2013, 12(3): 1321-1339. doi: 10.3934/cpaa.2013.12.1321

Global well-posedness for the Kawahara equation with low regularity

1. 

Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

Received  March 2012 Revised  July 2012 Published  September 2012

We consider the global well-posedness for the Cauchy problem of the Kawahara equation which is one of fifth order KdV type equations. We first establish the local well-posedness in a more suitable function space for the global well-posedness by a variant of the Fourier restriction norm method introduced by Bourgain. Next, we extend this local solution globally in time by the I-method. In the present paper, we can apply the I-method to the modified Bourgain space in which the structure of the nonlinear term is reflected.
Citation: Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
References:
[1]

I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schrödinger equation,, J. Funct. Anal., 233 (2006), 228. doi: 10.1016/j.jfa.2005.08.004.

[2]

J. Bourgain, Fourier restriction phenomena for certain lattice subset applications to nonlinear evolution equation. II. The KdV-equation,, Geom. Funct. Anal., 3 (1993), 209. doi: 10.1007/BF01895688.

[3]

W. Chen and Z. Guo, Global well-posedness and I method for the fifth-order Korteweg-de Vries equation,, J. Anal. Math., 114 (2011), 121. doi: 10.1007/s11854-011-0014-y.

[4]

W. Chen, J. Li, C. Miao and J. Wu, Low regularity solution of two fifth-order KdV type equations,, J. Anal. Math., 107 (2009), 221. doi: 10.1007/s11854-009-0009-0.

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index,, Electron. J. Differential Equations, 26 (2001), 1.

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schödinger equation,, Math. Res. Lett., 9 (2002), 659.

[7]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, J. Amer. Math. Soc., 16 (2003), 705.

[8]

S. Cui, D. Deng and S. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^2$ initial data,, Acta Math. Sin., 22 (2006), 1457. doi: 10.1007/s10114-005-0710-6.

[9]

Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}$, , J. Math. Pures Appl., 91 (2009), 583. doi: 10.1016/j.matpur.2009.01.012.

[10]

T. K. Kato, Local well-posedness for Kawahara equation,, Adv. Differential Equations, 16 (2011), 257.

[11]

T. K. Kato, Well-posedness for the fifth order KdV equation,, Funkcial. Ekvac., 55 (2012), 17.

[12]

T. Kawahara, Oscillatory solitary waves in dispersive media,, J. Phys. Soc. Japan, 33 (1972), 260. doi: 10.1143/JPSJ.33.260.

[13]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, Indiana Univ. Math. J., 40 (1991), 33.

[14]

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. doi: 10.1002/cpa.3160460405.

[15]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation,, J. Amer. Math. Soc, 9 (1996), 573.

[16]

N. Kishimoto, Well-podeness of the Cauchy problem for the Korteweg-de Vries equation at critical regularity,, Differential Integral Equations, 22 (2009), 447.

[17]

N. Kishimoto and K. Tsugawa, Local well-posedness for quadratic Schrödinger equations and "good'' Boussinesq equation,, Differential Integral Equations, 23 (2010), 463.

[18]

T. Tao, Multilinear weighted convolution of $L^2$ functions and application to nonlinear dispersive equations,, Amer. J. Math., 123 (2001), 839. doi: 10.1353/ajm.2001.0035.

[19]

H. Wang, S. Cui and D. Deng, Global existence of solutions for the Kawahara equation in Sobolev space of negative indices,, Acta. Math. Sin., 23 (2007), 1435. doi: 10.1007/s10114-007-0959-z.

[20]

W. Yan and Y. Li, The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity, Math. Method Appl., Sci., 33 (2010), 1647. doi: 10.1002/mma.1273.

show all references

References:
[1]

I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schrödinger equation,, J. Funct. Anal., 233 (2006), 228. doi: 10.1016/j.jfa.2005.08.004.

[2]

J. Bourgain, Fourier restriction phenomena for certain lattice subset applications to nonlinear evolution equation. II. The KdV-equation,, Geom. Funct. Anal., 3 (1993), 209. doi: 10.1007/BF01895688.

[3]

W. Chen and Z. Guo, Global well-posedness and I method for the fifth-order Korteweg-de Vries equation,, J. Anal. Math., 114 (2011), 121. doi: 10.1007/s11854-011-0014-y.

[4]

W. Chen, J. Li, C. Miao and J. Wu, Low regularity solution of two fifth-order KdV type equations,, J. Anal. Math., 107 (2009), 221. doi: 10.1007/s11854-009-0009-0.

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index,, Electron. J. Differential Equations, 26 (2001), 1.

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schödinger equation,, Math. Res. Lett., 9 (2002), 659.

[7]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, J. Amer. Math. Soc., 16 (2003), 705.

[8]

S. Cui, D. Deng and S. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^2$ initial data,, Acta Math. Sin., 22 (2006), 1457. doi: 10.1007/s10114-005-0710-6.

[9]

Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}$, , J. Math. Pures Appl., 91 (2009), 583. doi: 10.1016/j.matpur.2009.01.012.

[10]

T. K. Kato, Local well-posedness for Kawahara equation,, Adv. Differential Equations, 16 (2011), 257.

[11]

T. K. Kato, Well-posedness for the fifth order KdV equation,, Funkcial. Ekvac., 55 (2012), 17.

[12]

T. Kawahara, Oscillatory solitary waves in dispersive media,, J. Phys. Soc. Japan, 33 (1972), 260. doi: 10.1143/JPSJ.33.260.

[13]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, Indiana Univ. Math. J., 40 (1991), 33.

[14]

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. doi: 10.1002/cpa.3160460405.

[15]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation,, J. Amer. Math. Soc, 9 (1996), 573.

[16]

N. Kishimoto, Well-podeness of the Cauchy problem for the Korteweg-de Vries equation at critical regularity,, Differential Integral Equations, 22 (2009), 447.

[17]

N. Kishimoto and K. Tsugawa, Local well-posedness for quadratic Schrödinger equations and "good'' Boussinesq equation,, Differential Integral Equations, 23 (2010), 463.

[18]

T. Tao, Multilinear weighted convolution of $L^2$ functions and application to nonlinear dispersive equations,, Amer. J. Math., 123 (2001), 839. doi: 10.1353/ajm.2001.0035.

[19]

H. Wang, S. Cui and D. Deng, Global existence of solutions for the Kawahara equation in Sobolev space of negative indices,, Acta. Math. Sin., 23 (2007), 1435. doi: 10.1007/s10114-007-0959-z.

[20]

W. Yan and Y. Li, The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity, Math. Method Appl., Sci., 33 (2010), 1647. doi: 10.1002/mma.1273.

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