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

Huashui Zhan; Yuan Zhi; Xiaohua Niu; Huashui Zhan; Yuan Zhi; Xiaohua Niu

doi:10.3934/math.2022977

AIMS Mathematics

2022, Volume 7, Issue 10: 17747-17766. doi: 10.3934/math.2022977

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Research article

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

1.
School of Mathematics and Statistics, Xiamen University of Technology, Xiamen 361024, China
2.
School of Sciences, Jimei University, Xiamen 361021, China

Received: 03 June 2022 Revised: 19 July 2022 Accepted: 21 July 2022 Published: 03 August 2022
MSC : 35B35, 35K20, 35K55

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.
- the partial boundary value condition,
- variable diffusion,
- non-Newtonian fluid equation,
- submanifold,
- trace
Citation: Huashui Zhan, Yuan Zhi, Xiaohua Niu. On a non-Newtonian fluid type equation with variable diffusion coefficient[J]. AIMS Mathematics, 2022, 7(10): 17747-17766. doi: 10.3934/math.2022977

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Abstract

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.

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