# Mini course on "Fluid mechanic issues for flows with rough density and vacuum"

Abstract : We are concerned with the Navier-Stokes equations with variable and rough (viz. only bounded) density in dimension two. For simplicity, we assume that the fluid domain is the two-dimensional torus.

In the incompressible case, we establish that for any initial data such that the density is bounded and the velocity is in the Sobolev space H^1, there exists a unique global-in-time solution. It is shown that the flow of that solution is almost C^{1,1}, which allows to propagate the « density patch » structure with no loss of regularity. Compared to prior works, the main achievement here is global existence with uniqueness even though the density has no regularity and is not required to be bounded away from zero.

As regards the global existence, we establish a similar result for viscous compressible flows if the second viscosity coefficient is large enough, together with the convergence of the constructed solutions to that of the corresponding incompressible inhomogeneous Navier-Stokes equations. In the particular case where the pressure law is linear, we get uniqueness even though the velocity field is not expected to be locally-Lipschitz with respect to the space variable.

## Schedule

Wednesday | 10.04.2019, 16:00-17:30, | SR 1.067 |

Thursday | 11.04.2019, 11:00-12:30, | SR 1.067 |

Friday | 12.04.2019, 11:00-12:30, | SR 3.061 |