| In this paper, we study the blow-up and dynamics of strong solutions of theCamassa-Holm equation, and prove a blow-up criterion and two theorems on thebehavior of global solutions. The blow-up criterion is a sufcient condition givenby the behavior of solutions along the trajectories at small time. The resultsare parallel to those of Chae on the3-D incompressible Euler equation, with thevorticity these replaced by momentum density. Also the motivations are diferent.On the dynamics of global solutions, Chae was considering the possibility ofdeducing contradictions in the dynamics of some solution with smooth initialdata, assuming its global existence. This could solve the problem on whethersmooth initial data can guarantee global smooth solutions. Our concerns arediferent, as it is well-known that smooth initial data do not guarantee globalsmooth solution for the Camassa-Holm equation. However, because the Camassa-Holm equation is structurally simple, the results give a pretty informative pictureon the evolutions of the momentum density and other quantities. |
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