Higher Order Fractional Differentiability for the Stationary Stokes System

Abstract

This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system

$$\left\{ {\matrix{{{\rm{div}}\,{\cal A}\left( {x,D{\bf{u}}} \right) - \nabla \pi = {\rm{div}}\,\,{\bf{F}},} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {{\rm{div}}\,{\bf{u}} = 0,} \hfill & {{\rm{in}}\,\,\Omega .} \hfill \cr } } \right.$$

In terms of the difference quotient method, our first result reveals that if F ∈ B ^β_p,q,loc (Ω,ℝⁿ)for p = 2 and $1 \le q \le {{2n} \over {n - 2\beta }}$, then such extra Besov regularity can transfer to the symmetric gradient Du and its pressure π with no losses under a suitable fractional differentiability assumption on $x \mapsto {\cal A}\left( {x,\xi } \right)$. Furthermore, when the vector field ${\cal A}\left( {x,D{\bf{u}}} \right)$ is simplified to the full gradient ∇u, we improve the aforementioned Besov regularity for all integrability exponents p and q by establishing a new Campanato-type decay estimates for (∇u, π).