[Paper Review] A Blow-Up Criterion for the Compressible Navier-Stokes equations

Xiangdi Huang, Zhouping Xin|arXiv (Cornell University)|Feb 16, 2009

Navier-Stokes equation solutions19 references14 citations

TL;DR

This paper establishes a blow-up criterion for strong solutions to the 3D compressible Navier-Stokes equations, proving that singularities occur when the time integral of the $L^∞$-norm of the velocity gradient diverges. The analysis relies on Moser iteration for momentum estimates and space-time $L^2$ bounds on the density gradient, allowing initial vacuum and extending the Beale-Kato-Majda criterion to compressible flows.

ABSTRACT

In this paper, we obtain a blow up criterion for strong solutions to the 3-D compressible Naveri-Stokes equations just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible flow. The key ingredients in our analysis are the a priori super-norm estimate of the momentum by a Moser-iteration and an estimate of the space-time square mean of the gradient of the density. In addition, initial vacuum is allowed in our case.

Motivation & Objective

• To establish a blow-up criterion for strong solutions of the 3D compressible Navier-Stokes equations that depends only on the velocity gradient.
• To extend the Beale-Kato-Majda criterion from incompressible to compressible flows, even in the presence of initial vacuum.
• To prove global existence of strong solutions under a condition that prevents blow-up, using regularity estimates on density and velocity.
• To demonstrate that the maximal existence time $T^*$ is determined by the integrability of $\|\nabla u\|_{L^\infty}$ over time.
• To provide a framework for extending local solutions beyond $T^*$ by showing that boundedness of $\|\nabla u\|_{L^\infty}$ implies continuation of regularity.

Proposed method

• Apply Moser iteration to derive a priori super-norm estimates for the momentum $\rho u$, controlling its $L^\infty$-norm via $L^q$-bounds.
• Establish space-time $L^2$ bounds on the gradient of density $\nabla \rho$ to control the pressure and forcing terms.
• Use the continuity equation to derive a transport equation for $\partial_i \rho$, which is then tested with $|\partial_i \rho|^{q_0-2} \partial_i \rho$ to obtain $L^{q_0}$ estimates.
• Derive a differential inequality for $\|\nabla \rho\|_{L^{q_0}}$ involving $\|\nabla u\|_{L^\infty}$ and the forcing term $F$, leading to uniform bounds.
• Employ interpolation and embedding theorems to close the estimates and ensure $u \in L^2(0,T; W^{2,q_0})$, enabling continuation of the solution beyond $T^*$.
• Use the compatibility condition and limit arguments to show that the solution can be extended past $T^*$ if $\int_0^T \|\nabla u\|_{L^\infty} dt < \infty$, contradicting maximality if the integral diverges.

Experimental results

Research questions

• RQ1 Under what condition on the velocity gradient does a strong solution to the 3D compressible Navier-Stokes equations fail to exist globally?
• RQ2 Can the blow-up criterion be formulated solely in terms of $\|\nabla u\|_{L^\infty}$, similar to the Beale-Kato-Majda criterion for incompressible flow?
• RQ3 How can regularity estimates on $\nabla \rho$ and $\rho u$ be used to control the growth of $\|\nabla u\|_{L^\infty}$?
• RQ4 Does the presence of initial vacuum affect the possibility of global existence, and can the blow-up criterion still be established?
• RQ5 Is it possible to extend a local strong solution beyond its maximal time of existence $T^*$ if the integral of $\|\nabla u\|_{L^\infty}$ remains finite?

Key findings

• The maximal existence time $T^*$ of a strong solution to the 3D compressible Navier-Stokes equations is characterized by the divergence of $\int_0^T \|\nabla u\|_{L^\infty} dt$ as $T \to T^*$.
• The blow-up criterion holds even when initial vacuum is present, which is a significant extension over previous results requiring positive initial density.
• A priori estimates via Moser iteration yield uniform bounds on $\|\rho u\|_{L^\infty}$, which are essential for controlling the momentum equation.
• The space-time $L^2$-norm of $\nabla \rho$ is bounded uniformly up to $T^*$, which allows control of the pressure and forcing terms.
• The solution can be extended beyond $T^*$ if $\int_0^T \|\nabla u\|_{L^\infty} dt < \infty$, implying that blow-up occurs precisely when this integral diverges.
• The result holds for both bounded domains and $\mathbb{R}^3$, and extends to the periodic case $T^2$ with minor modifications using Desjardin's estimate.