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

Dominic Breit

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 4, page 493-508
  • ISSN: 0010-2628

Abstract

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We discuss regularity results concerning local minimizers u : ℝ n ⊃ Ω → ℝ n of variational integrals like ∫ Ω { F ( · , ε ( w ) ) - f · w } d x 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 , α on an open subset Ω 0 of Ω with full measure if q < p n + 2 n (for n = 2 we have Ω 0 = Ω ). 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.

How to cite

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Breit, Dominic. "Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials." Commentationes Mathematicae Universitatis Carolinae 54.4 (2013): 493-508. <http://eudml.org/doc/260584>.

@article{Breit2013,
abstract = {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 \}\lbrace F(\cdot ,\varepsilon (w))-f\cdot w\rbrace \,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.},
author = {Breit, Dominic},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Stokes problem; generalized Newtonian fluids; regularity; nonautonomous functionals; local minimizer; Stokes problem; generalized Newtonian fluids; regularity; nonautonomous functionals; local minimizer},
language = {eng},
number = {4},
pages = {493-508},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials},
url = {http://eudml.org/doc/260584},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Breit, Dominic
TI - Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 4
SP - 493
EP - 508
AB - 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 }\lbrace F(\cdot ,\varepsilon (w))-f\cdot w\rbrace \,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.
LA - eng
KW - Stokes problem; generalized Newtonian fluids; regularity; nonautonomous functionals; local minimizer; Stokes problem; generalized Newtonian fluids; regularity; nonautonomous functionals; local minimizer
UR - http://eudml.org/doc/260584
ER -

References

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