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Zbl 0909.35101
Lions, P.-L.; Masmoudi, N.
Incompressible limit for a viscous compressible fluid.
(English)
[J] J. Math. Pures Appl., IX. Sér. 77, No.6, 585-627 (1998). ISSN 0021-7824

The authors give the detailed proofs for results partially announced in {\it 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 \aligned &\frac{\partial \rho}{\partial t}+\text{div}(\rho u),\qquad \rho>0,\\ &\frac{\partial }{\partial t}(\rho u)+\text{div}(\rho u\times u) -\mu_\varepsilon\Delta u-\xi_\varepsilon\nabla\text{div}u+\frac a{\varepsilon^2} \nabla\rho^\gamma=0,\endaligned \tag 1 where $u$ is the velocity of a fluid, $\rho$ is the density, $a>0$, $\gamma>1$ are given numbers, $\mu_\varepsilon$ and $\xi_\varepsilon$ are normalized coefficients satisfying $$\mu_\varepsilon\to\mu,\quad \xi_\varepsilon\to\xi\quad \text{ as } \varepsilon \text{ goes to }0_+,\quad \mu>0 \text{ and } \mu+\xi>0\quad \text{or }\mu=0.$$ The limit of (1) as $\varepsilon\to 0$ is the Navier-Stokes system $$\frac{\partial u}{\partial t}+\text{div}(u\times u) -\mu\Delta u+\nabla\pi=0, \quad \text{div }u=0,\tag 2$$ or, when $\mu=0$, the Euler equations $$\frac{\partial u}{\partial t}+\text{div}(u\times u) +\nabla\pi=0, \quad\text{div }u=0,\tag 3$$ where $\rho$ goes to 1 and $\pi$ is the limit of $\rho^\gamma-1/\varepsilon^2$.\par 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.
[I.Sh.Mogilevskij (Ferrara)]
MSC 2000:
*35Q30 Stokes and Navier-Stokes equations
76N10 Compressible fluids, general

Keywords: global solution

Citations: Zbl 0795.76068

Cited in: Zbl 0934.76080

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