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The incompressible navier-stokes equations in vacuum

Raphaël Danchin 1 Piotr Bogusław Mucha 2 
1 UMR8050
LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
Abstract : 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.
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Submitted on : Wednesday, June 21, 2017 - 8:45:41 AM
Last modification on : Thursday, September 29, 2022 - 2:21:15 PM
Long-term archiving on: : Friday, December 15, 2017 - 10:10:14 PM


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  • HAL Id : hal-01523740, version 2
  • ARXIV : 1705.06061


Raphaël Danchin, Piotr Bogusław Mucha. The incompressible navier-stokes equations in vacuum. 2017. ⟨hal-01523740v2⟩



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