PERTURBATION OF THE DOMAIN AND REGULARITY OF THE SOLUTIONS OF THE BIPOLAR FLUID-FLOW EQUATIONS IN POLYGONAL DOMAINS

In this paper, we consider the isothermal, incompressible, viscous, bi polar fluid how equation, which is a mathematical model of viscous flu id motion with non-linear and higher order viscosity. We investigate t he question of stability of the solutions to the bipolar equations wit h respect to perturbations of the boundary of the domain, and show tha t in general the solutions are not stable with respect to perturbation s of the boundary by Lipschitz curves. We also study the regularity of the solution to the bipolar equations in a polygonal domain Omega. We show that near a corner of the polygonal domain, if F epsilon L(2)(Om ega), then any weak solution w epsilon H-2(Omega) boolean AND H-0(1)(O mega) may be written as the sum w = w(reg) + w(sing), where w(reg) eps ilon H-loc(4) is the regular part, and w(sing) is the singular part wh ich is not in H-loc(4) and whose precise behavior depends on the inter ior angle of the corner. We also provide an explicit characterization of the local singularities in terms of the interior angle of the corne r. The local singularities are calculated using the symbolic manipulat ion software package (Maple), and sharp a-priori estimates are then us ed to show that there are no other singularities.