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.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

Cited literature [28 references]  Display  Hide  Download

https://hal-upec-upem.archives-ouvertes.fr/hal-01523740
Contributor : Raphaël Danchin <>
Submitted on : Wednesday, June 21, 2017 - 8:45:41 AM
Last modification on : Friday, October 4, 2019 - 1:28:57 AM
Long-term archiving on : Friday, December 15, 2017 - 10:10:14 PM

Files

drop-arxiv.pdf
Files produced by the author(s)

Identifiers

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

Citation

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

Share

Metrics

Record views

171

Files downloads

635