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On the decay and Gevrey regularity of the solutions to the Navier-Stokes equations in general two-dimensional domains

Raphaël Danchin

TL;DR

The paper develops an elementary, energy-method framework to derive time-decay estimates for derivatives of all orders of finite-energy solutions to the 2D incompressible Navier–Stokes equations in general domains. By combining energy estimates with the Ladyzhenskaya inequality, it obtains Gevrey-type regularity and decay results that hold for small data and extend to large data via renormalization techniques, without relying on Fourier analysis. It furthermore establishes short-time Gevrey regularity for large data through a Stokes–fluctuation decomposition and demonstrates faster decay under assumed $L^2$-decay of the solution. The work provides explicit a priori bounds and decay rates for time derivatives in 2D, applicable to general domains, and highlights a unified method linking decay, regularity, and domain-general analysis.

Abstract

The present paper is devoted to the proof of time decay estimates for derivatives at any order of finite energy global solutions of the Navier-Stokes equations in general two-dimensional domains. These estimates only depend on the order of derivation and on the L2 norm of the initial data. The same elementary method just based on energy estimates and Ladyzhenskaya inequality also leads to Gevrey regularity results.

On the decay and Gevrey regularity of the solutions to the Navier-Stokes equations in general two-dimensional domains

TL;DR

The paper develops an elementary, energy-method framework to derive time-decay estimates for derivatives of all orders of finite-energy solutions to the 2D incompressible Navier–Stokes equations in general domains. By combining energy estimates with the Ladyzhenskaya inequality, it obtains Gevrey-type regularity and decay results that hold for small data and extend to large data via renormalization techniques, without relying on Fourier analysis. It furthermore establishes short-time Gevrey regularity for large data through a Stokes–fluctuation decomposition and demonstrates faster decay under assumed -decay of the solution. The work provides explicit a priori bounds and decay rates for time derivatives in 2D, applicable to general domains, and highlights a unified method linking decay, regularity, and domain-general analysis.

Abstract

The present paper is devoted to the proof of time decay estimates for derivatives at any order of finite energy global solutions of the Navier-Stokes equations in general two-dimensional domains. These estimates only depend on the order of derivation and on the L2 norm of the initial data. The same elementary method just based on energy estimates and Ladyzhenskaya inequality also leads to Gevrey regularity results.
Paper Structure (4 sections, 5 theorems, 80 equations)

This paper contains 4 sections, 5 theorems, 80 equations.

Table of Contents

  1. The case of small data
  2. The case of large data
  3. Small time Gevrey regularity in the case of large data
  4. Faster decay

Key Result

Theorem 1

Let $\Omega$ be a domain of $\mathbb{R}^d$ (with $d=2,3$) and denote by $L^2_\sigma(\Omega)$ the completion of the set ${\mathcal{V}}_\sigma$ of smooth divergence free vector-fields compactly supported in $\Omega$ for the $L^2(\mathbb{R}^d;\mathbb{R}^d)$ norm. Let $H^1_{0,\sigma}(\Omega)$ be the com

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 1.1
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1