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

TL;DR

This paper proves that the highest-order energy of 2D incompressible elastodynamics systems remains uniformly bounded over time, improving previous results by leveraging advanced energy methods and structural properties.

Contribution

It establishes the uniform boundedness of the highest-order energy in 2D elastodynamics, surpassing prior growth estimates and utilizing novel analytical techniques.

Findings

Highest-order energy is uniformly bounded for all time.

Decay rate of solutions can be enhanced to subcritical levels.

The analysis combines ghost weight energy method and structural properties.

Abstract

This paper concerns the time growth of the highest-order energy of the systems of incompressible isotropic elastodynamics in two space dimensions. The global well-posedness of smooth solutions near equilibrium was first obtained by Lei [31] where the highest-order generalized energy may have certain growth in time. We improve above result and show that the highest-order generalized energy is uniformly bounded for all the time. The two dimensional incompressible elastodynamics is a system of nonlocal quasilinear wave equations where the unknowns decay as $\langle t\rangle^{-\frac12}$. This suggests the problem is supercritical in the sense that the decay rate is far from integrable. Surprisingly, we showed that in the highest-order energy estimate, the temporal decay can be strongly enhanced to be subcritical $\langle t \rangle^{-\frac54}$. The analysis is based on the ghost weight energy method by Alinhac [5], the inherent strong null structure by Lei [31] and the inherent div-curl structure of the system.