The incompressible navier-stokes equations in vacuum

Raphaël Danchin (1) , Piotr Bogusław Mucha (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 Instytut Matematyki Stosowanej i Mechaniki

We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier-Stokes equations supplemented with H^1 initial velocity and only bounded nonnegative density. In contrast with all the previous works on that topics, we do not require regularity or positive lower bound for the initial density, or compatibility conditions for the initial velocity, and still obtain unique solutions. Those solutions are global in the two-dimensional case for general data, and in the three-dimensional case if the velocity satisfies a suitable scaling invariant smallness condition. As a straightforward application, we provide a complete answer to Lions' question in [25], page 34, concerning the evolution of a drop of incompressible viscous fluid in the vacuum.

Domaines: Equations aux dérivées partielles [math.AP]

Raphaël Danchin, Piotr Bogusław Mucha. The incompressible navier-stokes equations in vacuum. Communications on Pure and Applied Mathematics, 2019, 72 (7), pp.1351-1385. ⟨10.1002/cpa.21806⟩. ⟨hal-01523740v2⟩