American Institute of Mathematical Sciences

Advanced
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), 429-447. 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), 145-163. 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), 278-283. 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 Poincaré, Anal. Non Linéaire, 22 (2005), 783-797. 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 Poincaré, Anal. Non Linéaire, 6 (1995), 673-725.  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), 209-262. 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), 705-749. 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), 173-218. 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), 1-16. 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), 359-369. 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), 522-529. 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), 217-227. 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), 1325-1334.  Google Scholar

[14]

N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation, SIAM J. Math. Anal., 22 (1991), 1738-1743. 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), 575-591. 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), 457-460. 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), 95-110. doi: 10.1051/m2an:2007010.  Google Scholar

[18]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 3 (1986), 455-467.  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), 573-603. 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), 1-21. 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), 669-677. 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), 899-917. 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), 890-908. 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), 429-447. 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), 145-163. 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), 278-283. 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 Poincaré, Anal. Non Linéaire, 22 (2005), 783-797. 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 Poincaré, Anal. Non Linéaire, 6 (1995), 673-725.  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), 209-262. 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), 705-749. 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), 173-218. 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), 1-16. 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), 359-369. 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), 522-529. 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), 217-227. 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), 1325-1334.  Google Scholar

[14]

N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation, SIAM J. Math. Anal., 22 (1991), 1738-1743. 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), 575-591. 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), 457-460. 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), 95-110. doi: 10.1051/m2an:2007010.  Google Scholar

[18]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 3 (1986), 455-467.  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), 573-603. 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), 1-21. 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), 669-677. 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), 899-917. 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), 890-908. doi: 10.1353/ajm.2001.0035.  Google Scholar

[1]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
[2]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521
[3]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781
[4]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803
[5]

I. Baldomá, Àlex Haro. One dimensional invariant manifolds of Gevrey type in real-analytic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 295-322. doi: 10.3934/dcdsb.2008.10.295
[6]

Anatole Katok, Federico Rodriguez Hertz. Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 609-615. doi: 10.3934/dcds.2010.27.609
[7]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[8]

Borys Alvarez-Samaniego, Pascal Azerad. Existence of travelling-wave solutions and local well-posedness of the Fowler equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 671-692. doi: 10.3934/dcdsb.2009.12.671
[9]

Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic & Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029
[10]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[11]

Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082
[12]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[13]

Michael S. Jolly, Anuj Kumar, Vincent R. Martinez. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces. Communications on Pure & Applied Analysis, 2022, 21 (1) : 101-120. doi: 10.3934/cpaa.2021169
[14]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[15]

Belkacem Said-Houari. Long-time behavior of solutions of the generalized Korteweg--de Vries equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 245-252. doi: 10.3934/dcdsb.2016.21.245
[16]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635
[17]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
[18]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[19]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605
[20]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy