Some Qualitative Studies On The Camassa-Holm Equation

In this paper, we make some research on a recently derived one-dimensional shallow water equation-Camassa-Holm equation, which can be obtained by opproximating directly the Hamiltonian for Euler's equation in the shallow water regime.Firstly, we review the local well-posedness in the Sobolev space H~s(R) with any s > 3/2 for the Camassa-Holm equation proved by Y. Li and P. Olver in [19]. We make some supplementary proof to make it more clear. The method used is regularizing the original equation. Then we prove the solution of the regularized version a cauchy sequence in H~s(R). We finish the proof by taking the limit of the regularized equation. For the periodic case, the proof is somewhat similar.Secondly, we prove a new inequality by Fourier series. Using this inequality, a new criteria guaranteeing the development of singularities in finite time for strong solutions for the periodic case is obtained. Then the same technique is applied to rod equaion.The last part of the paper illustrates that the solution has infinite propogation speed and as a result a new blow-up criteria whose advantage can be explained by a example. Besides, some fruitful results about weak solutions are also introduced.