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Global refined Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes equations with large density

Hammadi Abidi, Guilong Gui, Ping Zhang

Abstract

We investigate the global unique Fujita-Kato solution to the 3-D inhomogeneous incompressible Navier-Stokes equations with initial velocity $u_0$ being sufficiently small in critical spaces and with initial density being bounded from above and below. We first prove the global existence of Fujita-Kato solution to the system if we assume in addition that the initial velocity is in the critical Sobolev space. While under the additional assumptions that the initial velocity is in the critical Besov space and initial density is in a critical Besov space, we prove that the solutions are controlled by the norm of the initial data. Our results not only improve the smallness condition in the previous references for the initial velocity concerning the global Fujita-Kato solution of the system but also improve the exponential-in-time growth estimate for the solution in the paper [Abidi-Gui-Zhang, ARMA 2012] to be the uniform-in-time estimate.

Global refined Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes equations with large density

Abstract

We investigate the global unique Fujita-Kato solution to the 3-D inhomogeneous incompressible Navier-Stokes equations with initial velocity being sufficiently small in critical spaces and with initial density being bounded from above and below. We first prove the global existence of Fujita-Kato solution to the system if we assume in addition that the initial velocity is in the critical Sobolev space. While under the additional assumptions that the initial velocity is in the critical Besov space and initial density is in a critical Besov space, we prove that the solutions are controlled by the norm of the initial data. Our results not only improve the smallness condition in the previous references for the initial velocity concerning the global Fujita-Kato solution of the system but also improve the exponential-in-time growth estimate for the solution in the paper [Abidi-Gui-Zhang, ARMA 2012] to be the uniform-in-time estimate.

Paper Structure

This paper contains 5 sections, 23 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorems
  4. The proof of Theorem
  5. Tools box on Littlewood-Paley analysis and Lorentz spaces

Key Result

Theorem 1.1

Let $(\rho_0, u_0)$ satisfy $u_0\in \dot{B}^{\frac{1}{2}}_{2, 1}(\mathbb{R}^3)$ with $\mathop{\rm div}\nolimits\,u_0=0,$ and Then there exists a constant $\varepsilon_0>0$ depending only on $c_0,\,C_0$ such that if the system 1.2 has a unique global solution $(\rho, \, u,\, \nabla\Pi)$ with $\rho\in L^{\infty}(\mathbb{R}^+\times \mathbb{R}^3)$ and $u\in C([0, +\infty); \dot{B}^{\frac{1}{2}}_{2,

Theorems & Definitions (47)

  • Theorem 1.1: DW2023Zhang2020
  • Theorem 1.2: Theorem 1.3 in HSWZ2024
  • Theorem 1.3
  • Theorem 1.4: Regularity for Lipschitz flows
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 37 more