Perturbation of the domain and regularity of the solutions of the bipolar fluid flow equations in polygonal domains

Abstract

In this paper, we consider the isothermal, incompressible, viscous, bipolar fluid flow equation, which is a mathematical model of viscous fluid motion with non-linear and higher order viscosity. We investigate the question of stability of the solutions to the bipolar equations with respect to perturbations of the boundary of the domain, and show that in general the solutions are not stable with respect to perturbations of the boundary by Lipschitz curves. We also study the regularity of the solution to the bipolar equations in a polygonal domain Ω. We show that near a corner of the polygonal domain, if F ϵ L 2(Ω) , then any weak solution w ϵ H 2(Ω) ∩ H 0 1(Ω) may be written as the sum w = w reg + w sing , where w reg ϵ H loc 4 is the regular part, and w sing is the singular part which is not in H loc 4 and whose precise behavior depends on the interior angle of the corner. We also provide an explicit characterization of the local singularities in terms of the interior angle of the corner. The local singularities are calculated using the symbolic manipulation software package (Maple), and sharp a-priori estimates are then used to show that there are no other singularities.