Well-posedness of weak solutions to electrorheological fluid equations with degeneracy on the boundary
In this article we study the electrorheological fluid equation $$ {u_t}= \hbox{div} ({\rho^\alpha}{| {\nabla u} |^{p(x) - 2}}\nabla u), $$ where $\rho (x) = \hbox{dist} (x,\partial \Omega )$ is the distance from the boundary, $p(x)\in C^{1}(\overline{\Omega})$, and $p^{-}=\min_{x\in \overline...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2017-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2017/13/abstr.html |
| Summary: | In this article we study the electrorheological fluid equation
$$
{u_t}= \hbox{div} ({\rho^\alpha}{| {\nabla u} |^{p(x) - 2}}\nabla u),
$$
where $\rho (x) = \hbox{dist} (x,\partial \Omega )$
is the distance from the boundary, $p(x)\in C^{1}(\overline{\Omega})$, and
$p^{-}=\min_{x\in \overline{\Omega}}p(x)>1$. We show how the degeneracy of
$\rho^{\alpha}$ on the boundary affects the well-posedness of the weak solutions.
In particular, the local stability of the weak solutions is established without
any boundary value condition. |
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| ISSN: | 1072-6691 |