Well-posedness and weak rotation limit for the Ostrovsky equation

We consider the Cauchy problem of the Ostrovsky equation. We first prove the time local well-posedness in the anisotropic Sobolev space Hs,a with s>−a/2−3/4 and 0⩽a⩽−1 by the Fourier restriction norm method. This result include the time local well-posedness in Hs with s>−3/4 for both positive and negative dissipation, namely for both βγ>0 and βγ<0. We next consider the weak rotation limit. We prove that the solution of the Ostrovsky equation converges to the solution of the KdV equation when the rotation parameter γ goes to 0 and the initial data of the KdV equation is in L2. To show this result, we prove a bilinear estimate which is uniform with respect to γ.

[1]  John P. Boyd,et al.  Analytical and numerical studies of weakly nonlocal solitary waves of the rotation-modified Korteweg-de Vries equation , 2001 .

[2]  S. Cui,et al.  Well-posedness of the Cauchy problem of Ostrovsky equation in anisotropic Sobolev spaces , 2007 .

[3]  Y. Stepanyants,et al.  Nonlinear surface and inter-nal waves in rotating uids , 1990 .

[4]  N. Tzvetkov,et al.  Remark on the local ill-posedness for KdV equation , 1999 .

[5]  Existence of periodic traveling wave solutions for the Ostrovsky equation , 2008 .

[6]  J. Ginibre,et al.  Existence and uniqueness of solutions for the generalized Korteweg de Vries equation , 1990 .

[7]  P. Isaza,et al.  Cauchy problem for the Ostrovsky equation in spaces of low regularity , 2006 .

[8]  Terence Tao,et al.  Sharp global well-posedness for KdV and modified KdV on ℝ and , 2003 .

[9]  O. A. Gilman,et al.  Approximate Analytical and Numerical Solutions of the Stationary Ostrovsky Equation , 1995 .

[10]  J. Bona,et al.  The initial-value problem for the Korteweg-de Vries equation , 1975, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[11]  Luis Vega,et al.  On the ill-posedness of some canonical dispersive equations , 2001 .

[12]  J. Ginibre,et al.  On the Cauchy Problem for the Zakharov System , 1997 .

[13]  P. Isaza,et al.  Global Cauchy problem for the Ostrovsky equation , 2007 .

[14]  Yue Liu,et al.  On the stability of solitary waves of a generalized Ostrovsky equation , 2006, Analysis and Mathematical Physics.

[15]  E. Benilov.  On the surface waves in a shallow channel with an uneven bottom , 1992 .

[16]  Kenji Nakanishi,et al.  Counterexamples to Bilinear Estimates Related with the KDV Equation and the Nonlinear Schrödinger Equation , 2001 .

[17]  Yue Liu,et al.  Stability of solitary waves and weak rotation limit for the Ostrovsky equation , 2004 .

[18]  Yue Liu,et al.  Cauchy problem for the Ostrovsky equation , 2004 .

[19]  Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations , 2002, math/0203044.

[20]  Yue Liu,et al.  Symmetry and Uniqueness of the Solitary-Wave Solution for the Ostrovsky Equation , 2010 .

[21]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[22]  Y. Stepanyants,et al.  On the existence of stationary solitary waves in a rotating fluid , 1991 .

[23]  Yue Liu,et al.  On the stability of solitary waves for the ostrovsky equation , 2007 .

[24]  Felipe Linares,et al.  Local and global well-posedness for the Ostrovsky equation , 2006 .

[25]  C. Kenig,et al.  Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle , 1993 .

[26]  H. Takaoka,et al.  Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$ , 2001 .

[27]  Luis Vega,et al.  A bilinear estimate with applications to the KdV equation , 1996 .

[28]  Zhaohui Huo,et al.  LOW-REGULARITY SOLUTIONS FOR THE OSTROVSKY EQUATION , 2006, Proceedings of the Edinburgh Mathematical Society.

[29]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[30]  S. Levandosky,et al.  Stability and weak rotation limit of solitary waves of the Ostrovsky equation , 2007 .

[31]  Jean Bourgain,et al.  Periodic Korteweg de Vries equation with measures as initial data , 1997 .

[32]  Yue Liu,et al.  STABILITY OF SOLITARY WAVES FOR THE OSTROVSKY EQUATION , 2007 .

[33]  Guilong Gui,et al.  On the Cauchy Problem for the Ostrovsky Equation with Positive Dispersion , 2007 .