Density-Dependent Incompressible Fluids in Bounded Domains

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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