In this paper, we consider a non-Newtonian fluid and electrorheological fluid mixed-type equation $v_t=\mathrm{div}(b(x)|v{|}^{\alpha (x)}{\left|\mathrm{\nabla }v\right|}^{p-2}\mathrm{\nabla }v)+f(x,t,v),$where p>1, $b(x)\ge 0$ and $\alpha (x)\ge 0$. The existence of weak solution is investigated by means of the parabolically regularized method. Due to $|\mathrm{\nabla }v|\in L^{\mathrm{\infty }}(0,T;L_{loc}^p(\mathrm{\Omega }))$, the stability of weak solution is established by choosing a suitable test function and the local stability is also discussed without any boundary value condition.

中文翻译：

---

非牛顿流体与电流变流体混合型方程的稳定性

在本文中，我们考虑了一个非牛顿流体和电流变流体混合型方程$v_{吨}=\mathrm{div}(b(X)|v{|}^{\alpha (X)}{\left|\mathrm{\nabla }v\right|}^{p-2}\mathrm{\nabla }v)+F(X,吨,v),$其中p > 1，$b(X)\ge 0$和$\alpha (X)\ge 0$. 采用抛物正则化方法研究弱解的存在性。由于$|\mathrm{\nabla }v|\in {\mathrm{大号}}^{\mathrm{\infty }}(0,吨;{\mathrm{大号}}_{l○C}^p(\mathrm{\Omega }))$，通过选择合适的测试函数来建立弱解的稳定性，并且在没有任何边界值条件的情况下也讨论了局部稳定性。