On an electrorheological fluid equation with orientated convection term

Abstract

Abstract A kind of electrorheological fluid equations with orientated convection terms is considered. If the diffusion coefficient a(x,t)∈C1(QT‾) $a(x,t)\in C^{1}(\overline{Q_{T}})$ is degenerate on the boundary ∂Ω, not only the uniqueness of weak solution is proved, but also the stability of the solutions can be proved without any boundary condition, provided that there are some restrictions on the diffusion coefficient a(x,t) $a(x,t)$ and the convective coefficient b→(x,t) $\vec{b}(x,t)$. Moreover, the large time behavior of weak solution is studied.

Keywords

• The electrorheological fluid equation
• Orientated convection term
• Partial boundary value condition
• Stability
• Large time behavior