TY  - JOUR
AU  - Zhan, Huashui
AU  - Zhi, Yuan
AU  - Niu, Xiaohua
PY  - 2022
TI  - On a non-Newtonian fluid type equation with variable diffusion coefficient
JO  - AIMS Mathematics
SP  - 17747
EP  - 17766
VL  - 7
IS  - 10
AB  - 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    ∂  Ω, it is found that a partial boundary value condition imposed on a submanifold of    ∂  Ω  ×  (  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.
UR  - https://doi.org/10.3934/math.2022977
DO  - 10.3934/math.2022977