In this paper, we study the barotropic compressible Navier--Stokes equations in a bounded plane domain $\O$. Nonzero velocities are prescribed on the boundary of $\O$, and the density is prescribed on that part of the boundary corresponding to entering velocity. This causes a weak singularity in the solution at the junction of incoming and outgoing flows. We prove the existence of the solution $(\u,p)$ of the system \[ \left\{ \begin{array}{lcl} -\mu\D\u-\nu\nab\divu+\r(p)(\u\cdot \nab)\u + \nab p =0 \quad\mbox{in\ }\O,\\ \mbox{div}(\r\u) = 0 \quad \mbox{in\ }\O,\\ \u=\u_{0}(x,y)\quad \mbox{on\ }\G,\\ p=p_{0}(x,y)\quad \mbox{on\ }\G_{\rm in} \end{array}\right.\ \] in the Sobolev space $H^{2,q}\times H^{1,q}$$(2 < q < 3)$. The proof follows from an analysis of the linearized problem and a fixed-point argument.
[1]
D. Whittaker,et al.
A Course in Functional Analysis
,
1991,
The Mathematical Gazette.
[2]
H. Beirão da Veiga,et al.
An L(p)-Theory for the n-Dimensional, Stationary, Compressible, Navier-Stokes Equations, and the Incompressible Limit for Compressible Fluids. The Equilibrium Solutions.
,
1987
.
[3]
S. Agmon,et al.
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
,
1959
.
[4]
D. Gilbarg,et al.
Elliptic Partial Differential Equa-tions of Second Order
,
1977
.