Cauchy Problem for Incompressible Neo-Hookean Materials

Abstract

In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension $$d \geqq 2$$ . The Cauchy problem can be formulated in terms of maps $$x(t,\cdot )$$ with domain a reference space $${\mathbb {R}}^d_\xi $$ , and with values in space $${\mathbb {R}}^d_x$$ . Initial data consists of initial deformation $$\phi (\xi ) = x(0,\xi )$$ and velocity $$\psi (\xi ) = \partial x(t,\xi )/\partial t |_{t=0}$$ . We consider the initial deformations of the form $$x(0, \xi ) = A \xi + \varphi (\xi )$$ , where A is a constant $$SL(d, {\mathbb {R}})$$ matrix. We assume that $$\varphi $$ and $$\psi $$ are in Sobolev spaces $$(\varphi , \psi ) \in H^{s+1}({\mathbb {R}}^d)\times H^{s}({\mathbb {R}}^d)$$ . If $$s>s_{crit}= d/2+1$$ , well-posedness is well-known. We are here interested primarily in the low regularity case, $$s \le s_{crit}$$ . For $$d = 2, 3$$ , we prove existence and uniqueness for $$s_0 < s\le s_{crit}$$ , and we can prove the well-posedness, but for a smaller range, $$s_1 < s \le s_{crit}$$ , where, if $$d = 2$$ , $$s_0 = 7/4$$ and $$s_1 = 7/4 + (\sqrt{65}-7)/8$$ , and if $$d=3$$ , then $$s_0=2$$ and $$s_1 = 1 + \sqrt{3/2}$$ . For the full range (in s) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of $$\varphi $$ . A key observation in the proof is that the equations of evolution for the vorticities decompose into a first-order hyperbolic system, for which a Strichartz estimate holds, and a coupled transport system. This allows one to set up a bootstrap argument to prove local existence and uniqueness. Continuous dependence on initial data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement $$s > s_{crit}$$ is necessary.