Vanishing Shear Viscosity Limit and Incompressible Limit of Two Dimensional Compressible Dissipative Elastodynamics

We study the vanishing shear viscosity limit and incompressible limit of two dimensional compressible dissipative elastodynamics systems near the equilibrium. The large value of the volume viscosity forces the limit system to be incompressible, and the vanishing shear viscosity indicates that the limit system is inviscid. Due to the weak dispersive estimate in dimensions two and the lack of the null structures in nonlinear terms, the generalised energy method (Klainerman and Sideris in Commun Pure Appl Math 49(3):615–666, 1996; Sideris in Ann Math 151(2):849–874, 2000; Sideris and Thomases in Commun Pure Appl Math 58(6):750–788, 2005), combined with the “ghost weight" method in (Alinhac in Invent Math 145(3):597–618, 2001), guarantees the global-in-time convergence from compressible dissipative elastodynamics to incompressible elastodynamics with the help of vector fields.