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On the temporal estimates for the incompressible Navier-Stokes equations and the Hall-magnetohydrodynamic equations

Hantaek Bae, Jinwook Jung, Jaeyong Shin

TL;DR

This paper analyzes the long-time behavior of the incompressible Navier–Stokes and Hall–MHD equations in $\mathbb{R}^3$ and derives decay rates for weak solutions and for data in Lei–Lin spaces using a refined Fourier-splitting method with initial data in pseudo-measures $\mathcal{Y}^\sigma$. It proves that Leray weak solutions enjoy improved decay under low-frequency control, and extends the analysis to Hall–MHD, obtaining invariant spaces and decay for both velocity and magnetic fields. The results unify the NSE and Hall–MHD decay framework and provide explicit rates for higher-order and Lei-Lin norms, with small-data conditions yielding global strong solutions. The work advances understanding of dissipation and the large-time dynamics in parabolic PDEs with Hall effects.

Abstract

In this paper, we derive decay rates for solutions to the incompressible Navier-Stokes equations and Hall-magnetohydrodynamic equations. We first improve the decay rate of weak solutions to these equations by refining the Fourier splitting method with initial data in the space of pseudo-measures. Additionally, we investigate these equations with initial data in the Lei-Lin spaces and establish decay rates for those solutions.

On the temporal estimates for the incompressible Navier-Stokes equations and the Hall-magnetohydrodynamic equations

TL;DR

This paper analyzes the long-time behavior of the incompressible Navier–Stokes and Hall–MHD equations in and derives decay rates for weak solutions and for data in Lei–Lin spaces using a refined Fourier-splitting method with initial data in pseudo-measures . It proves that Leray weak solutions enjoy improved decay under low-frequency control, and extends the analysis to Hall–MHD, obtaining invariant spaces and decay for both velocity and magnetic fields. The results unify the NSE and Hall–MHD decay framework and provide explicit rates for higher-order and Lei-Lin norms, with small-data conditions yielding global strong solutions. The work advances understanding of dissipation and the large-time dynamics in parabolic PDEs with Hall effects.

Abstract

In this paper, we derive decay rates for solutions to the incompressible Navier-Stokes equations and Hall-magnetohydrodynamic equations. We first improve the decay rate of weak solutions to these equations by refining the Fourier splitting method with initial data in the space of pseudo-measures. Additionally, we investigate these equations with initial data in the Lei-Lin spaces and establish decay rates for those solutions.
Paper Structure (19 sections, 11 theorems, 147 equations)

This paper contains 19 sections, 11 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. The incompressible Navier-Stokes equations
  3. Hall-magnetohydrodynamic equations
  4. Preliminaries
  5. Main Lemmas
  6. Incompressible Naiver-Stokes equations
  7. Proof of Theorem \ref{['Theorem 1']}
  8. Proof of Corollary \ref{['Corollary 1']}
  9. Proof of Theorem \ref{['Theorem 2']}
  10. Hall-magnetohydrodynamic equations
  11. Proof of Theorem \ref{['Theorem 3']}
  12. Proof of Corollary \ref{['Corollary 3']}
  13. Proof of Theorem \ref{['Theorem 4']}
  14. Proof of Theorem \ref{['Theorem 5']}
  15. Bounds in $\mathcal{Y}^\sigma$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $u_{0}\in L^{2}\cap \mathcal{Y}^{\sigma}$ with $\mathop{\mathrm{div}}\nolimits u_{0}=0$ and $\sigma \in [-1,1]$. Let $u$ be a weak solution of (NSE) satisfying (NSE energy). Then, $u\in L^\infty([0,\infty);L^{2}\cap \mathcal{Y}^{\sigma})$ and $u$ decays in time as follows:

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.1
  • Theorem 1.5
  • Lemma 2.1
  • ...and 7 more