Higher differentiability for solutions to stationary $p$-Stokes systems under sub-quadratic growth conditions

We consider weak solutions $(u,\pi ):{\mathbb{R}}^n\supset \mathrm{\Omega }\to {\mathbb{R}}^n\times \mathbb{R}$ to stationary $p$-Stokes systems of the type $$\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}(a(\mathcal{E}u))+\mathrm{\nabla }\pi =f\\ \mathrm{d}\mathrm{i}\mathrm{v}(u)=0,\end{array}$$ in $\mathrm{\Omega },$ where the function $a(\xi )$ satisfies $p$-growth conditions in $\xi$. By $\mathcal{E}u$ we denote the symmetric part of the gradient $Du$. In this setting, we establish results on the fractional higher differentiability of both the symmetric part of the gradient $Du$ and of the pressure $\pi$.