Recent results on 2D density patches for inhomogeneous Navier-Stokes

This talk is about the dynamics of a patch given by two fluids of different constant densities, evolving by the inhomogeneous Navier-Stokes equations. The main question to address is whether the regularity of the boundary of the initial patch is preserved in time. Using classical Sobolev spaces for the velocity, we establish the propagation of C1+γ, W2,∞and C^{2+\gamma}regularities with0<\gamma<1$. The results are based on new cancelations found for time dependent singular integrals given by the linear nonhomogeneous heat kernel acting on quadratic terms.