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Incompressible limit for a viscous compressible fluid. (English) Zbl 0909.35101

The authors give the detailed proofs for results partially announced in P.-L. Lions [C. R. Acad. Sci., Paris, Sér. I 317, 1197-1202 (1993; Zbl 0795.76068)]. The main object of the study is the system

$\begin{array}{cc}& \frac{\partial \rho }{\partial t}+\text{div}\left(\rho u\right),\phantom{\rule{2.em}{0ex}}\rho >0,\hfill \\ & \frac{\partial }{\partial t}\left(\rho u\right)+\text{div}\left(\rho u×u\right)-{\mu }_{\epsilon }{\Delta }u-{\xi }_{\epsilon }\nabla \text{div}u+\frac{a}{{\epsilon }^{2}}\nabla {\rho }^{\gamma }=0,\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $u$ is the velocity of a fluid, $\rho$ is the density, $a>0$, $\gamma >1$ are given numbers, ${\mu }_{\epsilon }$ and ${\xi }_{\epsilon }$ are normalized coefficients satisfying

${\mu }_{\epsilon }\to \mu ,\phantom{\rule{1.em}{0ex}}{\xi }_{\epsilon }\to \xi \phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}\epsilon \phantom{\rule{4.pt}{0ex}}\text{goes}\phantom{\rule{4.pt}{0ex}}\text{to}\phantom{\rule{4.pt}{0ex}}{0}_{+},\phantom{\rule{1.em}{0ex}}\mu >0\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\mu +\xi >0\phantom{\rule{1.em}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}\mu =0·$

The limit of (1) as $\epsilon \to 0$ is the Navier-Stokes system

$\frac{\partial u}{\partial t}+\text{div}\left(u×u\right)-\mu {\Delta }u+\nabla \pi =0,\phantom{\rule{1.em}{0ex}}\text{div}\phantom{\rule{4.pt}{0ex}}u=0,\phantom{\rule{2.em}{0ex}}\left(2\right)$

or, when $\mu =0$, the Euler equations

$\frac{\partial u}{\partial t}+\text{div}\left(u×u\right)+\nabla \pi =0,\phantom{\rule{1.em}{0ex}}\text{div}\phantom{\rule{4.pt}{0ex}}u=0,\phantom{\rule{2.em}{0ex}}\left(3\right)$

where $\rho$ goes to 1 and $\pi$ is the limit of ${\rho }^{\gamma }-1/{\epsilon }^{2}$.

The authors prove the convergence results for the periodic case, in the whole space or in a bounded domain with Dirichlet or other boundary conditions. The stationary problem and the problem related to the linearized system are discussed too. The presented convergence results are valid globally in time and without restrictions upon the initial conditions. The authors mention a number of open questions together with open related problems.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 76N10 Compressible fluids, general
global solution