Existence, uniqueness, and stability of solutions to the initial-boundary value problem for bipolar viscous fluids

Abstract

We obtain, via a Galerkin argument, the existence of a unique weak solution to the initial-boundary value problem for an incompressible bipolar viscous fluid satisfying nonhomogeneous boundary conditions. The analysis depends on the derivation of several key a priori estimates. Regularity results are also established and the solution is proven to be asymptotically stable when the forcing function and initial and boundary data decay in an appropriate sense.