The incompressible navier-stokes equations in vacuum

2017/05/17 by Raphaël Danchin, Danchin, Raphaël, Piotr B. Mucha +1 · 8 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1705.06061

openalex publication_date 2017/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are concerned with the existence and uniqueness issue for the inhomogeneous incompressible Navier-Stokes equations supplemented with H¹ 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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