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This work is devoted to the study of the initial boundary value problem for a general non isothermal model of capillary fluids derived by J. E Dunn and J. Serrin (1985) in [9, 16], which can be used as a phase transition model. We distinguish two cases, when the physical coefficients depend only on the density, and the general case. In the first case we can work in critical scaling...

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in ${ℝ}^N$ with $N\ge 2$. We address the question of well-posedness for and initial data having critical Besov regularity in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon (where $u_0\in B_{p,r}^{\frac{N}{p}-1}$ with $1\le p\<+\infty ,1\le r\le +\infty$). This improves the classical analysis where $u_0$ is considered belonging in $B_{p,1}^{\frac{N}{p}-1}$ such that the velocity $u$ remains Lipschitz....