Low-Regularity Solutions for the Ostrovsky Equation

Abstract

The well-posedness of the Ostrovsky equation is considered. Local well-posedness for data iñ H s (R) (s − 1 8 ) and global well-posedness for data inL 2 (R) are obtained.

The global attractor of the damped forced Ostrovsky equation

“…These lemmas can be also found in [9], however for completeness, we give the proofs of them here. First let us use the following notations…”

“…In this section, we use the approaches in [9] to obtain the local well-posedness of the problem (1.1)-(1.2). By theL 2 energy equation for the solution, we can obtain the global well-posedness and the existence of the bounded absorbing sets inL 2 .…”

“…Moreover, the global-in-time solutions were constructed in the space C(R,L 2 (R)) for the small initial data, and the asymptotics of the solution was computed for x/t = const and t → ∞. Huo and Jia [9] obtained that the Cauchy problem of (1.4) is locally well-posed inH s (R) (s − 1 8 ) without the condition γ > 0 by the so-called Fourier restriction norm (the Bourgain function spaces). The method was first introduced by J. Bourgain [2] to study the KdV and nonlinear Schrödinger equations in the periodic case.…”

Sharp well-posedness of the Cauchy problem for a generalized Ostrovsky equation with positive dispersion

“…Some authors have investigated the stability of the solitary waves or soliton solutions of (1.2); for instance, see [16][17][18]. Many people have studied the Cauchy problem for (1.2), for instance, see [17,[19][20][21][22][23][24][25][26][27][28][29][30]. The result of [23,25,31] showed that s = - 3 4 is the critical regularity index for (1.2).…”

The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity

“…It draws much attention of physicists and mathematician. Many people have investigated the Cauchy problem for (1.1), for instance, see [7,[9][10][11][12][13][14][15][16][17][21][22][23][28][29][30]. By using the Fourier restriction norm method introduced in [3,4], Isaza and Mejía [13] proved that (1.1) is locally well-posed in H s (R) with s > − 3 4 in the negative dispersion case and is locally well-posed in H s (R) with s > − 1 2 in the positive dispersion case.…”