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Existence and uniqueness for viscous steady compressible motions. (English) Zbl 0644.76086
The aim of this paper is to investigate the existence and uniqueness of steady flow for a viscous, isothermal, compressible fluid. Although the isothermal assumption might seem restrictive from the physical viewpoint, it is made here in order to clarify the underlying idea of the method. Relaxing this assumption complicates the formalism but presents no conceptual difficulties. Under suitable assumptions on the body force and on the various dimensionless numbers entering the problem, we prove that for a bounded 3-dimensional domain there exists at least one steady solution, satisfying the equations almost everywhere and continuously assuming the boundary data. Moreover, this solution is unique in a sense which we shall make precise.
The paper is divided into three parts. In the first part we briefly recall the fundamental equations and some analytical tools concerning the Stokes problem. The second part is devoted to the existence proof. We state the existence theorem. Using an iterative procedure and a regularity result for positive symmetric systems, we complete the proof. The theorem applies when the body force and the Mach number are sufficiently small and the viscosity number is sufficiently large. Moreover, assuming that the total mass is positive (which is a natural physical assumption) we prove that the density has a strictly positive lower bound. In the third part, we state and prove a uniqueness theorem for solutions whose density is bounded from above by a fixed positive constant h; the proof uses an a priori bound for the velocity field in terms of h and the data of the problem. It is worth remarking that our uniqueness conditions reduce to the “classical” restrictions for uniqueness of steady solutions of the incompressible Navier-Stokes equations in the limit as the Mach number approaches zero. The fluid is assumed throughout to be isothermal and ideal and the boundary conditions to be homogeneous.

##### MSC:
 76N15 Gas dynamics, general 35Q99 Partial differential equations of mathematical physics and other areas of application 35B45 A priori estimates in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs
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