Inspired by a pioneer work of Andersson and Kapitanski (Arch Ration Mech Anal 247(2):Paper No. 21, 76 pp, 2023), we prove the local well-posedness of the Cauchy problem of incompressible neo-Hookean equations if the initial deformation and velocity belong to \(H^{\frac{n+2}{2}+}({\mathbb {R}}^n) \times H^{\frac{n}{2}+}({\mathbb {R}}^n)\) (\(n=2,3\)), 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 Andersson and Kapitanski (Arch Ration Mech Anal 247(2):Paper No. 21, 76 pp, 2023).

受 Andersson 和 Kapitanski 的开创性工作（Arch Ration Mech Anal 247(2):Paper No. 21, 76 pp, 2023）的启发，我们证明了不可压缩新胡克方程的柯西问题的局部适定性，如果初始变形和速度属于\(H^{\frac{n+2}{2}+}({\mathbb {R}}^n) \times H^{\frac{n}{2}+}({ \mathbb {R}}^n)\) ( \(n=2,3\) )，其中\(\frac{n+2}{2}\)和\(\frac{n}{2}\ )分别是 Sobolev 空间中变形和速度的比例不变指数。我们的新观察依赖于两个方面：减少变形和速度的二阶波椭圆系统；以及该耦合系统中固有的“波图类型”零形式。特别是，具有“波图类型”零形式的波性质使我们能够证明非线性项的 Klainerman-Macedon 类型的双线性估计。因此，与 Andersson 和 Kapitanski 相比，我们可以降低3D 中的\(\frac{1}{2}\)阶正则性和2D 中的\(\frac{3}{4}\)阶正则性以获得适定性《Ration Mech Anal》247(2)：论文第 21 号，第 76 页，2023 年）。