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Global well-posedness of 2-D incompressible anisitropic Navier-Stokes equations with variable density

Hammadi Abidi, Guilong Gui, Ping Zhang

Abstract

We establish the global well-posedness for two-dimensional inhomogeneous, incompressible, anisotropic Navier-Stokes systems. Two specific models are analyzed: one with partial dissipation (referred to as (AINS)) and one with only horizontal dissipation (referred to as (HINS)), under the assumption that the initial density is bounded away from zero and infinity. For the (AINS) system posed in the whole plane $\mathbb{R}^2$, we prove the existence and uniqueness of global solutions for finite-energy initial data, employing time-weighted energy estimates and a duality argument. For the (HINS) system on the domain $\mathbb{T} \times \mathbb{R}$, global well-posedness is established for sufficiently small initial velocity and sufficiently small density variation. By exploiting the anisotropic dissipation structure, employing Poincaré-type inequalities to obtain exponential decay for the oscillatory part of the velocity field, and carefully balancing the growth of the density gradient, we overcome the principal analytical challenges.

Global well-posedness of 2-D incompressible anisitropic Navier-Stokes equations with variable density

Abstract

We establish the global well-posedness for two-dimensional inhomogeneous, incompressible, anisotropic Navier-Stokes systems. Two specific models are analyzed: one with partial dissipation (referred to as (AINS)) and one with only horizontal dissipation (referred to as (HINS)), under the assumption that the initial density is bounded away from zero and infinity. For the (AINS) system posed in the whole plane , we prove the existence and uniqueness of global solutions for finite-energy initial data, employing time-weighted energy estimates and a duality argument. For the (HINS) system on the domain , global well-posedness is established for sufficiently small initial velocity and sufficiently small density variation. By exploiting the anisotropic dissipation structure, employing Poincaré-type inequalities to obtain exponential decay for the oscillatory part of the velocity field, and carefully balancing the growth of the density gradient, we overcome the principal analytical challenges.
Paper Structure (9 sections, 10 theorems, 267 equations)

This paper contains 9 sections, 10 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. The global well-posedness of the system (AINS)
  3. Time-weighted energy estimates with bounded density
  4. The proof of Theorem
  5. Global well-posedness of (HINS)
  6. Estimate of $\|(u_{\neq}, \partial_1 u_{\neq})\|_{L^{\infty}_T(L^2)}$
  7. Estimate of $\|(\omega_{\neq}, \partial_1\omega_{\neq})\|_{L^{\infty}_T(L^2)}$
  8. Proof of Theorem
  9. Preliminary

Key Result

Theorem 1.1

Let $\rho_0$ satisfy t.1 and let $u_0 \in L^2(\mathbb{R}^2)$ be a solenoidal vector field. Then the system eqns-ains-1 admits a unique global solution $(\rho, u, \nabla \Pi)$ such that and Moreover, the following estimates hold: Here and below, we always denote $D_t\buildrel\hbox{\footnotesize def}\over = \partial_t+u\cdot\nabla$ to be the material derivative.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Proposition 2.1: $L^2$ estimate of $u$
  • proof
  • Proposition 2.2: $H^1$ estimate of $u$
  • proof
  • Proposition 2.3: $\dot H^2$ estimate of $u$
  • proof
  • proof : Proof of Theorem
  • ...and 9 more