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

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 $\frac{1}{2}$-order regularity in 3D and $\frac{3}{4}$-order regularity in 2D for well-posedness compared with \cite{AK}.