Stabilization effect of elasticity on three-dimensional compressible vortex sheets

Abstract

We are concerned with the vortex sheet solutions for the three-dimensional compressible isentropic elastic flows. This is a nonlinear hyperbolic problem with a characteristic free boundary. Compared with the analysis in two dimensions, this added dimension leads to more complicated frequency interactions between the effects of elasticity and the fluid velocity, making the stability analysis more challenging. Through a very delicate examination of the Lopatinskiĭ determinant of the linearized boundary value problem, necessary and sufficient conditions are established for the linear stability of the planar vortex sheet solutions. These conditions are closely related to the geometric properties of the elastic deformation gradient and provide the first stability criterion justifying the stabilization effect of elasticity on the compressible vortex sheets in the three-dimensional elastodynamics. In contrast to the two-dimensional isentropic elastic fluids, we find that the stability can only hold in the subsonic region for the three-dimensional vortex sheets.