Uniform Bound of the Highest-Order Energy of the 2D Incompressible Elastodynamics

This paper concerns the time growth of the highest-order energy of the systems of incompressible isotropic Hookean elastodynamics in two space dimensions. The global well-posedness of smooth solutions near equilibrium was proved by Lei [Comm. Pure Appl. Math., 69 (2016), pp. 2072–2106] where the highest-order generalized energy may have certain growth in time. We develop novel energy estimate strategies based on the inherent structures and show that the highest-order generalized energy is uniformly bounded for all the time.