Electron. J. Differential Equations, Vol. 2017 (2017), No. 13, pp. 1-15.

Well-posedness of weak solutions to electrorheological fluid equations with degeneracy on the boundary

Huashui Zhan, Jie Wen

Abstract:
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.

Submitted August 12, 2016. Published January 12, 2017.
Math Subject Classifications: 35L65, 35K85, 35R35.
Key Words: Electrorheological fluid equation; boundary degeneracy; Holder's inequality; local stability.

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Huashui Zhan
School of Applied Mathematics
Xiamen University of Technology
Xiamen, Fujian 361024, China
email: 2012111007@xmut.edu.cn
Jie Wen
School of Sciences
Jimei University
Xiamen, Fujian 361021, China
email: 1195103523@qq.com

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