Well-posedness And Large Deviation Principle For The Stochastic Modified Camassa-Holm Equation

In this paper,it mainly studies the existence and uniqueness of the solution of the modified Camassa-Holm equation,the well-posedness of the solution of the stochastic modified Camassa-Holm equation,and the large deviation principle of the solution of the stochastic modified Camassa-Holm equation.The main contents are:In the chapter 1,it mainly explains the physical background and current research status of the equations studied,and introduces some concepts and inequalities involved in the article.In the chapter 2,By adding the dissipation term to the modified Camassa-Holm equation,the existence space of the modified Camassa-Holm equation solution can be improved,and the existence and uniqueness of the solution can be established in the low regularity space.First,the Sobolev embedding theorem,Holder inequality,and Fourier transform are used to establish the estimation of the nonlinear terms.Secondly,by the principle of contraction mapping,the local existence and uniqueness of the solution can be obtained.Finally,by the energy estimation of the solution,the existence of the global solution is obtained.In the chapter 3,we study the effect of a noise on the stochastic modified Camassa-Holm equation.Firstly,derived the stochastic modified Camassa-Holm equation by the stochastic variational principle.Then,we prove the well-posedness of the stochastic modified Camassa-Holm equation by the iterative process.Under the condition of the small noise intensity,we can also get the regularization of the noise on the solution.In the chapter 4,we study the large deviation principle of the solution of the stochastic modified Camassa-Holm equation with Wiener random terms.Firstly,on the basis of the well-posedness of the proof in Chapter 3,the weak convergent method is used to obtain the principle that the solution of the regularized stochastic Camassa-Holm equation satisfies the large deviation.Secondly,by establishing the exponential equivalence between the distribution of the solution of the regularized equation and the solution of the original equation,the large deviation principle of the solution of the original equation is obtained.In the chapter 5,we have summarized the main content of the previous chapters.