Flows of weakly compressible viscoelastic fluids through a regular bounded domain with inflow-outflow boundary conditions

We study steady isothermal motions of a nonlinear weakly compressible viscoelastic fluids of Oldroyd type in a bounded domain $\Omega\subset\mathbb{R}^2$, with given non-zero velocities on the boundary of $\Omega$. We suppose that the pressure $p$ and the extra-stress tensor $\tau$ are prescribed on the part of the boundary corresponding to entering velocities. A uniqueness and existence result for the solution $(\mathbf u,p,\tau)$ is established in $W^{2,q}(\Omega)\times W^{1,q}(\Omega)\times W^{1,q}(\Omega)$ with $ 2 < q < 3$. The proof follows from an analysis of a linearized problem. The fixed point theorem is used to establish the existence of a solution. The solutions of two transport equations for $p$ and $\tau$ are obtained by integration along the streamlines.