[Paper Review] The incompressible navier-stokes equations in vacuum

Raphaël Danchin, Piotr B. Mucha|arXiv (Cornell University)|May 17, 2017

Navier-Stokes equation solutions27 references6 citations

TL;DR

This paper establishes global existence and uniqueness of solutions to the incompressible Navier-Stokes equations in vacuum with only $H^1$ initial velocity and bounded nonnegative initial density, without requiring a positive lower bound on density or compatibility conditions. The key contribution is proving unique global solutions in 2D for general data and in 3D under a scaling-invariant smallness condition on the initial velocity, resolving a long-standing question on the evolution of a viscous fluid drop in vacuum.

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.

Motivation & Objective

• Address the long-standing open problem of whether a viscous incompressible fluid drop can evolve uniquely in vacuum without requiring positive initial density or compatibility conditions.
• Extend the existence and uniqueness theory for the inhomogeneous incompressible Navier-Stokes equations beyond classical assumptions of positive density and smooth initial data.
• Provide a complete answer to Lions’ question (2006, p.34) on the evolution of a drop of incompressible viscous fluid in vacuum.
• Establish well-posedness in critical regularity spaces for initial velocity in $H^1$ and bounded nonnegative density, even when density is discontinuous or vanishes.

Proposed method

• Formulate the inhomogeneous incompressible Navier-Stokes system with initial density $\rho_0 \in L^\infty$, $\rho_0 \geq 0$, and initial velocity $v_0 \in H^1_0(\Omega)$, with no lower bound on $\rho_0$.
• Use a fixed-point argument in a carefully chosen function space $X_T$ involving $L^4(0,T;L^2)$, $L^2(0,T;H^1_0)$, and $L^{4/3}(0,T;L^{3/2})$ for time derivatives.
• Apply maximal regularity estimates for the Stokes operator with time-dependent coefficients, leveraging the theory of singular integrals and $L^p$-boundedness of the divergence operator.
• Construct a solution operator via a time-dependent matrix $A(t,x)$ approximating the identity, and use a Banach fixed-point theorem in the space $X_T$ to prove existence and uniqueness.
• Establish energy estimates and conservation laws (mass, momentum, energy) under minimal assumptions, relying on renormalization techniques for the continuity equation.
• Adapt results from [7] and [25] on renormalized solutions and maximal regularity to handle the lack of regularity in density and velocity.

Experimental results

Research questions

• RQ1 Can unique global solutions exist for the incompressible Navier-Stokes equations in vacuum when the initial density is only bounded and nonnegative, without a positive lower bound?
• RQ2 Is it possible to prove existence and uniqueness of solutions without requiring compatibility conditions between the initial velocity and density?
• RQ3 Does the classical smallness condition on the initial velocity in 3D still guarantee global existence when the density is not bounded away from zero?
• RQ4 Can the evolution of a viscous fluid drop in vacuum be uniquely described under minimal regularity assumptions on initial data?
• RQ5 What is the minimal regularity required for initial velocity and density to ensure well-posedness of the inhomogeneous incompressible Navier-Stokes equations?

Key findings

• Global unique solutions exist for the inhomogeneous incompressible Navier-Stokes equations in 2D for any initial data with $v_0 \in H^1_0(\Omega)$ and $\rho_0 \in L^\infty(\Omega)$, $\rho_0 \geq 0$, without requiring $\inf \rho_0 > 0$.
• In 3D, unique global solutions exist if the initial velocity satisfies the scaling-invariant smallness condition $\mu^{-2}\|v_0\|_{L^2}\|\nabla v_0\|_{L^2} \ll 1$, even with $\rho_0$ only bounded and nonnegative.
• The solution preserves the total mass and energy balance, with $\|\sqrt{\rho(t)}v(t)\|_{L^2}^2 + 2\mu\int_0^T \|\nabla v\|_{L^2}^2 \, dt \leq \|\sqrt{\rho_0}v_0\|_{L^2}^2$.
• The density remains bounded and its $L^\infty$ norm is conserved: $\operatorname{ess\,inf}_x \rho(t,x) = \operatorname{ess\,inf}_x \rho_0(x)$ and $\operatorname{ess\,sup}_x \rho(t,x) = \operatorname{ess\,sup}_x \rho_0(x)$.
• The solution is unique in the space $X_T = L^2(0,T;H^1_0) \cap L^4(0,T;L^2)$ with time derivative in $L^{4/3}(0,T;L^{3/2})$, under the smallness condition.
• The paper provides a complete answer to Lions’ question (2006, p.34) on the evolution of a viscous fluid drop in vacuum, confirming unique global existence under minimal assumptions.