Stability of non-Newtonian fluid and electrorheological fluid mixed-type equation

Abstract

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.

Keywords:

• Non-Newtonian fluid
• mixed-type equation
• variable exponent
• stability
• weak solution

2010 Mathematics Subject Classifications:

• 35K55
• 35B35
• 35R35

Acknowledgments

This work is supported by Natural Science Foundation of Fujian Province No. 2019J01858.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by Natural Science Foundation of Fujian Province No. 2019J01858.