[Paper Review] Asymptotic stability of Landau solutions to Navier-Stokes system

Grzegorz Karch, Dominika Pilarczyk|arXiv (Cornell University)|Apr 18, 2011

Navier-Stokes equation solutions19 citations

TL;DR

This paper establishes the asymptotic stability of Landau solutions—explicit, axisymmetric, homogeneous stationary solutions of degree −1—to the 3D incompressible Navier-Stokes system under $ L^2 $-perturbations. Using a perturbation framework with a singular external force modeled by a Dirac delta, the authors prove that weak solutions starting near these solutions converge to them in $ L^2 $ as $ t \to \infty $, provided the parameter $ c $ in the solution satisfies $ |c| > 1 $ and is sufficiently large.

ABSTRACT

It is known that the three dimensional Navier-Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions, which are axisymmetric and homogeneous of degree -1. We show that these solutions are asymptotically stable under any $L^2$-perturbation.

Motivation & Objective

• To establish the global asymptotic stability of explicit stationary solutions (Landau solutions) to the 3D incompressible Navier-Stokes system.
• To analyze the long-time behavior of weak solutions perturbed from these solutions in the $ L^2 $-topology.
• To show that solutions starting near Landau solutions converge to them as $ t \to \infty $, despite the presence of a singular external force.

Proposed method

• Formalizing the Landau solutions as generalized solutions with a Dirac delta forcing term $ F = (b(c)\delta_0, 0, 0) $, where $ b(c) $ is explicitly computed.
• Constructing a weak solution framework for the perturbed Navier-Stokes system with initial data $ u_0 = v_c + w_0 $, $ w_0 \in L^2 $, and $ |c| $ large.
• Applying a semigroup approach using the Stokes operator $ \mathcal{L} $ and estimating the evolution of the perturbation $ w $ via $ L^2 $-duality and heat kernel bounds.
• Using the strong energy inequality to control the $ L^2 $-norm of the perturbation and proving its decay via time-averaging and convolution estimates.
• Employing Sobolev and Young-type inequalities to control nonlinear terms involving $ w \cdot \nabla w $, leveraging $ \|w\|_{4} \leq C \|\nabla w\|_{2}^{3/4} \|w\|_{2}^{1/4} $.
• Applying a bootstrap argument across $ L^p $-spaces for $ p \in (6/5, 2) $, refining decay estimates through repeated application of convolution and $ L^p $-decay estimates.

Experimental results

Research questions

• RQ1 Can Landau solutions to the 3D Navier-Stokes system be shown to be asymptotically stable under $ L^2 $-perturbations?
• RQ2 What is the long-time behavior of weak solutions initiated near Landau solutions, particularly in the presence of a singular forcing term?
• RQ3 Does the perturbation $ w_0 \in L^2 $ lead to a solution that decays to zero in $ L^2 $, implying convergence to the stationary Landau solution?
• RQ4 How does the decay rate of the perturbation depend on the initial $ L^p $-norm of $ w_0 $ for $ p \in (6/5, 2) $?
• RQ5 Is the asymptotic stability robust across different $ L^p $-initial data, and can decay estimates be systematically improved?

Key findings

• The Landau solutions $ (v_c, p_c) $ are asymptotically stable in $ L^2({\mathbb{R}}^3) $ under $ L^2 $-perturbations of the initial data, provided $ |c| $ is sufficiently large.
• For initial perturbations $ w_0 \in L^p({\mathbb{R}}^3) $ with $ p \in (6/5, 2) $, the solution $ w(t) $ satisfies $ \|w(t)\|_{2} \to 0 $ as $ t \to \infty $, with decay rate $ \|w(t)\|_{2} \leq C t^{-3/2(1/p - 1/2)} $ for $ p \in [3/2, 2) $.
• For $ p \in [4/3, 3/2) $, the decay rate improves to $ \|w(t)\|_{2} \leq C t^{-3/8} $, and further refinement via iterative estimates yields improved decay in this range.
• The strong energy inequality ensures that $ \|w(t)\|_{2} $ is non-increasing a.e., which is crucial for proving the $ L^2 $-decay of the time average and hence the pointwise decay.
• The singular forcing $ F = (b(c)\delta_0, 0, 0) $ is explicitly characterized, with $ b(c) = \frac{8\pi c}{3(c^2 - 1)} \left(2 + 6c^2 - 3c(c^2 - 1)\log\left(\frac{c+1}{c-1}\right)\right) $, and $ b(c) \to 0 $ as $ |c| \to \infty $.
• The proof relies on a novel use of $ L^2 $-duality and heat semigroup estimates, particularly $ \|e^{-t\mathcal{L}}\psi\|_{4/3} \leq C t^{-1/2} \|\psi\|_2 $, to control nonlinear terms in the perturbation equation.