We consider weak solutions $(u,\pi ):\mathrm{\Omega }\to {\mathbb{R}}^n\times \mathbb{R}$ to stationary p‐Stokes systems of the type

$$\left\{\begin{array}{c}-\mathrm{div}a(x,\mathcal{E}u)+\nabla \pi +[Du]u=f,\hfill \\ \mathrm{div}u=0\hfill \end{array}\right.$$

in $\mathrm{\Omega }\subset {\mathbb{R}}^n$, where the function $a(x,\xi )$ satisfies p‐growth conditions in ξ and depends Hölder continuously on x. By $\mathcal{E}u$ we denote the symmetric part of the gradient $Du$ and we write $[Du]u$ for the convective term. In this setting, we establish results on the fractional higher differentiability of both the symmetric part of the gradient $\mathcal{E}u$ and of the pressure π. As an application, we deduce dimension estimates for the singular set of the gradient $Du$, thereby improving known results on partial $C^{1,\alpha }$‐regularity for solutions to stationary p‐Stokes systems.

中文翻译：

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固定p-Stokes系统解决方案的更高可微性

我们考虑较弱的解决方案 $（ü，\pi ）：\mathrm{\Omega }\to {\mathbb{[R}}^{ñ}\times \mathbb{[R}$到类型的固定式p- Stokes系统

$$\left\{\begin{array}{c}-\mathrm{div}\mathrm{一种}（X，\mathcal{Ë}ü）+\nabla \pi +[dü]ü=F，\hfill \\ \mathrm{div}ü=0\hfill \end{array}\right.$$

在 $\mathrm{\Omega }\subset {\mathbb{[R}}^{ñ}$，这里的功能 $\mathrm{一种}（X，\xi ）$满足ξ中p的增长条件，并连续取决于x的Hölder 。通过$\mathcal{Ë}ü$ 我们表示梯度的对称部分 $dü$ 然后我们写 $[dü]ü$对流期。在这种情况下，我们建立了梯度两个对称部分的分数高微分的结果$\mathcal{Ë}ü$和压力π。作为应用，我们推导了梯度的奇异集的维估计$dü$，从而改善了部分已知结果 $C^{\mathrm{1个}，\alpha }$固定p- Stokes系统解决方案的正规性。