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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 N (N2) with C 2+ε 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 N3 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, and initial velocities in W 2-2 q,q with q>N

MSC:
76D03Existence, uniqueness, and regularity theory
35Q30Stokes and Navier-Stokes equations
76D05Navier-Stokes equations (fluid dynamics)