[Paper Review] Regularity of Leray-Hopf solutions to Navier-Stokes equations (I)-Critical interior regularity in weak spaces

Jian Zhai|ArXiv.org|Nov 30, 2006

Navier-Stokes equation solutions10 references3 citations

TL;DR

This paper establishes the critical regularity of Leray-Hopf solutions to the 3D incompressible Navier-Stokes equations under a smallness condition in the weak-L^2(0,T;L^∞(R^3)) norm. By introducing a novel embedding inequality in Lorentz spaces and analyzing the vorticity equation via dyadic decomposition and maximal function estimates, the authors prove that sufficiently small initial data in this critical weak space implies global regularity, solving an open problem in [8].

ABSTRACT

We consider the interior regularity of Leray-Hopf solutions to Navier-Stokes equations on critical case L^2_w(0,T;L^\infty(R^3)). Particularly, an open problem proposed in [KK] was solved.

Motivation & Objective

• To resolve an open problem on the critical regularity of Leray-Hopf solutions to the 3D incompressible Navier-Stokes equations in weak spaces.
• To establish a smallness condition in the weak-L^2(0,T;L^∞(R^3)) norm that guarantees global regularity of solutions.
• To develop a new embedding inequality in Lorentz spaces to control nonlinear terms in the vorticity equation.
• To extend prior results on regularity in Lorentz spaces to the critical case involving weak-L^∞(R^3) and weak-L^2(0,T).

Proposed method

• Introduces a new embedding inequality in Lorentz spaces, specifically in $ L^{2,∞}(0,1) \times V(Q) $, to control nonlinear interactions.
• Uses Littlewood-Paley dyadic decomposition to localize frequency components of the velocity and vorticity fields.
• Applies maximal function estimates and Hardy-Young inequality to control the sum of dyadic frequency interactions.
• Analyzes the vorticity equation via paraproduct decomposition and estimates nonlinear terms using the weak-L^∞ norm of velocity.
• Employs the $ \|v\|_{Q} $-norm, defined via frequency-localized $ L^2 $-norms and $ L^2(0,T;H^1) $-control, to quantify regularity.
• Derives a priori estimates by bounding $ \|v\|_{Q}^2 $ in terms of $ \|u\|_{L^{2,\infty}(0,1;L^\infty)}\|v\|_{Q}^2 + \|v_0\|_{L^2}^2 $, leading to a smallness condition.

Experimental results

Research questions

• RQ1 Can the critical regularity of Leray-Hopf solutions be established under a smallness condition in the weak-L^2(0,T;L^∞(R^3)) norm?
• RQ2 Is there a new embedding inequality in Lorentz spaces that allows control of nonlinear terms in the vorticity equation?
• RQ3 Can the open problem posed in [8] regarding local regularity in the critical weak space be resolved?
• RQ4 Does the smallness of $ \|u\|_{L^{2,\infty}(0,T;L^\infty)} $ imply global regularity for Leray-Hopf solutions?
• RQ5 Can the method used in [T] be extended to the critical local case, and if not, what new tools are required?

Key findings

• The paper proves that if $ \|u\|_{L^{2,\infty}(0,T;L^\infty(\mathbb{R}^3))} \leq \epsilon $ for some small $ \epsilon > 0 $, then the Leray-Hopf solution $ u $ is regular in $ \mathbb{R}^3 \times (0,T] $, solving an open problem from [8].
• A new embedding inequality in Lorentz spaces is established: for $ f \in L^{2,\infty}(0,1) $ and $ v \in V(Q) $, the sum $ \sum_{q \geq -1} \sum_{j \sim q} \int_0^1 |f(t)| \|\Delta_j v(t)\|_{L^2} \|\nabla \Delta_q v(t)\|_{L^2} dt \leq C \|f\|_{L^{2,\infty}} \|v\|_Q^2 $.
• The regularity result is extended to the full scaling-critical class: for $ r \in [3,\infty] $, if $ \|u\|_{L^{s}_{w}(0,T;L^{r}_{w}(\mathbb{R}^3))} \leq \epsilon $ with $ \frac{2}{s} + \frac{3}{r} = 1 $, then $ u $ is regular.
• The proof establishes a priori control of the $ \|v\|_Q $-norm via $ \|v\|_Q^2 \leq C \|u\|_{L^{2,\infty}(0,1;L^\infty)} \|v\|_Q^2 + \|v_0\|_{L^2}^2 $, leading to regularity when $ \|u\|_{L^{2,\infty}(0,1;L^\infty)} < 1/(2C) $.
• The result implies that if $ |u(x,t)| \leq C / (T-t)^{1/2} $ in a right-closed time interval, then $ u $ is bounded and regular in $ \mathbb{R}^3 \times (T-R,T] $.
• The method overcomes limitations of prior approaches by using Lorentz space techniques and frequency localization, enabling treatment of the critical local case where earlier methods failed.