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Global regularity and sharp decay to the 2D Hypo-Viscous compressible Navier-Stokes equations

Chen Liang, Zhaonan Luo, Zhaoyang Yin

TL;DR

This work analyzes the 2D isentropic hypo-viscous compressible Navier–Stokes equations with fractional dissipation $(-\Delta)^\beta$ for $\tfrac12\le\beta<1$. By transforming to $(a,u)$ with $\rho=a+1$, the authors prove global regularity for small initial data in $H^s$ ($s>1$) using an energy-dissipation framework and bootstrap arguments, revealing a parabolic smoothing effect. They then establish sharp long-time decay rates through a combination of improved Fourier splitting, time-weighted energy methods, and Littlewood-Paley/Besov techniques, obtaining decay $\|\Lambda^{s_1}(a,u)\|_{L^2} \lesssim (1+t)^{-(s_1+1)/(2\beta)}$ for $s_1\in[0,s]$, with matching lower bounds under appropriate conditions and extensions to $L^1$ data. The results provide a complete picture of global regularity and optimal decay for this hypo-viscous 2D compressible flow, complementing the known theory for the fully viscous and higher-dimensional cases. The methods combine rigorous energy estimates with harmonic-analysis tools to yield sharp decay in both Sobolev and Besov frameworks, highlighting the role of fractional dissipation in long-time behavior.

Abstract

In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial data are close to the constant equilibrium state in $H^s$ framework with $s>1$. Furthermore, by virtue of improved Fourier splitting method and the Littlewood-Paley decomposition theory, we then establish the optimal time decay rate for low regularity data.

Global regularity and sharp decay to the 2D Hypo-Viscous compressible Navier-Stokes equations

TL;DR

This work analyzes the 2D isentropic hypo-viscous compressible Navier–Stokes equations with fractional dissipation for . By transforming to with , the authors prove global regularity for small initial data in () using an energy-dissipation framework and bootstrap arguments, revealing a parabolic smoothing effect. They then establish sharp long-time decay rates through a combination of improved Fourier splitting, time-weighted energy methods, and Littlewood-Paley/Besov techniques, obtaining decay for , with matching lower bounds under appropriate conditions and extensions to data. The results provide a complete picture of global regularity and optimal decay for this hypo-viscous 2D compressible flow, complementing the known theory for the fully viscous and higher-dimensional cases. The methods combine rigorous energy estimates with harmonic-analysis tools to yield sharp decay in both Sobolev and Besov frameworks, highlighting the role of fractional dissipation in long-time behavior.

Abstract

In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial data are close to the constant equilibrium state in framework with . Furthermore, by virtue of improved Fourier splitting method and the Littlewood-Paley decomposition theory, we then establish the optimal time decay rate for low regularity data.
Paper Structure (6 sections, 9 theorems, 131 equations)

This paper contains 6 sections, 9 theorems, 131 equations.

Table of Contents

  1. Introduction and the main results
  2. Model and synopsis of related studies
  3. Main results
  4. Preliminaries
  5. The global regularity to the global small solution to (\ref{['eq1']})
  6. The time decay rate of the global small solution to (\ref{['eq1']})

Key Result

Theorem 1.1

Let $\frac{1}{2}\le\beta<1$, $s>1$. Let $(a,u)$ be a local strong solution of (eq1) with the initial data $(a_{0},u_{0})\in H^{s}$. There exists a small constant $\delta$ such that if then the system (eq1) admits a unique global strong solution $(a,u)\in C([0,\infty),H^{s})$. Moreover, we obtain thar for all $t>0$, there holds where $k$ is a sufficiently small constant.

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 4.1
  • proof
  • ...and 5 more