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 \({\mathbb R}^N\) (\(N \geq 2\)) 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.
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 \(W^{2-\tfrac{2}{q},q}\) with \(q>N\)


76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids


Zbl 0376.76021
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