Higher differentiability for solutions of stationary p ‐Stokes systems

We consider weak solutions( u , π ) : Ω → R n × R to stationary p ‐Stokes systems of the type− div a ( x , E u ) + ∇ π + [ D u ] u = f ,div u = 0in Ω ⊂ R n , where the function a ( x , ξ ) satisfies p ‐growth conditions in ξ and depends Hölder continuously on x . By E u we denote the symmetric part of the gradient D u and we write [ D u ] u for the convective term. In this setting, we establish results on the fractional higher differentiability of both the symmetric part of the gradient E u and of the pressure π. As an application, we deduce dimension estimates for the singular set of the gradient D u , thereby improving known results on partial C 1 , α ‐regularity for solutions to stationary p ‐Stokes systems.