On a non-Newtonian fluid type equation with variable diffusion coefficient

Abstract

<abstract><p>Since the non-Newtonian fluid type equations arise from a broad and in-depth background, many research achievements have been gained from 1980s. Different from the usual non-Newtonian fluid equation, there is a nonnegative variable diffusion in the equations considered in this paper. Such a variable diffusion reflects the characteristic of the medium which may not be homogenous. By giving a generalization of the Gronwall inequality, the stability and the uniqueness of weak solutions to the non-Newtonian fluid equation with variable diffusion are studied. Since the variable diffusion may be degenerate on the boundary $ \partial \Omega $, it is found that a partial boundary value condition imposed on a submanifold of $ \partial\Omega\times (0, T) $ is enough to ensure the well-posedness of weak solutions. The novelty is that the concept of the trace of $ u(x, t) $ is generalized by a special way.</p></abstract>