Global Existence of Compressible Dissipative Elastodynamics Systems with Zero Shear Viscosity in Two Dimensions

In this paper, we study the two dimensional compressible dissipative elastodynamics systemwith zero shear viscosity, and prove global stability around a constant equilibrium. Due to the low pointwise dispersive estimate, 1/root t, for the divergence-free projection of solutions in dimension two, and the lack of null structures in convective terms, energy estimates, combined with Green's functions and dispersive estimates, for both solutions and their potentials, are obtained to control the interactions in nonlinear terms.