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Gain of regularity for a Korteweg-de Vries–Kawahara type equation. (English) Zbl 1064.35172

The author investigates the existence of local and global solutions together with the gain of regularity of the initial value problem associated with the Korteweg-de Vries–Kawahara (KdVK) equation perturbed by a dispersive term which appears in several fluid dynamical problems. In Section 1 the KdVK equation under study is presented, i.e. \[ u_t+ \eta u_{xxxxx}+ uu_x= 0\tag{1} \] with \(-\infty< x<+\infty\), \(t> 0\) and \(\eta\in\mathbb{R}\).
The main Theorem 1.1, which is proved in Section 8, states that if the initial data \(u(x, 0)\) decays faster than polynomially on \(\mathbb{R}^+= \{x\in\mathbb{R};x> 0\}\) and posesses certain initial Sobolev regularity, then the solution \(u(x, t)\in C^\infty\) for all \(t> 0\).
In Section 2 some preliminaries are described, while in Section 3 the main inequality is demonstrated. It Section 4 the author proves an important a priori estimate. In Section 5 a basic local-in-time existence and uniqueness theorem is shown. Section 6 contains the proof of a fundamental global existence theorem, while in Section 7 a series of estimates for solutions of (1) in weighted Sobolev norms is developed. These provide a starting point for the a priori gain of regularity.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
47J35 Nonlinear evolution equations
35B65 Smoothness and regularity of solutions to PDEs
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