Abstract.
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 \(C^{2+\epsilon}\) boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in W1,q for some q > N, and the initial velocity has \(\epsilon\) fractional derivatives in Lr 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 ≥ 3 if the initial velocity is small.
Similar qualitative results were obtained earlier in dimension N = 2, 3 by O. Ladyzhenskaya and V. Solonnikov in [18] for initial densities in W1,∞ and initial velocities in \(W^{2 - \tfrac{2}{q},q} \) with q > N.
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Danchin, R. Density-Dependent Incompressible Fluids in Bounded Domains. J. math. fluid mech. 8, 333–381 (2006). https://doi.org/10.1007/s00021-004-0147-1
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DOI: https://doi.org/10.1007/s00021-004-0147-1