Variants of the Stokes Problem: the Case of Anisotropic Potentials

Abstract

We investigate the smoothness properties of local solutions of the nonlinear Stokes problem

$\begin{eqnarray*}

-\diverg \{T(\eps(v))\} + \nabla \pi &=& g \msp \mbox{on $\Omega$,}\\

\diverg v&\equiv & 0 \msp \mbox{on $\Omega$,}

\end{eqnarray*}$

where v: Ω → ℝⁿ is the velocity field, $\pi$: Ω → ℝ$ denotes the pressure function, and g: Ω → ℝⁿ represents a system of volume forces, Ω denoting an open subset of ℝⁿ. 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 ≤ q < \infty such that

\lambda (1+|\eps|^{2})^{\frac{p-2}{2}} |\sigma|^{2} \leq D^{2}f(\eps)(\sigma ,\sigma) \leq \Lambda (1+|\eps|^{2})^{\frac{q-2}{2}} |\sigma|^{2}

holds with suitable constants λ, Λ > 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 ¹ _{p, loc} (Ω;ℝⁿ) are of class C ^1,α on an open subset of Ω with full measure. If n = 2, then the set of interior singularities is empty.