Source-linked AI summary
Serrin Type Criterion for the Three-Dimensional Viscous Compressible Flows
Xiangdi Huang, Jing Li, Zhouping Xin
TL;DR
The paper asks how strong or smooth solutions of the 3D compressible Navier–Stokes equations can break down. It extends Serrin-type analysis to compressible flow and shows that velocity Serrin conditions, density or divergence control, and suitable viscosity or non-vacuum assumptions yield global continuation criteria.
Problem
The paper investigates the main mechanism for possible breakdown of strong or smooth solutions to the 3D compressible Navier–Stokes equations.
Method
The paper extends Serrin’s incompressible criterion to compressible flow and analyzes density-gradient and velocity-gradient estimates under density, divergence, viscosity, and non-vacuum assumptions.
Results
The criteria establish global existence for strong or smooth solutions under Serrin’s velocity condition with density or divergence control, while suitable viscosity or absence of vacuum removes that velocity condition.
Takeaways & Limitations
The results provide Serrin-type blowup criteria for 3D viscous compressible flows and improve earlier criteria in the absence of vacuum.
Takeaways & Limitations
The stated results concern viscous barotropic flows; extension to viscous heat-conductive flows is left for future work.
Abstract
from arXiv · showhide
We extend the well-known Serrin's blowup criterion for the three-dimensional (3D) incompressible Navier-Stokes equations to the 3D viscous compressible cases. It is shown that for the Cauchy problem of the 3D compressible Navier-Stokes system in the whole space, the strong or smooth solution exists globally if the velocity satisfies the Serrin's condition and either the supernorm of the density or the $L^1(0,T;L^\infty)$-norm of the divergence of the velocity is bounded. Furthermore, in the case that either the shear viscosity coefficient is suitably large or there is no vacuum, the Serrin's condition on the velocity can be removed in this criteria.
1 Introduction
The paper studies possible breakdown of strong or smooth solutions for the 3D viscous compressible Navier–Stokes equations and extends Serrin-type criteria from incompressible flow. It establishes criteria based on velocity integrability, density control, and divergence control, with weakened requirements under additional assumptions.
- The study concerns the main mechanism for possible breakdown of strong or smooth solutions to the 3D compressible Navier–Stokes equations.
- The paper extends the incompressible Serrin criterion to 3D compressible Navier–Stokes equations.When div u ≡ 0, the stated criteria reduce to the classical incompressible Serrin criterion.
- Theorem 1.1 also applies to classical solutions of the 3D compressible viscous flows.
- The velocity Serrin condition can be replaced when the viscosity coefficients satisfy an additional condition or when the initial density is away from vacuum.Theorem 1.2 uses the additional viscosity condition, while Theorem 1.3 assumes initial density away from vacuum.
- The proof’s key step is an L∞(0,T;Lp) estimate for the density gradient, followed by bounds on the velocity gradient under divergence control.
2 Preliminaries
The preliminaries establish local strong-solution existence and collect interpolation, Beal–Kato–Majda-type, and elliptic estimates used later.
- Local existence: Local existence and uniqueness hold for a positive time even when the initial density may vanish in an open set.The result applies under the stated initial-data conditions.
- Basic inequalities: The Gagliardo–Nirenberg inequality supplies interpolation estimates for functions in H1 and intersections involving Lq and D1,r.The stated parameter ranges are p ∈ [2, 6], q ∈ (1, ∞), and r ∈ (3, ∞).
- Basic inequalities: A Beal–Kato–Majda-type inequality is introduced to estimate ∥∇u∥L∞ and density-gradient norms.It is stated for 3 < q < ∞ and ∇u ∈ L2 ∩ D1,q.
- Proof ingredients: The proof sketches a cutoff decomposition and Poisson-formula argument to estimate the gradient terms needed for the inequality.The cutoff ηδ is one near the origin, vanishes outside radius 2δ, and has gradient bounded by Cδ^-1.
3 Proof of Theorem 1.1
The proof derives a priori bounds under the Serrin condition, controls density and velocity regularity, and uses continuation to rule out finite-time breakdown.
- Energy estimates: Standard energy estimates and the momentum equation tested by ut provide the initial higher-order bounds.The estimates combine integration by parts, Cauchy’s inequality, interpolation, and Gronwall’s inequality.
- A priori estimates: Under the Serrin relation, interpolation and Gronwall estimates control the material derivatives and establish the required velocity estimates.The argument repeatedly uses the Gagliardo–Nirenberg inequality together with previously obtained bounds.
- Density and velocity regularity: The density-gradient argument uses elliptic estimates, the effective viscous flux, vorticity, and a logarithmic Gronwall quantity to bound f(t).Here G is the effective viscous flux, ω is vorticity, and f(t) ≜ e + ∥∇ρ∥Lq.
- Continuation: The proof also obtains bounded density and divergence-integrability consequences, with additional estimates available under the no-vacuum condition.The continuation step applies the local existence lemma at the limiting time.
- Estimate dependence: The constants in the estimates depend on the viscosity, pressure parameters, time horizon, and initial data, but are independent of p where stated.This dependence is recorded explicitly in the proof.
4 Proof of Theorems 1.2 and 1.3
The proofs of Theorems 1.2 and 1.3 reduce the claims to Theorem 1.1 plus auxiliary estimates obtained under alternative coefficient or vacuum assumptions.
- Proof of Theorem 1.2: Theorem 1.2 is derived from Theorem 1.1, Lemma 3.4, and Lemma 4.1.Lemma 4.1 assumes (1.6) and failure of (1.15).
- Proof of Theorem 1.2: A Hoff-type multiplication of the momentum equation by q|u|q−2u yields the estimate needed for Lemma 4.1.The estimate is closed using Lemma 3.4 and Gronwall’s inequality.
- Proof of Theorem 1.3: Theorem 1.3 follows from Theorem 1.1 and Lemma 4.2.Lemma 4.2 is stated under condition (1.7) with failure of (1.14).
- Proof of Theorem 1.3: The proof of Lemma 4.2 uses the momentum equation multiplied by 4|u|2u together with earlier estimates.The resulting argument invokes (3.28) and (3.27).