Summary: This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of () with boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in for some , and the initial velocity has fractional derivatives in for some and 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 regardless of the size of the data, or in dimension if the initial velocity is small.
Similar qualitative results were obtained earlier in dimension 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 and initial velocities in with