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The Incompressible Navier‐Stokes Equations in Vacuum
Author(s) -
Danchin Raphaël,
Mucha Piotr Bogusław
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21806
Subject(s) - compressibility , bounded function , mathematics , uniqueness , scaling , invariant (physics) , mathematical analysis , incompressible flow , fluid mechanics , navier–stokes equations , mathematical physics , mechanics , physics , geometry
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 to all the previous works on those topics, we do not require regularity or a 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 his 1996 book Mathematical Topics in Fluid Mechanics , vol. 1, Incompressible Models , concerning the evolution of a drop of incompressible viscous fluid in the vacuum. © 2018 Wiley Periodicals, Inc.