Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains

Christophe Prange, Jin Tan

Abstract

In the well-known book of Lions [{\em Mathematical topics in fluid mechanics. Incompressible models}, 1996], global existence results of finite energy weak solutions of the inhomogeneous incompressible Navier-Stokes equations (INS) were proved without assuming positive lower bounds on the initial density, hence allowing for vacuum. Uniqueness, regularity and persistence of boundary re\-gularity of density patches were listed as open problems. A breakthrough on Lions' problems was recently made by Danchin and Mucha [The incompressible Navier-Stokes equations in vacuum, {\em Comm. Pure Appl. Math.}, 72 (2019), 1351--1385] in the case where the fluid domain is either bounded or the torus. However, the case of unbounded domains was left open because of the lack of Poincaré-type inequalities. In this paper, we obtain regularity and uniqueness of Lions' weak solutions for (INS) with \emph{only bounded and nonnegative initial density} and additional regularity only assumed for the initial velocity, in the whole-space case $\mathbb R^d$, $d=2$ or $3$. In particular, our result allows us to study the evolution of a vacuum bubble embedded in an incompressible fluid, as well as a patch of a homogeneous fluid embedded in the vacuum, which provides an answer to Lions' question in the whole-space case.

Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains

Abstract

In the well-known book of Lions [{\em Mathematical topics in fluid mechanics. Incompressible models}, 1996], global existence results of finite energy weak solutions of the inhomogeneous incompressible Navier-Stokes equations (INS) were proved without assuming positive lower bounds on the initial density, hence allowing for vacuum. Uniqueness, regularity and persistence of boundary re\-gularity of density patches were listed as open problems. A breakthrough on Lions' problems was recently made by Danchin and Mucha [The incompressible Navier-Stokes equations in vacuum, {\em Comm. Pure Appl. Math.}, 72 (2019), 1351--1385] in the case where the fluid domain is either bounded or the torus. However, the case of unbounded domains was left open because of the lack of Poincaré-type inequalities. In this paper, we obtain regularity and uniqueness of Lions' weak solutions for (INS) with \emph{only bounded and nonnegative initial density} and additional regularity only assumed for the initial velocity, in the whole-space case , or . In particular, our result allows us to study the evolution of a vacuum bubble embedded in an incompressible fluid, as well as a patch of a homogeneous fluid embedded in the vacuum, which provides an answer to Lions' question in the whole-space case.
Paper Structure (38 sections, 22 theorems, 262 equations, 1 figure)

This paper contains 38 sections, 22 theorems, 262 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A brief state of the art
  3. Global-in-time finite-energy weak solutions
  4. Uniqueness in the absence of vacuum
  5. Uniqueness in the presence of vacuum
  6. Main results
  7. A few remarks on the main results
  8. Main novelties
  9. Some further estimates
  10. Main difficulties and strategy for the proofs
  11. Dealing with the vacuum
  12. The compatibility condition for the far-field vacuum case
  13. Outline of the proofs
  14. Some notations and assumptions
  15. Outline of the paper
  16. ...and 23 more sections

Key Result

Theorem 1

Assume that the initial data $(\rho_0, u_0)$ satisfies condition initialcond and one of the conditions cond1-cond3, then there exists a global weak solution $(\rho, u)$ of system INS in the sense of Definition defweaksolu. Furthermore, one has for all $0\leq \alpha_0\leq \beta_0<\infty$ And if $\rho_0-\rho^\infty\in L^p({\mathbb R}^d)$ for some $1\leq p<\infty,\, \rho^\infty\in [0, \infty),$ then

Figures (1)

  • Figure 1: Relationships between the results in the paper

Theorems & Definitions (40)

  • Definition 1.1: PLL
  • Theorem : PLL
  • Theorem A: existence and uniqueness in 2D
  • Theorem B: existence and uniqueness in 3D
  • Remark 1.2
  • Theorem C: solution to Lions' problem
  • Remark 1.3
  • Proposition 2.1: gradient estimate
  • proof
  • Remark 2.2: boundedness in space of the velocity
  • ...and 30 more