zbMATH — the first resource for mathematics

Global existence of solutions for isentropic compressible Navier-Stokes equations. (Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques.) (French. Abridged English version) Zbl 0778.76086
Summary: We present global existence results for weak solutions of isentropic, compressible Navier-Stokes equations in arbitrary dimensions \((N\geq 2)\) with arbitrary initial data (finite mass and total energy). In the case of a pressure law \(p(\rho)=a\rho^{\gamma}(a>0)\), the result is shown under the condition \(\gamma\geq \gamma_ 0(N)\) with \(\gamma_ 0(2)=3/2\) for example. The proof relies upon “stability” properties of sequences of solutions and on appropriate approximations based upon, for instance, the study of stationary problems. And we give, for these stationary problems, existence results of weak solutions and regularity results if \(N=2\) or if \(N=3\).

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations