The global attractor for a viscous weakly dissipative generalized two-component μ-Hunter-Saxton system

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton (gμHS2) system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative (gμHS2) system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative (gμHS2) system, we show that the semi-group of the solution operator {S(t)}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S(t)}t≥0 possesses a global attractor in the Sobolev space H2(S)×H2(S).