Dirichlet boundary value problem for equations describing flows of a nonlinear viscoelastic fluid in a bounded domain

Abstract

We consider a boundary value problem for a mathematical model describing a stationary isotermic flow of a nonlinear viscoelastic liquid with a varying viscosity depending on the shear rate in a bounded three- or two-dimensional domain with a sufficiently smooth boundary. We assume that the viscosity function is continuous and bounded. The considered model is a system of strongly nonlinear third order partial differential equations. The boundary of the flow region is subject to the homogeneous Dirichlet boundary condition, which corresponds to the standard condition of adhesion on the solid walls of a vessel. This boundary value problem is considered in a weak (generalized) sense. A weak solution is a pair of functions “velocity-pressure” satisfying the equations of motion in the distribution sense. Using the regularization method via introducing terms with an additional viscosity into the equations, we construct a family of auxiliary approximating problems. We provide an interpretation of the problems of this family in the form of operator equation with a continuous nonlinearoperator satisfying the $\alpha$-monotonicity condition. On the base of a solvability theorem for equations with $\alpha$-operators, we prove the existence of at least one solution for each positive value of the additional viscosity. We obtain estimates for the norms of solutions independent of the additional viscosity parameter. The solution to the original boundary value problem is obtained as the limit of the sequence of solutions to approximating problems as the additional viscosity tends to zero. The passage to the limit is carried out on the base of well-known results on the compactness of the embedding of Sobolev spaces and Lebesgue theorem on dominated convergence. In addition, we establish an energy-type estimate for the vector velocity function.