耗散修正的Camassa-Holm方程解的存在唯一性

Existence and uniqueness of solution to the dissipation modified Camassa-Holm equation

通过对修正的Camassa-Holm方程添加耗散项ε?_x^4u,改进了其解的存在空间,证明了其在低正则性空间上解的存在唯一性。首先,通过Sobolev嵌入定理、H9lder不等式及傅里叶变换建立了非线性项的估计;其次,由压缩映射原理证明了解的局部存在唯一性;最后,由解的能量估计证明了整体解的存在性。结果表明:对于初值u_0∈L^2(R),耗散修正的Camassa-Holm方程在空间C([0,T]:L^2(R))∩L^2((0,T):H_2(R))存在唯一的局部解;进一步,对于初值u_0∈H_2(R),耗散修正的Camassa-Holm方程在空间C([0,T]:L^2(R))∩L^2((0,T):H_2(R))存在整体解。

By adding a dissipative term to the modified Camassa Holm equation, the existence space of its solution is improved, and the existence and uniqueness of the solution in the low regularity space are proven. Firstly, by Sobolev embedding theorem, H61der inequality and Fourier transform are used to estimate the nonlinear term. Secondly, the local existence and uniqueness of the solution are proven by the contraction mapping principle. Finally, the existence of the global solution is proven by estimating the energy of the solution. The results show that there is a unique local solution to the modified Camassa Holm equation in space C（[O,T]：L2（R）） ∩L^2 （（0,T）：H^2 （R）） for the initial value u0∈ L^2 （R） , and further,for the initial value u0 ∈ H^2 （R） , there is a global solution in space C（[0,T] ：L^2 （R）） ∩L^2 （（0,T） ： H^2 （R））.