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Global well-posedness for the Navier-Stokes system in new critical mixed-norm Besov spaces

Leithold L. Aurazo-Alvarez, Wladimir Neves

TL;DR

The paper develops global well-posedness results for the incompressible Navier-Stokes equations in new mixed-norm Besov-type spaces, including both Besov-Lebesgue and Fourier-Besov-Lebesgue variants. The authors formulate a mild solution framework using the Stokes semigroup and Leray projection, then employ Bony's paraproduct and a fixed-point argument to obtain global solutions for small initial data in these spaces. Two main theorems establish existence, uniqueness, and continuous dependence of global mild solutions, expanding prior results and introducing a broader class of admissible initial data beyond the classical BMO-based spaces. The findings highlight the role of anisotropic, mixed-norm function spaces in fluid dynamics and provide new optimal spaces that accommodate directional regularity, with potential implications for low-regularity turbulence models and anisotropic flows.

Abstract

In this work, we proved the existence of a unique global mild solution of the d-dimensional incompressible Navier-Stokes equations, for small initial data in Besov type spaces based on mixed-Lebesgue spaces; namely, mixed-norm Besov-Lebesgue spaces and also mixed-norm Fourier-Besov-Lebesgue spaces. The main tools are the Bernstein's type inequalities, Bony's paraproduct to estimate the bilinear term and a fixed point scheme in order to get the well-posedness. Our results complement and cover previous and recents result on (Fourier-)Besov spaces and, for instance, provide a new class of initial data possibly not included in BMO^{-1}(R^3) but continuously included in \dot{B}^{-1}_{\infty,infty}(R^3).

Global well-posedness for the Navier-Stokes system in new critical mixed-norm Besov spaces

TL;DR

The paper develops global well-posedness results for the incompressible Navier-Stokes equations in new mixed-norm Besov-type spaces, including both Besov-Lebesgue and Fourier-Besov-Lebesgue variants. The authors formulate a mild solution framework using the Stokes semigroup and Leray projection, then employ Bony's paraproduct and a fixed-point argument to obtain global solutions for small initial data in these spaces. Two main theorems establish existence, uniqueness, and continuous dependence of global mild solutions, expanding prior results and introducing a broader class of admissible initial data beyond the classical BMO-based spaces. The findings highlight the role of anisotropic, mixed-norm function spaces in fluid dynamics and provide new optimal spaces that accommodate directional regularity, with potential implications for low-regularity turbulence models and anisotropic flows.

Abstract

In this work, we proved the existence of a unique global mild solution of the d-dimensional incompressible Navier-Stokes equations, for small initial data in Besov type spaces based on mixed-Lebesgue spaces; namely, mixed-norm Besov-Lebesgue spaces and also mixed-norm Fourier-Besov-Lebesgue spaces. The main tools are the Bernstein's type inequalities, Bony's paraproduct to estimate the bilinear term and a fixed point scheme in order to get the well-posedness. Our results complement and cover previous and recents result on (Fourier-)Besov spaces and, for instance, provide a new class of initial data possibly not included in BMO^{-1}(R^3) but continuously included in \dot{B}^{-1}_{\infty,infty}(R^3).

Paper Structure

This paper contains 13 sections, 12 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Mixed-norm Besov-Lebesgue spaces
  5. Mixed-norm Fourier-Besov-Lebesgue spaces
  6. The Stokes semigroup and Mild solutions
  7. The main theorem
  8. Some auxiliary tools and linear estimates
  9. The Bony's paraproduct
  10. An abstract lemma
  11. The linear estimates
  12. The bilinear estimates
  13. Proof of the Main Theorem

Key Result

Lemma 2.2

For $\vec{p}_{1},\vec{p}_{2},\vec{p}_{3}\in [1,\infty]^{d}$, such that $\frac{1}{\vec{p}_{3}}=\frac{1}{\vec{p}_{1}}+\frac{1}{\vec{p}_{2}}$ we have for all $g_{1}\in L^{\vec{p_{1}}}(\mathbb{R}^{d})$ and $g_{2}\in L^{\vec{p_{2}}}(\mathbb{R}^{d})$. For $\vec{p}\in[1,\infty]^{d}$ we have for all $\phi \in L^{1}(\mathbb{R}^{d})$ and $g\in L^{\vec{p}}(\mathbb{R}^{d})$. The normed space $\left(L^{\vec{

Theorems & Definitions (26)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 3.1
  • Theorem 3.2
  • ...and 16 more