Local well-posedness for incompressible neo-Hookean Elastic equations in almost critical Sobolev spaces

TL;DR

This paper establishes local well-posedness for incompressible neo-Hookean elastic equations in near-critical Sobolev spaces, utilizing a reduction to a wave-elliptic system and null form estimates to improve regularity thresholds.

Contribution

It introduces a novel reduction to a wave-elliptic system and employs null form estimates, lowering regularity requirements for well-posedness in 2D and 3D.

Findings

Lowered regularity thresholds for well-posedness in 3D and 2D.

Established bilinear estimates of Klainerman-Machedon type.

Proved local well-posedness in almost critical Sobolev spaces.

Abstract

Inspired by a pioneer work of Andersson-Kapitanski \cite{AK}, we prove the local well-posedness of the Cauchy problem of incompressible neo-Hookean equations if the initial deformation and velocity belong to $H^{s+1}(\mathbb{R}^n) \times H^{s}(\mathbb{R}^n), s>\frac{n+1}{2}$ ($n=2,3$). Moreover, if the initial data is small, then we can lower the regularity to $s>\frac{n}{2}$, where $\frac{n+2}{2}$ and $\frac{n}{2}$ is respectively a scaling-invariant exponent for deformation and velocity in Sobolev spaces. Our new observation relies on two folds: a reduction to a second-order wave-elliptic system of deformation and velocity; and a "wave-map type" null form intrinsic in this coupled system. In particular, the wave nature with "wave-map type" null form allows us to prove a bilinear estimate of Klainerman-Machedon type for nonlinear terms. So we can lower $\frac12$-order regularity in 3D and $\frac34$-order regularity in 2D for well-posedness compared with \cite{AK}.