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Zbl 1142.76354
Danchin, R.
Density-dependent incompressible fluids in bounded domains. (English)
[J] J. Math. Fluid Mech. 8, No. 3, 333-381 (2006). ISSN 1422-6928; ISSN 1422-6952/e

Summary: This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of ${\Bbb R}^N$ ($N \geq 2$) with $C^{2+\varepsilon}$ boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in $W^{1, q}$ for some $q > N$, and the initial velocity has $\epsilon$ fractional derivatives in $L^{r}$ for some $r > N$ and $\epsilon$ arbitrarily small. Assuming in addition that the initial density is bounded away from $0$, we prove existence and uniqueness on a short time interval. This result is shown to be global in dimension $N=2$ regardless of the size of the data, or in dimension $N \geq 3$ if the initial velocity is small. \par Similar qualitative results were obtained earlier in dimension $N=2, 3$ by {\it O. A. Ladyshenskaya} and {\it V. A. Solonnikov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 52, 52--109 (1975; Zbl 0376.76021)] for initial densities in $W^{1,\infty}$ and initial velocities in $W^{2-\tfrac{2}{q},q}$ with $q>N$

MSC 2000:: *76D03 Existence, uniqueness, and regularity theory
35Q30 Stokes and Navier-Stokes equations
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: Incompressible inhomogeneous viscous fluids; maximal regularity; local and global existence theory; non-stationary Stokes equations

Citations: Zbl 0376.76021

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