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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

Qifan Li 1,

1. 

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

Received  November 2010 Revised  February 2011 Published  December 2011

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation,, SIAM J. Math. Anal., 22 (1991), 1738.  doi: 10.1137/0522107.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I,, Ann. Inst. Henri Poincar\'e, 3 (1986), 455.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

G. Staffilani, On solutions for periodic generalized KdV equations,, Internat. Math. Res. Notices, 18 (1997), 899.  doi: 10.1155/S1073792897000585.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation,, SIAM J. Math. Anal., 22 (1991), 1738.  doi: 10.1137/0522107.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I,, Ann. Inst. Henri Poincar\'e, 3 (1986), 455.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

G. Staffilani, On solutions for periodic generalized KdV equations,, Internat. Math. Res. Notices, 18 (1997), 899.  doi: 10.1155/S1073792897000585.  Google Scholar

[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.  Google Scholar

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