On the \(C^{1,\gamma}(\overline \varOmega)\cap W^{2,2}(\varOmega) \) regularity for a class of electro-rheological fluids
The stationary boundary value problem for a class of electro-rheological fluids is studied in the paper. The velocity \(v(x)\) and the pressure \(\pi(x)\) satisfy to the equations in a bounded smooth domain \(\Omega\subset \mathbb R^n\), \(n\geq 2\): \[ \begin{cases} &-\nabla\cdot S(Dv)+(v\cdot\nabla)v+\nabla\pi=f,\quad \text{div}\,v=0, \quad x\in\Omega,\\ &v|_{\partial\Omega}=0. \end{cases} \] Here \(S(Dv)\) is the extra-stress tensor of the form \[ S(Dv)=\big(1+|Dv|\big)^{p\,(x)-2}Dv,\quad Dv=\frac{1}{2}\big(\nabla v+\nabla v^T\big), \] where \(p\,(x)\) is a given continuous function such that \(1<p_\infty\leq p\,(x)\leq p_0\), \(p_0>2\). The author prove the existence and uniqueness of a smooth solution \(v\in C^{1,\gamma}(\bar{\Omega})\cap W^{2,2}(\Omega)\), \(\pi\in C^{0,\gamma}(\bar{\Omega})\cap W^{1,2}(\Omega)\) to the small \(f\) without further restrictions on the constants \(p_\infty,p_0\).
scientific article; zbMATH DE number 5563851
scientific article