[Paper Review] Blow-up of critical norms for the 3-D Navier-Stokes equations

Wendong Wang, Zhifei Zhang|arXiv (Cornell University)|Oct 9, 2015

Navier-Stokes equation solutions16 references3 citations

TL;DR

This paper establishes a new interior regularity criterion for the 3D Navier-Stokes equations based on the behavior of a single velocity component, proving that if the vertical component $ u_3 $ is bounded in a critical Besov space and the horizontal component $ u_h $ is bounded in $ BMO^{-1} $ with $ u_h(T) o VMO^{-1} $, then smooth solutions can be extended beyond time $ T $. The result improves prior criteria by allowing one component to be large in a scaling-invariant norm while still ensuring global regularity via blow-up analysis and backward uniqueness methods.

ABSTRACT

Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in $\R^3 imes [0,T)$. It was proved that if $u_3\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$ for $3

Motivation & Objective

• To address the open problem of whether smooth solutions to the 3D Navier-Stokes equations can be extended beyond a finite time $ T $ when the solution remains bounded in a scaling-invariant space.
• To generalize the recent result by Gallagher, Koch, and Planchon, which required the full velocity field $ u $ to be bounded in $ \dot{B}^{-1+3/p}_{p,q} $, by relaxing the condition to only one component.
• To establish a new interior regularity criterion based on the vertical velocity component $ u_3 $, allowing the horizontal component $ u_h $ to be large in $ BMO^{-1} $, provided $ u_h(T) \in VMO^{-1} $.
• To prove that blow-up of the solution cannot occur if the vertical component $ u_3 $ remains small in a local scaling-invariant norm, even when $ u_h $ is large.
• To develop a refined blow-up analysis and backward uniqueness argument that leverages the structure of the limiting profile when $ u_3 $ vanishes in the blow-up limit.

Proposed method

• Introduces a new scaling-invariant quantity $ G(f,p,q;r) = r^{1-3/p-2/q}\|f\|_{L^q_t L^p_x(Q_r)} $ to measure local integrability of velocity components.
• Applies blow-up analysis to study the behavior of solutions near a potential singular point, constructing a limiting profile $ v $ that satisfies a reduced system with $ v_3 = 0 $.
• Uses the backward uniqueness method of Escauriaza, Seregin, and Šverák to rule out nontrivial singularities under the new criteria.
• Derives a system for the horizontal velocity $ v_h $ in the blow-up limit: $ \partial_t v_h - \Delta v_h + v_h \cdot \nabla_h v_h + \nabla_h \pi = 0 $, with $ \partial_{x_3}\pi = 0 $, which allows for vorticity-based estimates.
• Employs elliptic estimates and energy-type inequalities on the vorticity $ w_h = \partial_1 v_2 - \partial_2 v_1 $ and the vertical derivative $ d = \partial_3 v_h $ to control $ \|\nabla v_h\|_{L^\infty_t L^2_x} $.
• Uses cutoff functions and Sobolev embedding to control nonlinear terms in the energy estimates, leading to uniform bounds in smaller parabolic balls.

Experimental results

Research questions

• RQ1 Can the global regularity of 3D Navier-Stokes solutions be guaranteed when only one component, $ u_3 $, is bounded in a critical Besov space, while the horizontal component $ u_h $ is bounded in $ BMO^{-1} $?
• RQ2 Does the condition $ u_h(T) \in VMO^{-1} $, combined with $ u_3 \in L^\infty(0,T;\dot{B}^{-1+3/p}_{p,q}) $, suffice to extend smooth solutions beyond time $ T $?
• RQ3 Can a new interior regularity criterion be established that allows one component to be large in a scaling-invariant norm while still ensuring regularity?
• RQ4 What structural properties emerge in the blow-up limit when $ u_3 $ vanishes, and can these be used to rule out singularities?
• RQ5 Is it possible to improve the classical $ L^\infty(0,T;L^3) $ criterion by allowing $ u_h \in BMO^{-1} $ and $ u_3 \in \dot{B}^{-1+3/p}_{p,q} $?

Key findings

• If $ u_3 \in L^\infty(0,T;\dot{B}^{-1+3/p}_{p,q}) $ and $ u_h \in L^\infty(0,T;BMO^{-1}) $ with $ u_h(T) \in VMO^{-1} $, then the solution $ u $ can be extended beyond $ T $, improving upon the result of Gallagher, Koch, and Planchon.
• The blow-up limit $ v $ of a potential singular solution satisfies $ v_3 = 0 $ and the reduced system $ \partial_t v_h - \Delta v_h + v_h \cdot \nabla_h v_h + \nabla_h \pi = 0 $, $ \partial_{x_3}\pi = 0 $, which is structurally similar to the 2D Navier-Stokes system.
• The vorticity $ w_h $ of the horizontal component satisfies $ \partial_t w_h - \Delta w_h + v_h \cdot \nabla_h w_h = 0 $, enabling $ L^2 $-based energy estimates that control $ \|w_h\|_{L^\infty_t L^2_x} $ in a smaller parabolic ball.
• The vertical derivative $ d = \partial_3 v_h $ satisfies a transport-diffusion equation that allows control of $ \|d\|_{L^\infty_t L^2_x} $ via energy estimates, leading to $ \|\nabla v_h\|_{L^\infty_t L^2_x} \leq C(M) $ in $ Q_{1/4} $.
• The result implies $ u \in C_w([0,T]; BMO^{-1}) $, so $ u(T) $ is well-defined in $ VMO^{-1} $, and the solution remains regular due to the smallness of $ u_3 $ in a localized scaling-invariant norm.
• The new criterion allows for two components to be large in their respective scaling-invariant norms, as long as $ u_3 $ is small in $ L^1(Q_1) $, which is a significant relaxation compared to classical interior regularity conditions.