Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity - Haspot, Boris (Annales de l'Institut Fourier, 2012)
This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous ﬂuids in RN with N 2. We address the question of well-posedness for large and small 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 u0 ∈ BpNp,r−1 with 1 p < +∞, 1 r +∞). This improves the classical analysis where u0 is considered belonging in BpNp,1−1 such that the velocity u remains Lipschitz. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin when the velocity u is not necessary Lipschitz but only log Lipschitz. Furthermore it gives a ﬁrst kind of answer to the problem of self-similar solution.