Global Well-posedness for Incompressible Hookean Elastodynamics in the Critical Besov Spaces

TL;DR

This paper proves the global well-posedness of incompressible Hookean elastodynamics equations with small initial data in critical Besov spaces, using wave maps type nonlinearities and adapted function spaces.

Contribution

It identifies wave maps type nonlinearities in elastodynamics and establishes global well-posedness in critical Besov spaces for the first time.

Findings

Global well-posedness for small data in critical Besov spaces

Identification of wave maps type nonlinearities

Use of $U^2$-type spaces for iteration

Abstract

We identify the wave maps type nonlinearities of incompressible Hookean elastodynamics equations in Lagerangian coordinates, and iterate them in the adapted $U^2$-type spaces to prove the small data global well-posedness in the critical Besov space $\dot{B}^{\frac{n}{2}+1}_{2,1}(\mathbb{R}^n)\times \dot{B}^{\frac{n}{2}}_{2,1}(\mathbb{R}^n)\ (n\ge 2)$.