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Density-dependent incompressible fluids in bounded domains. (English) Zbl 1142.76354

Summary: This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of ${ℝ}^{N}$ ($N\ge 2$) with ${C}^{2+\epsilon }$ 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 $ϵ$ fractional derivatives in ${L}^{r}$ for some $r>N$ and $ϵ$ 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 $N\ge 3$ if the initial velocity is small.

Similar qualitative results were obtained earlier in dimension $N=2,3$ by O. A. Ladyshenskaya and 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 ${W}^{2-\frac{2}{q},q}$ with $q>N$

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