Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 3

Degenerate non-Newtonian fluid equation on the half space

Pages: 215 – 233

DOI: http://dx.doi.org/10.4310/DPDE.2018.v15.n3.a4

Authors

Huashui Zhan (School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian, China)

Zhaosheng Feng (Department of Mathematics, University of Texas at Rio Grande Valley, Edinburg, Tx., U.S.A.)

Abstract

The degenerate non-Newtonian fluid equation\[\frac{\partial u}{ \partial t} - \mathrm{div} (a(x) {\lvert \nabla u \rvert}^{p-2} \nabla u)- \sum^{N}_{i=1} f_i (x) D_i u = g(u, x, t), (x, t) \in \mathbb{R}^N_{+} \times (0, T)\]arises in several scientific fields. When $a(x)$ and $p$ satisfy certain conditions, the existence of solution of this equation is established. When $a^{-\frac{1}{p}} (x) f_i (x) \leq c$ for $i \in \lbrace 1, 2, \dotsm , N \rbrace$, by choosing a suitable test function, the local stability of the solutions is discussed without any boundary value condition.

Keywords

non-Newtonian fluid equation, half space, boundary value condition, local stability

2010 Mathematics Subject Classification

Primary 35B35, 35K55. Secondary 76A05.

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This work is supported by National Science Foundation of Fujian Province under No. 2015J01592 and Natural Science Foundation of Xiamen University of Technology. It is also partially supported by UTRGV Faculty Research Council Award 110000327.

Received 28 May 2017