. doi: 10.1023/A:1009002920348.
[13]	Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions,, Differential Integral Equations, 15 (2002), 1325.
[14]	N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation,, SIAM J. Math. Anal., 22 (1991), 1738. doi: 10.1137/0522107.
[15]	H. Hayashi, Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and Szegö spaces on a sector,, Duke Math. J., 62 (1991), 575. doi: 10.1215/S0012-7094-91-06224-1.
[16]	H. Kalisch, Rapid convergence of a Galerkin projection of the KdV equation,, C. R. Math. Acad. Sci. Paris, 341 (2005), 457. doi: 10.1016/j.crma.2005.09.006.
[17]	H. Kalisch and X. Raynaud, On the rate of convergence of a collocation projection of the KdV equation,, M2AN Math. Model. Numer. Anal., 41 (2007), 95. doi: 10.1051/m2an:2007010.
[18]	T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I,, Ann. Inst. Henri Poincar\'e, 3 (1986), 455.
[19]	C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equations,, J. Amer. Math. Soc., 9 (1996), 573. doi: 10.1090/S0894-0347-96-00200-7.
[20]	C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,, Duke Math. J., 71 (1993), 1. doi: 10.1215/S0012-7094-93-07101-3.
[21]	I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations,, Proc. Amer. Math. Soc., 137 (2009), 669. doi: 10.1090/S0002-9939-08-09693-7.
[22]	G. Staffilani, On solutions for periodic generalized KdV equations,, Internat. Math. Res. Notices, 18 (1997), 899. doi: 10.1155/S1073792897000585.
[23]	T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations,, Amer. J. Math., 123 (2001), 890. doi: 10.1353/ajm.2001.0035.

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References:

[1]	A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with $l_p$ initial data,, J. Differential Equations, 215 (2005), 429. doi: 10.1016/j.jde.2004.12.012.
[2]	A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $R^n$,, J. Differential Equations, 240 (2007), 145. doi: 10.1016/j.jde.2007.05.022.
[3]	M. Bjørkavåg and H. Kalisch, Exponential Convergence of a Spectral Projection of the KdV Equation,, Physics Letters A, 365 (2007), 278. doi: 10.1016/j.physleta.2006.12.085.
[4]	J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV-equation,, Ann. Inst. Henri Poincar\'e, 22 (2005), 783. doi: 10.1016/j.anihpc.2004.12.004.
[5]	A. de Bouard, N. Hayashi and K. Kato, Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations,, Ann. Inst. Henri Poincar\'e, 6 (1995), 673.
[6]	J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Parts II,, Geometric Funct. Anal., 3 (1993), 209. doi: 10.1007/BF01895688.
[7]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1.
[8]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations and applications,, J. Functional Anal., 211 (2004), 173. doi: 10.1016/S0022-1236(03)00218-0.
[9]	A. B. Ferrari and E. S. Titi, Gevrey regularity for nonlinear analytic parabolic equations,, Comm. Partial Differential Equations, 23 (1998), 1. doi: 10.1080/03605309808821336.
[10]	C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations,, J. Functional Anal., 87 (1989), 359. doi: 10.1016/0022-1236(89)90015-3.
[11]	J. Gorsky and A. A. Himonas, Construction of non-analytic solutions for the generalized KdV equation,, J. Math. Anal. Appl., 303 (2005), 522. doi: 10.1016/j.jmaa.2004.08.055.
[12]	Z. Grujić, Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation,, J. Dynam. Differential Equations, 12 (2000), 217. doi: 10.1023/A:1009002920348.
[13]	Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions,, Differential Integral Equations, 15 (2002), 1325.
[14]	N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation,, SIAM J. Math. Anal., 22 (1991), 1738. doi: 10.1137/0522107.
[15]	H. Hayashi, Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and Szegö spaces on a sector,, Duke Math. J., 62 (1991), 575. doi: 10.1215/S0012-7094-91-06224-1.
[16]	H. Kalisch, Rapid convergence of a Galerkin projection of the KdV equation,, C. R. Math. Acad. Sci. Paris, 341 (2005), 457. doi: 10.1016/j.crma.2005.09.006.
[17]	H. Kalisch and X. Raynaud, On the rate of convergence of a collocation projection of the KdV equation,, M2AN Math. Model. Numer. Anal., 41 (2007), 95. doi: 10.1051/m2an:2007010.
[18]	T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I,, Ann. Inst. Henri Poincar\'e, 3 (1986), 455.
[19]	C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equations,, J. Amer. Math. Soc., 9 (1996), 573. doi: 10.1090/S0894-0347-96-00200-7.
[20]	C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,, Duke Math. J., 71 (1993), 1. doi: 10.1215/S0012-7094-93-07101-3.
[21]	I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations,, Proc. Amer. Math. Soc., 137 (2009), 669. doi: 10.1090/S0002-9939-08-09693-7.
[22]	G. Staffilani, On solutions for periodic generalized KdV equations,, Internat. Math. Res. Notices, 18 (1997), 899. doi: 10.1155/S1073792897000585.
[23]	T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations,, Amer. J. Math., 123 (2001), 890. doi: 10.1353/ajm.2001.0035.

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