[Paper Review] Free boundary regularity of vacuum states for incompressible viscous flows in unbounded domains

Christophe Prange, Jin Tan|arXiv (Cornell University)|Oct 13, 2023

Navier-Stokes equation solutionsMathematics3 citations

TL;DR

This paper establishes uniqueness and free boundary regularity for finite-energy weak solutions of the incompressible Navier-Stokes equations with vacuum in unbounded domains ($\mathbb{R}^d$, $d=2,3$), under only bounded, nonnegative initial density and minimal regularity on the initial velocity. It resolves Lions' open problem on persistence of interface regularity for vacuum bubbles and density patches in the whole-space case by overcoming the lack of Poincaré inequalities through novel interpolation and weighted energy estimates.

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.

Motivation & Objective

• To resolve the open problem posed by Lions regarding the persistence of free boundary regularity for vacuum states in the incompressible Navier-Stokes equations on $\mathbb{R}^d$.
• To establish uniqueness and regularity of weak solutions when the initial density is bounded and nonnegative, without requiring a positive lower bound or additional regularity beyond $L^2$ for the initial momentum.
• To extend the results of Danchin and Mucha from bounded and periodic domains to the whole-space case, where Poincaré-type inequalities fail.
• To provide a rigorous framework for studying the evolution of vacuum bubbles and density patches in incompressible viscous flows.

Proposed method

• Derives a new interpolation inequality in $L^2$ for functions in $\mathcal{D}^{1,2}(\mathbb{R}^d)$, decomposing functions into $L^1$ and $L^2$ components to control $L^2$ norms via weighted $L^2$ and gradient norms.
• Applies weighted energy estimates using a weight $\eta$ that satisfies either the almost no vacuum (aNV) or vacuum bubble (VB) condition, enabling control of $L^2$ norms of velocity and density perturbations.
• Uses the structure of the inhomogeneous incompressible Navier-Stokes equations to derive a priori estimates that preserve regularity of the free boundary associated with the vacuum region.
• Employs a decomposition of the velocity field and density into regular and singular parts to handle the degeneracy of the momentum equation in vacuum regions.
• Establishes a new weighted $L^2$ estimate for the velocity field by combining Gagliardo-Nirenberg inequalities with Young's inequality and the structure of the initial data.
• Proves that the free boundary of a vacuum region or a density patch remains $C^1$-regular for all time under the given assumptions, using the regularity of the solution and the structure of the transport equation for density.

Figure 1. Relationships between the results in the paper

Experimental results

Research questions

• RQ1 Does the free boundary of a vacuum region or a density patch remain regular under the time evolution of the incompressible Navier-Stokes equations in $\mathbb{R}^d$?
• RQ2 Can uniqueness and regularity of weak solutions be established for the incompressible Navier-Stokes equations with only bounded, nonnegative initial density and $L^2$ initial momentum in the whole-space case?
• RQ3 Is it possible to extend the regularity results from bounded or periodic domains to unbounded domains like $\mathbb{R}^d$ without relying on Poincaré-type inequalities?
• RQ4 What conditions on the initial data ensure that the interface between a vacuum and a fluid region remains smooth over time?
• RQ5 Can the persistence of $C^1$-regularity of the free boundary be proven for vacuum bubbles or density patches in the absence of lower bounds on density?

Key findings

• The paper establishes the uniqueness of finite-energy weak solutions to the incompressible Navier-Stokes equations in $\mathbb{R}^d$ ($d=2,3$) with bounded, nonnegative initial density and $L^2$ initial momentum, under the standard energy conditions.
• It proves that the free boundary of a vacuum region or a density patch remains $C^1$-regular for all time, resolving Lions' open problem in the whole-space case.
• The authors construct a new interpolation estimate in $L^2$ that controls the $L^2$ norm of a function via its weighted $L^2$ norm and gradient norm, even when the weight $\eta$ is degenerate.
• The method overcomes the lack of Poincaré inequalities in unbounded domains by using a decomposition of functions into $L^1$ and $L^2$ components and applying weighted estimates.
• The result applies to physically relevant configurations such as a vacuum bubble embedded in a fluid or a homogeneous fluid patch in vacuum, both of which are shown to preserve interface regularity.
• The paper provides a complete answer to Lions' question on the persistence of boundary regularity in the whole-space case, extending prior results from bounded and periodic domains.