Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global Gevrey solution of 3D anisotropic Navier-Stokes system in a strip domain

Wei-Xi Li, Zhan Xu, Ping Zhang

TL;DR

The paper tackles the 3D incompressible anisotropic Navier–Stokes equations on a strip with horizontal diffusion only, where boundary terms and lack of vertical diffusion pose major regularity challenges. By enforcing Gevrey regularity in the vertical direction and developing a sophisticated Gevrey-energy framework with auxiliary norms, the authors prove global well-posedness for small Gevrey data and establish instantaneous space–time smoothing, including an enhanced Gevrey radius in the horizontal direction. A key strategy is the decomposition into a linear part and a nonlinear remainder, used across time and spatial variables to obtain refined radius estimates, notably showing a near-initial-time Gevrey radius in x_2 of order at least C√(t|ln t|). The results significantly advance understanding of anisotropic diffusion with boundaries, providing quantitative Gevrey regularity, smoothing effects, and radius refinements that are relevant for geophysical fluid models and related PDE analyses.

Abstract

We investigate the three-dimensional (3D) incompressible anisotropic Navier-Stokes system with dissipation only in the horizontal variables, posed in a strip domain. To overcome the difficulties arising from the boundary terms and the absence of vertical dissipation, we impose a Gevrey-class regularity condition in the vertical direction. For the remaining directions, we prove that the solution exhibits space-time analytic or Gevrey-class regularization. Furthermore, the solution is shown to possess an enhanced Gevrey regularity in the direction of strong diffusion, which is unconstrained by boundaries.

Global Gevrey solution of 3D anisotropic Navier-Stokes system in a strip domain

TL;DR

The paper tackles the 3D incompressible anisotropic Navier–Stokes equations on a strip with horizontal diffusion only, where boundary terms and lack of vertical diffusion pose major regularity challenges. By enforcing Gevrey regularity in the vertical direction and developing a sophisticated Gevrey-energy framework with auxiliary norms, the authors prove global well-posedness for small Gevrey data and establish instantaneous space–time smoothing, including an enhanced Gevrey radius in the horizontal direction. A key strategy is the decomposition into a linear part and a nonlinear remainder, used across time and spatial variables to obtain refined radius estimates, notably showing a near-initial-time Gevrey radius in x_2 of order at least C√(t|ln t|). The results significantly advance understanding of anisotropic diffusion with boundaries, providing quantitative Gevrey regularity, smoothing effects, and radius refinements that are relevant for geophysical fluid models and related PDE analyses.

Abstract

We investigate the three-dimensional (3D) incompressible anisotropic Navier-Stokes system with dissipation only in the horizontal variables, posed in a strip domain. To overcome the difficulties arising from the boundary terms and the absence of vertical dissipation, we impose a Gevrey-class regularity condition in the vertical direction. For the remaining directions, we prove that the solution exhibits space-time analytic or Gevrey-class regularization. Furthermore, the solution is shown to possess an enhanced Gevrey regularity in the direction of strong diffusion, which is unconstrained by boundaries.

Paper Structure

This paper contains 16 sections, 29 theorems, 343 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['thm:main1']}: global existence and uniqueness
  3. Auxiliary norms and statement of a priori estimate
  4. Estimate on pure spatial derivatives
  5. Estimate on space-time derivatives and completing the proof of Theorem \ref{['prop:wellposedness']}
  6. Proof of Theorem \ref{['thm:smoothing']}: smoothing effect for the time variable
  7. Proof of Theorem \ref{['thm:smoothing']}: smoothing effect for the variable $x_2$
  8. Technical lemmas
  9. Proof of Proposition \ref{['prop:x2']}
  10. Proof of Theorem \ref{['thm:smoothing']}: smoothing effect for the all variables
  11. Refined Gevrey radius for the variable $x_2$
  12. Decomposition of the solution and proof of Theorem \ref{['thm:radius']}
  13. Proof of Proposition \ref{['lem:uL']}: estimate on the linear part
  14. Proof of Proposition \ref{['prop:v']}: estimate on the nonlinear part
  15. Proof of some inequalities
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $\sigma \ge 1$ and let the Gevrey space $G_{\rho,\sigma,N}$ be given by Definition def:re1. Assume the initial datum $u_0 \in G_{\rho_0,\sigma, 6}$ for some $\rho_0 > 0$ with $\operatorname{div} u_0 = 0$, compatible with the boundary condition in system ANS. There exists a small constant $\varep then the anisotropic Navier-Stokes system ANS admits a unique global solution $u \in L^{\infty}([0,

Theorems & Definitions (75)

  • Definition 1.1
  • Theorem 1.2: Global well-posedness
  • Theorem 1.3: Regularization effect
  • Theorem 1.4: Refined radius estimate
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 65 more