On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

Abstract

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain $\Omega ={\Omega }_0\times (0,L)\in {\mathbb{R}}^3$. We show existence of a solution $(v,\rho )\in W_p^2(\Omega )\times W_p^1(\Omega )$, where $v$ is the velocity of the fluid and $\rho$ is the density, that is a small perturbation of a constant flow $(\bar{v}\equiv [1,0,0],\bar{\rho }\equiv 1)$. We also show that this solution is unique in a class of small perturbations of $(\bar{v},\bar{\rho })$. The term $u\cdot \nabla w$ in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence $(v^n,{\rho }^n)$ that is bounded in $W_p^2(\Omega )\times W_p^1(\Omega )$ and satisfies the Cauchy condition in a larger space $L_{\infty }(0,L;L_2({\Omega }_0))$ what enables us to deduce that the weak limit of a subsequence of $(v^n,{\rho }^n)$ is in fact a strong solution to our problem.

27 pages, 1 figure. Proof of Theorem 1 corrected, some misprints removed