ON THE EXISTENCE OF SHOCK WAVES IN COMPRESSIBLE NEO-HOOKK ELASTIC MATERIALS

The talk discusses the results on the structural stability of shock waves in the elastodynamics of compressible neo-Hookean materials. By nonlinear structural stability of a shock wave we mean the local in time existence and uniqueness of the solution with a shock wave-discontinuity of the hyperbolic system of conservation laws, which is necessary for the real existence of a shock wave as a physical structure. We describe finding of the conditions for structural stability for shock waves in the two-dimensional elasto-dynamics using both the energy method and the spectral analysis of the corresponding linearized problem with a free boundary. Recent results on structural stability in the general three-dimensional case will be briefly discussed too