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Gevrey regularity solution for initial data in Triebel-Lizorkin-Lorentz spaces via single norm defined by nonlinearity of frequency

Qixiang Yang, Hongwei Li

TL;DR

The paper advances the theory of Navier-Stokes regularity by introducing a critical single-norm iterative space ${^{m'}_{m} \dot{F}}^{\frac{n}{p}-1,q}_{p,r}$ built from frequency decomposition and leveraging Fefferman-Stein tools. It proves global well-posedness in the critical Triebel-Lizorkin-Lorentz setting and establishes Gevrey regularity of mild solutions via the Fourier multiplier $e^{(-t\Delta)^\gamma}$, with $0<\gamma<1$. This framework encompasses and extends Besov-Lorentz and Triebel-Lizorkin spaces, yielding spatial analyticity and unifying gradient estimates of arbitrary order. By combining Meyer wavelet characterizations with sharp nonlinear estimates, the work provides a robust approach to control large-value point distributions and to propagate regularity globally in time for small initial data. The results broaden the landscape of critical-space well-posedness and Gevrey regularity beyond previous Besov-Lorentz or $BMO^{-1}$-type settings, offering a versatile tool for dissipative PDEs in a broad function-space context.

Abstract

The properties of solutions to Navier-Stokes equations, including well-posedness and Gevrey regularity, are a class of highly interesting problems. We are inspired by the location result on Triebel-Lizorkin-Lorentz space of Hobus and Saal in 2019. In order to overcome the difficulties they encountered when dealing with global well-posedness, we introduce the single norm iterative space ${^{m'}_{m} \dot{F}}^{\frac{n}{p}-1,q}_{p,r}$ and utilize tools such as the Fefferman-Stein inequality to investigate the properties of our iterative spaces. As a result, we establish the global well-posedness of Navier-Stokes equations in critical Triebel-Lizorkin-Lorentz space and obtain the Gevrey regularity of the mild solution. Regarding that there're many regularity studies focused on Besov spaces, such as Bae-Biswas-Tadmor(2012) and Liu-Zhang (2024), our Triebel-Lizorkin-Lorentz spaces contain more general initial value spaces, including part of Besov spaces and all of Triebel-Lizorkin spaces, etc.. Furthermore, compared with Germain-Pavlović-Staffilani (2007), our Gevrey estimation also implies spatial analyticity and is more convenient to unify the estimates of gradient of any order.

Gevrey regularity solution for initial data in Triebel-Lizorkin-Lorentz spaces via single norm defined by nonlinearity of frequency

TL;DR

The paper advances the theory of Navier-Stokes regularity by introducing a critical single-norm iterative space built from frequency decomposition and leveraging Fefferman-Stein tools. It proves global well-posedness in the critical Triebel-Lizorkin-Lorentz setting and establishes Gevrey regularity of mild solutions via the Fourier multiplier , with . This framework encompasses and extends Besov-Lorentz and Triebel-Lizorkin spaces, yielding spatial analyticity and unifying gradient estimates of arbitrary order. By combining Meyer wavelet characterizations with sharp nonlinear estimates, the work provides a robust approach to control large-value point distributions and to propagate regularity globally in time for small initial data. The results broaden the landscape of critical-space well-posedness and Gevrey regularity beyond previous Besov-Lorentz or -type settings, offering a versatile tool for dissipative PDEs in a broad function-space context.

Abstract

The properties of solutions to Navier-Stokes equations, including well-posedness and Gevrey regularity, are a class of highly interesting problems. We are inspired by the location result on Triebel-Lizorkin-Lorentz space of Hobus and Saal in 2019. In order to overcome the difficulties they encountered when dealing with global well-posedness, we introduce the single norm iterative space and utilize tools such as the Fefferman-Stein inequality to investigate the properties of our iterative spaces. As a result, we establish the global well-posedness of Navier-Stokes equations in critical Triebel-Lizorkin-Lorentz space and obtain the Gevrey regularity of the mild solution. Regarding that there're many regularity studies focused on Besov spaces, such as Bae-Biswas-Tadmor(2012) and Liu-Zhang (2024), our Triebel-Lizorkin-Lorentz spaces contain more general initial value spaces, including part of Besov spaces and all of Triebel-Lizorkin spaces, etc.. Furthermore, compared with Germain-Pavlović-Staffilani (2007), our Gevrey estimation also implies spatial analyticity and is more convenient to unify the estimates of gradient of any order.

Paper Structure

This paper contains 12 sections, 12 theorems, 101 equations.

Table of Contents

  1. Motivations and Main Theorem
  2. Preliminaries
  3. Triebel-Lizorkin-Lorentz spaces and Meyer wavelets
  4. Critical work space
  5. Basic inequalities
  6. Properties of ${ ^{m'}_{m} \dot{F}}^{\frac{n}{p}-1, q}_{p, r}$ and its connection with $\dot{F}^{\frac{n}{p}-1, q}_{p, r}$
  7. Properties of work space
  8. Boundedness of $e^{(-t\Delta)^\gamma}e^{t\Delta}$
  9. Proof of theorem \ref{['mthmain']} (ii)
  10. The priori estimates of nonlinear terms
  11. The proof of theorem \ref{['bili']}(i)
  12. The proof of theorem \ref{['bili']}(ii)

Key Result

Theorem 1.1

(i) Given $1<p,r<\infty$, $m>1$, $1\leq q< \infty$ and $0\leq m'<\frac{1}{2}$, resp., $q=\infty$ and $0<m'<\frac{1}{2}$. (NS) have a unique mild solution in $({^{m'}_{m} \dot{F}}^{\frac{n}{p}-1,q}_{p,r} )^n$ for all initial data $f(x)$ with $\nabla \cdot f =0$ and $\|f\|_{(\dot{F}^{\frac{n}{p}-1,q}_ or (NS) have a unique solution $u(t,x)$ such that $e^{(-t\Delta)^\gamma}u\in({ ^{m'}_{m} \dot{F}}^

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 9 more