[Paper Review] A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations

Yongzhong Sun, Chao Wang+1 more|arXiv (Cornell University)|Jan 8, 2010

Navier-Stokes equation solutions19 references15 citations

TL;DR

This paper establishes a Beale-Kato-Majda-type blow-up criterion for the 3D compressible Navier-Stokes equations, proving that strong solutions remain regular as long as the density remains bounded. The proof relies on a priori estimates involving the effective viscous flux and Gronwall's inequality under the assumption of upper-bounded density, extending previous results to allow for initial vacuum and general domains.

ABSTRACT

We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier-Stokes equations. The initial vacuum is allowed. The main ingredient of the proof is extit{a priori} estimate for an important quantity under the assumption that the density is upper bounded, whose divergence can be viewed as the effective viscous flux.

Motivation & Objective

• To establish a blow-up criterion for strong solutions of the 3D compressible Navier-Stokes equations in the presence of initial vacuum.
• To extend existing Beale-Kato-Majda-type criteria to the compressible case by linking blow-up to the upper bound of the density.
• To prove that the maximal existence time of strong solutions is controlled by the uniform boundedness of the density in $ L^∞ $.
• To develop a priori estimates for the effective viscous flux and velocity gradients under the bounded density assumption.
• To generalize previous results on blow-up criteria by incorporating the Lamé system and using Sobolev embedding and Gronwall's inequality.

Proposed method

• Derive a priori estimates for the effective viscous flux by decomposing the velocity gradient into symmetric and antisymmetric parts.
• Use the continuity equation and its derivative to control the evolution of $ \|\nabla \rho\|_{L^q} $ via energy-type estimates.
• Apply the BMO norm and logarithmic Sobolev inequality to bound $ \|\nabla v\|_{L^\infty} $ in terms of $ \|\nabla \rho\|_{L^q} $, leveraging the boundedness of $ \rho $.
• Employ the Lamé system to relate $ \nabla^2 w $ and $ \nabla^2 v $ to the velocity and density gradients.
• Use Gronwall's inequality on the $ L^q $-norm of $ \nabla \rho $, relying on the boundedness of $ \|\nabla w\|_{W^{1,q}} $ and $ \|\nabla^2 w\|_{L^q} $.
• Establish the key estimate $ \|\nabla \rho\|_{L^q} \leq C(1 + \ln(e + \|\nabla \rho\|_{L^q})) $ to close the argument via Gronwall's inequality.

Experimental results

Research questions

• RQ1 Under what conditions does a strong solution to the 3D compressible Navier-Stokes equations fail to exist globally in time?
• RQ2 Can the blow-up of solutions be characterized by the upper bound of the density rather than velocity gradient norms?
• RQ3 How does the presence of initial vacuum affect the regularity and maximal existence time of strong solutions?
• RQ4 What role does the effective viscous flux play in controlling the growth of density gradients?
• RQ5 Can a Beale-Kato-Majda-type criterion be established in the compressible case using only density bounds?

Key findings

• The maximal existence time $ T^* $ of a strong solution is finite if and only if the $ L^\infty $-norm of the density $ \rho $ becomes unbounded as $ t \to T^* $.
• The $ L^q $-norm of $ \nabla \rho $ remains uniformly bounded on $ [0, T^*) $ provided $ \|\rho\|_{L^\infty} $ is bounded, under the assumption $ \lambda < 7\mu $.
• The estimate $ \|\nabla v\|_{L^\infty} \leq C(1 + \ln(e + \|\nabla \rho\|_{L^q})) $ is crucial in controlling the velocity gradient growth.
• The effective viscous flux $ \nabla w $ satisfies $ \|\nabla w\|_{L^2(0,T;L^\infty)} \leq C $ and $ \|\nabla^2 w\|_{L^2(0,T;L^q)} \leq C $, with $ C $ depending on $ T $, $ M $, and initial data.
• The proof relies on a contradiction argument: if $ \|\rho\|_{L^\infty} < \infty $, then $ \|\nabla \rho\|_{L^q} $ remains bounded, implying $ T^* = \infty $, contradicting the assumption of finite-time blow-up.
• The result generalizes earlier criteria by replacing the $ L^1(0,T;L^\infty) $-norm of $ \nabla u $ with the $ L^\infty $-bound of $ \rho $, providing a new criterion for regularity.