Variants of the Stokes problem : the case of anisotropic potentials

Abstract

We investigate the smoothness properties of local solutions of the nonlinear Stokes problem -\mbox{div}\{T(\varepsilon(v))\}+\nabla\pi=g\mbox{ on }\Omega, \mbox{ div }v\equiv0\mbox{ on }\Omega, where v:\Omega\rightarrow\mathbb{R}^{n} is the velocity field, \pi:\Omega\rightarrow\mathbb{R} denotes the pressure function, and g:\Omega\rightarrow\mathbb{R}^{n} represents a system of volume forces, \Omega denoting an open subset of \mathbb{R}^{n}. The tensor t is assumed to be the gradient of some potential f acting on symmetric matrices. Our main hypothesis imposed on f is the existence of exponents 1<p\leq q<\infty such that \lambda(1+\left|\varepsilon\right|^{2})^{\frac{p-2}{2}}\left|\sigma\right|^{2}\leq D^{2}f(\varepsilon)(\sigma,\sigma)\leq\Lambda(1+\left|\varepsilon\right|^{2})^{\frac{q-2}{2}}\left|\sigma\right|^{2} holds with suitable constants \lambda,\Lambda>0, i.e. the potential f is of anisotropic power growth. Under natural assumptions on p and q we prove that velocity fields from the space W_{p,loc}^{1}(\Omega;\mathbb{R}^{n}) are of class C^{1,\alpha} on an open subset of \Omega with full measure. If n=2, then the set of interior singularities is empty.