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On the barotropic compressible Navier-Stokes equations. (English) Zbl 1149.35070
Authors’ abstract: “We consider barotropic compressible Navier-Stokes equations with density dependent viscosity coefficients that vanish on vacuum. We prove the stability of weak solutions in periodic domain $$\Omega = T^N$$ and in the whole space $$\Omega = \mathbb{R}^N$$, when $$N = 2$$ and $$N = 3$$. The pressure is given by $$p(\rho^Z) = \rho^\gamma$$ and our result holds for any $$\gamma > 1$$. Note that our notion of weak solutions is not the usual one. In particular we require some regularity on the initial density (which may still vanish). On the other hand, the initial velocity must satisfy only minimal assumptions (a little more than finite energy). Existence results for such solutions can be obtained from this stability analysis.”
The original existence results for this equation obtained by [P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 2: Compressible models. Oxford: Clarendon Press (1998; Zbl 0908.76004)] in $$2$$ and $$3$$ dimensions for large enough $$\gamma$$ were improved by E. Feireisl, A. Novotný and H. Petzeltová [J. Math. Fluid Mech. 3, No. 4, 358–392 (2001; Zbl 0997.35043)] to any dimension but still with dependent lower bound on $$\gamma$$. Under symmetry assumptions the lower bound $$\gamma>1$$ was established by, e.g., S. Jiang and P. Zhang [J. Math. Pures Appl. (9) 82, No. 8, 949–973 (2003; Zbl 1109.35088)]. In all these case viscosity is bounded from below. The authors improve these results as outlined above.

##### MSC:
 35Q30 Navier-Stokes equations 35B35 Stability in context of PDEs
##### Keywords:
compressible Navier-Stokes equation; existence; stability
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