Global well-posedness for systems of elasticity

Abstract

This is a survey paper. We mainly revisit those highlighted results on the global well-posedness of classical solutions near equilibrium for systems ofcompressible and incompressible elasticity. Due to the fact that the methods and ideas used for elasticity are closely related to those used for quasilinear wave equations, we will also revisit some corresponding highlighted results on quasilinear wave equations. We will try to pinpoint the key difficulties of those problems and the key methods to overcome them, and present direct and simple proofs as many as we can.We will also raise some open questions and make a few comments to attract the attention of young researchers.