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Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$

Zhibin Wang, Jiahong Wu, Ning Zhu

TL;DR

This work analyzes the 2D incompressible Navier–Stokes equations with dissipation acting only in the horizontal direction, modeled by $\partial_t u + \nu \Lambda_1^{2s} u + u\cdot\nabla u = -\nabla p$ on $\mathbb{R}^2$ with $0\le s<1$, placing the dynamics between viscous NS and Euler. The authors develop a triad of strategies tailored to $s$: (i) energy methods for $0\le s\le 3/4$, (ii) a coupled energy-decay framework for $3/4<s<11/12$ requiring negative horizontal regularity $\Lambda_1^{-\,\sigma}u_0\in L^2$ with $0<\sigma<1/2$, and (iii) a weighted-energy approach for $0<s<1$ using vertical weights $[x_2]^\gamma$ and $A_2$-weighted Riesz transform theory. They prove global stability and explicit decay rates in unweighted norms for the first regime, plus decay with horizontal negative regularity in the second, and extend to full range with optimal weighted decay using the $[x_2]^\gamma$ weights in the third, addressing the pressure via weighted estimates and eliminating the Leray projection bounds. The results illuminate how anisotropic and fractional dissipation can yield robust stabilization near equilibrium, with decay rates and anisotropic behavior depending on $s$ and the horizontal regularity budget. The combination of anisotropic product estimates, negative-regularity propagation, and weighted Calderón–Zygmund techniques provides a comprehensive framework for understanding stability in partially dissipative flows.

Abstract

The stability problem for the 2D Navier-Stokes equations with dissipation in only one direction on $\mathbb R^2$ is not fully understood. This dissipation is in the intermediate regime between the fully dissipative Navier-Stokes and the inviscid Euler. Navier-Stokes solutions in $\mathbb R^2$ decay algebraically in time while Euler solutions can grow rather rapidly in time. This paper solves the fundamental stability and large-time behavior problem on the anisotropic Navier-Stokes with fractional dissipation $Λ_1^{2s}$ for all $0\leq s<1$. The case $s=1$ corresponds to the standard one directional dissipation $\partial_1^2$. Different techniques are developed to treat different ranges of fractional exponents: $0\leq s\leq \frac34$, $\frac34<s<\frac{11}{12}$, and $\frac{11}{12} \leq s <1$. The final range is the most difficult case, for which we introduce the spatial polynomial $A_2$ weights and exploit the boundedness of Riesz transforms on weighted $L^2$-spaces.

Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$

TL;DR

This work analyzes the 2D incompressible Navier–Stokes equations with dissipation acting only in the horizontal direction, modeled by on with , placing the dynamics between viscous NS and Euler. The authors develop a triad of strategies tailored to : (i) energy methods for , (ii) a coupled energy-decay framework for requiring negative horizontal regularity with , and (iii) a weighted-energy approach for using vertical weights and -weighted Riesz transform theory. They prove global stability and explicit decay rates in unweighted norms for the first regime, plus decay with horizontal negative regularity in the second, and extend to full range with optimal weighted decay using the weights in the third, addressing the pressure via weighted estimates and eliminating the Leray projection bounds. The results illuminate how anisotropic and fractional dissipation can yield robust stabilization near equilibrium, with decay rates and anisotropic behavior depending on and the horizontal regularity budget. The combination of anisotropic product estimates, negative-regularity propagation, and weighted Calderón–Zygmund techniques provides a comprehensive framework for understanding stability in partially dissipative flows.

Abstract

The stability problem for the 2D Navier-Stokes equations with dissipation in only one direction on is not fully understood. This dissipation is in the intermediate regime between the fully dissipative Navier-Stokes and the inviscid Euler. Navier-Stokes solutions in decay algebraically in time while Euler solutions can grow rather rapidly in time. This paper solves the fundamental stability and large-time behavior problem on the anisotropic Navier-Stokes with fractional dissipation for all . The case corresponds to the standard one directional dissipation . Different techniques are developed to treat different ranges of fractional exponents: , , and . The final range is the most difficult case, for which we introduce the spatial polynomial weights and exploit the boundedness of Riesz transforms on weighted -spaces.
Paper Structure (23 sections, 18 theorems, 324 equations)

This paper contains 23 sections, 18 theorems, 324 equations.

Table of Contents

  1. Introduction
  2. Anisotropic Estimates and Tools
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['thm1']}
  5. Proof of Lemma \ref{['observ lem']}
  6. Proof of Theorem \ref{['thm3']}
  7. Estimate for $\|\Lambda_1^{-\sigma} u\|_{L^2}$
  8. Estimate for $\sum_{m=1}^2\|\Lambda_1^{-\sigma} \partial_m^k u\|_{L^2}$
  9. The decay of $u_1$
  10. The decay of $u_2$
  11. The decay of $\partial_2 u_1$
  12. The decay of $\partial_1 u_1$
  13. The decay of $\partial_1 u_2$
  14. Proof of Theorem \ref{['thm4']}
  15. Estimate for $\|\Lambda_1^{-\sigma} u(t)\|_{H^k}$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Consider the anisotropic Navier-Stokes system NS with viscosity coefficient $\nu>0$. Let $0 \le s \le \tfrac{3}{4}$ and assume the initial data $u_0 \in H^{k}(\mathbb{R}^2)$, $k \ge 3$, satisfies $\nabla\cdot u_0 = 0$. Then there exists a sufficiently small constant $\varepsilon>0$ such that if $\|u which satisfies the uniform bound

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4: Truncated Power Weight
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 20 more