Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials

Commentationes mathematicae Universitatis Carolinae

Dominic Breit¹ 

关键词: Stokes problem;    generalized Newtonian fluids;    regularity;    nonautonomous functionals;    slow flows;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

We discuss regularity results concerning local minimizers $u \mathbb R^n\supset \Omega\rightarrow\mathbb R^n$ of variational integrals like \begin{align*} \int_{\Omega}\{F(\cdot ,\varepsilon (w))-f\cdot w\}\,dx \end{align*} defined on energy classes of solenoidal fields. For the potential $F$ we assume a $(p,q)$-elliptic growth condition. In the situation without $x$-dependence it is known that minimizers are of class $C^{1,\alpha }$ on an open subset $\Omega_{0}$ of $\Omega$ with full measure if $q< p\,\frac{n+2}{n}$ (for $n=2$ we have $\Omega_{0}=\Omega$). In this article we extend this to the case of nonautonomous integrands. Of course our result extends to weak solutions of the corresponding nonlinear Stokes type system.

【 授权许可】

CC BY