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29 March 2024
 
  » arxiv » 0911.3404

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Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations
Hailiang Liu ; Zhaoyang Yin ;
Date 17 Nov 2009
AbstractThis paper is concerned with a class of nonlocal dispersive models -- the $ heta$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {it Acta Math. Appl. Sin.} Engl. Ser. {f 24}(3)(2008)423--440]: $$ (1-partial_x^2)u_t+(1- hetapartial_x^2)(frac{u^2}{2})_x =(1-4 heta)(frac{u_x^2}{2})_x, $$ including integrable equations such as the Camassa-Holm equation, $ heta=1/3$, and the Degasperis-Procesi equation, $ heta=1/4$, as special models. We investigate both global regularity of solutions and wave breaking phenomena for $ heta in mathbb{R}$. It is shown that as $ heta$ increases regularity of solutions improves: (i) $0 < heta < 1/4$, the solution will blow up when the momentum of initial data satisfies certain sign conditions; (ii) $1/4 leq heta < 1/2$, the solution will blow up when the slope of initial data is negative at one point; (iii) ${1/2} leq heta leq 1$ and $ heta=frac{2n}{2n-1}, nin mathbb{N}$, global existence of strong solutions is ensured. Moreover, if the momentum of initial data has a definite sign, then for any $ hetain 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 $ heta 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.}, {f 624} (2008)51--80.]
Source arXiv, 0911.3404
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