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 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  ≥  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 W^1,∞ and initial velocities in $W^{2 - \tfrac{2}{q},q}$ with q  >  N.