Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations

TL;DR

This paper studies a class of nonlocal dispersive equations, analyzing conditions for global regularity and wave breaking, and showing how solution behavior depends on the parameter , with new results on existence and blow-up phenomena.

Contribution

It provides a comprehensive analysis of the -equation, establishing new criteria for global existence and blow-up, and extends understanding of solution behavior across different parameter ranges.

Findings

Solutions blow up for certain initial momentum signs when <1/4.

Wave breaking occurs when initial slope is negative for 1/4 <1/2.

Global solutions exist for   1/2, including special cases like Camassa-Holm and Degasperis-Procesi equations.

Abstract

This paper is concerned with a class of nonlocal dispersive models -- the $\theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: $$ (1-\partial_x^2)u_t+(1-\theta\partial_x^2)(\frac{u^2}{2})_x =(1-4\theta)(\frac{u_x^2}{2})_x, $$ including integrable equations such as the Camassa-Holm equation, $\theta=1/3$, and the Degasperis-Procesi equation, $\theta=1/4$, as special models. We investigate both global regularity of solutions and wave breaking phenomena for $\theta \in \mathbb{R}$. It is shown that as $\theta$ increases regularity of solutions improves: (i) $0 <\theta < 1/4$, the solution will blow up when the momentum of initial data satisfies certain sign conditions; (ii) $1/4 \leq \theta < 1/2$, the solution will blow up when the slope of initial data is negative at one point; (iii) ${1/2} \leq \theta \leq 1$ and $\theta=\frac{2n}{2n-1}, n\in \mathbb{N}$, global existence of strong solutions is ensured. Moreover, if the momentum of initial data has a definite sign, then for any $\theta\in \mathbb{R}$ global smoothness of the corresponding solution is proved. Proofs are either based on the use of some global invariants or based on exploration of favorable sign conditions of quantities involving solution derivatives. Existence and uniqueness results of global weak solutions for any $\theta \in \mathbb{R}$ are also presented. For some restricted range of parameters results here are equivalent to those known for the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, {\it J. reine angew. Math.}, {\bf 624} (2008)51--80.]