Breaking and permanent waves for the periodic Emphasis Type="Italic"??/Emphasis-Degasperisa??Procesi equation with linear dispersion

摘要

Considered herein is the periodic ??-Degasperisa??Procesi equation, which is an evolution equation on the space of tensor densities over the Lie algebra of smooth vector fields. First two conditions on the initial data that lead to breaking waves in finite time are formulated. The first breaking-wave result relies on the refined analysis on the evolution of the Lyapunov function (V(t)=int_{mathbb{S}}u_{x}^{3}(t,x),dx); while the second result is based on the delicate comparison of the evolution of the solution u and its gradient (u_{x}). Second the existence of permanent waves is obtained by using an a??invarianta?? property of the momentum. Last the blow-up rate of breaking wave is determined by the argument of Constantin and Eschera??s well-known result on the evolution of the minimum of the gradient of the solution??u.