Global Weak Solution For A Weakly Dissipative Generalized Two-component μ-Hunter-Saxton System

In this paper, the global existence of weak solutions for a periodic two-component μ-Hunter-Saxton system with a weakly dissipative are studied. Firstly, the global existence for strong solutions to the system with smooth initial data is obtained. Then, we show that the limit of approximate solutions is a global weak solutions of the two-componentμ-Hunter-Saxton system.This paper is organized as follows:In the first chapter, we introduced aboutμ-Hunter-Saxton system and the problem in this paper.In the second chapter, firstly, we use the operator theory and green function to simply μ-Hunter-Saxton system to two systems. Then,we use Kato’s Theorem to prove the local well-posed for the Cauchy problem of the system. At last, the priori estimates of μ-Hunter-Saxton system are given.In the third chapter, firstly, we introduced the definition of mollification operator and its basic property. Then, we create smooth function sequence by mollification theorem to converge to the solution of system. Finally, we give some estimates.In the last chapter, Firstly, we use the basic energy estimation to the compactness of solution. Then, we prove the relative compactness of solution. At last, we prove global weak solution existence of system.