[Paper Review] Long time decay for global solutions to the Navier-Stokes equations in Sobolev-Gevery spaces

Hajer Orf|arXiv (Cornell University)|Mar 7, 2019

Advanced Mathematical Physics Problems5 references4 citations

TL;DR

This paper establishes long-time decay for global solutions to the 3D incompressible Navier-Stokes equations in Sobolev-Gevery spaces, specifically proving that the $̂̂̂\dot{H}^{1/2}_{a,1}$-norm of global solutions decays to zero as time tends to infinity. Using Fourier analysis and energy estimates in a weighted Sobolev-Gevery framework, the authors extend previous results on decay in $̂̂̂\dot{H}^{1/2}$ and $̂̂̂\dot{H}^{1}_{a,\sigma}$, showing polynomial decay rates in higher-order homogeneous Sobolev norms.

ABSTRACT

In this paper, we prove that if $u\in C([0,\infty), \dot{H}^{1/2}_{a,1}(\mathbb{R}^3))$ is a global solution of 3D incompressible Navier-Stokes equations, then $\|u\|_{\dot{H}^{1/2}_{a,1}}$ decays to zero as time approaches infinity. Fourier analysis and standard techniques are used.

Motivation & Objective

• To establish the long-time decay of global solutions to the 3D incompressible Navier-Stokes equations in the critical Sobolev-Gevery space $\dot{H}^{1/2}_{a,1}(\mathbb{R}^3)$.
• To extend previous results on decay in $\dot{H}^{1/2}$ and $\dot{H}^{1}_{a,\sigma}$ to the limiting space $\dot{H}^{1/2}_{a,1}$, which incorporates exponential-type weightings via the Gevrey class structure.
• To prove the stability of global solutions under small perturbations in the $\dot{H}^{1/2}_{a,1}$-norm, ensuring persistence of global existence for nearby initial data.
• To derive quantitative decay rates for higher-order Sobolev norms, showing $\|u(t)\|_{\dot{H}^s} = o(t^{-(s-1/2)/2})$ as $t \to \infty$ for $s \geq 1/2$.

Proposed method

• Utilizes the Fourier transform and frequency localization techniques to analyze the evolution of the solution in the $\dot{H}^{1/2}_{a,1}$-norm, defined via a weighted sum over dyadic frequency blocks.
• Employs energy estimates in the $\dot{H}^{1/2}_{a,1}$-space, deriving a differential inequality involving $\|u(t)\|_{\dot{H}^{1/2}_{a,1}}^2$ and $\|u(t)\|_{\dot{H}^{3/2}_{a,1}}^2$ to control growth.
• Applies Gronwall's inequality to the difference equation for perturbed solutions, using the integrability of $\|u\|_{\dot{H}^1_{a,1}}^4$ over time to control the growth of the difference in the $\dot{H}^{1/2}_{a,1}$-norm.
• Uses induction and interpolation in homogeneous Sobolev spaces to derive decay rates for $\|u(t)\|_{\dot{H}^s}$, starting from the base case $s = 1/2$.
• Establishes local and global well-posedness in $\dot{H}^{1/2}_{a,1}$ via fixed-point arguments in a suitable function space, leveraging the structure of the nonlinear term $u \cdot \nabla u$.
• Relies on the fact that $\|u(t)\|_{\dot{H}^{1/2}} \to 0$ as $t \to \infty$ (from Gallager, Iftimie, Planchon) as a key bootstrap input for higher-order decay.

Experimental results

Research questions

• RQ1 Does the $\dot{H}^{1/2}_{a,1}$-norm of global solutions to the 3D Navier-Stokes equations decay to zero as $t \to \infty$?
• RQ2 What is the decay rate of $\|u(t)\|_{\dot{H}^s}$ for $s \geq 1/2$ in the limit $t \to \infty$?
• RQ3 Can global solutions in $\dot{H}^{1/2}_{a,1}$ be shown to be stable under small perturbations in the same space?
• RQ4 Is the maximal time of existence for solutions in $\dot{H}^{1/2}_{a,1}$ infinite if the initial data is sufficiently small?
• RQ5 How does the structure of the nonlinear term $u \cdot \nabla u$ affect the long-time behavior in the weighted Gevrey-Sobolev framework?

Key findings

• The $\dot{H}^{1/2}_{a,1}$-norm of any global solution $u$ to the 3D incompressible Navier-Stokes equations satisfies $\lim_{t \to \infty} \|u(t)\|_{\dot{H}^{1/2}_{a,1}} = 0$.
• The solution $u$ decays polynomially in higher-order homogeneous Sobolev norms: $\|u(t)\|_{\dot{H}^s} = o(t^{-(s-1/2)/2})$ as $t \to \infty$ for all $s \geq 1/2$.
• If the initial data $\|u(0)\|_{\dot{H}^{1/2}_{a,1}} < 1/C$, then the solution exists globally in time ($T^* = \infty$).
• If the maximal time $T^*$ is finite, then $\int_0^{T^*} \|u(t)\|_{\dot{H}^{3/2}_{a,1}}^2 dt = \infty$, indicating blow-up in the $L^2$-norm of the higher-order energy.
• The global solution is stable: if $\|v^0 - u(0)\|_{\dot{H}^{1/2}_{a,1}}^2 \leq \frac{1}{4} \exp\left(-\frac{C}{2} \int_0^\infty \|u(z)\|_{\dot{H}^1_{a,1}}^4 dz\right)$, then $v$ exists globally and $\|v(t) - u(t)\|_{\dot{H}^{1/2}_{a,1}}^2$ remains bounded by a decaying exponential.
• The norm $\|u(t)\|_{\dot{H}^{1/2}_{a,1}}^2$ is expressed as a weighted sum $\sum_{k=0}^\infty \frac{(2a)^k}{k!} \|u(t)\|_{\dot{H}^{(1+k)/2}}^2$, which tends to zero as $t \to \infty$.