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

TL;DR

This paper proves the existence and uniqueness of steady viscous compressible fluid flow solutions in a cylindrical domain with inhomogeneous slip and inflow boundary conditions, using a sequence construction approach.

Contribution

It introduces a novel method to establish existence of solutions for a complex boundary value problem involving inhomogeneous slip conditions and a nonlinearity in the continuity equation.

Findings

Existence of a solution close to a constant flow state.

Uniqueness of the solution within small perturbations.

Development of a sequence-based approach for proving existence.

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 $(v,\rho) \in W^2_p(\Omega) \times W^1_p(\Omega)$, 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 $W^2_p(\Omega) \times W^1_p(\Omega)$ 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.