Well-posedness of the free boundary problem for incompressible elastodynamics

Abstract

The free boundary problem for the three dimensional incompressible elastodynamics system is studied under the Rayleigh–Taylor sign condition. Both the columns of the elastic stress FF⊤−I and the transpose of the deformation gradient F⊤−I are tangential to the boundary which moves with the velocity, and the pressure vanishes outside the flow domain. The linearized equation takes the form of wave equation in terms of the flow map in the Lagrangian coordinate, and the local-in-time existence of a unique smooth solution is proved using a geometric argument in the spirit of [19].