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Global regularity and sharp decay rates to the 1D hypo-viscous compressible Navier-Stokes equations

Chen Liang, Zhaonan Luo, Zhaoyang Yin

Abstract

In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium. Especially, we establish the global critical regularity in the Sobolev space $H^β$ with $\frac{1}{2}<β<1$. Furthermore, by bootstrap argument, Fourier splitting method and energy method, we then establish the optimal time decay rates under the extra low-frequency smallness assumption. We find the $L^2$ energy is self-closed, which motivates us to obtain the existence of global large solutions for initial data with high regularity. By a pure energy method, we also derive the optimal time decay rates when $\frac{1}{2}\leβ<\frac{3}{4}$. We find a phenomenon that $\|(a,u)\|_{L^2}$ still decays even if the initial data does not possess $L^2$ smallness. Notably, the low-frequency smallness assumption is removed in the case with $\frac{1}{2}\leβ<\frac{3}{4}$.

Global regularity and sharp decay rates to the 1D hypo-viscous compressible Navier-Stokes equations

Abstract

In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium. Especially, we establish the global critical regularity in the Sobolev space with . Furthermore, by bootstrap argument, Fourier splitting method and energy method, we then establish the optimal time decay rates under the extra low-frequency smallness assumption. We find the energy is self-closed, which motivates us to obtain the existence of global large solutions for initial data with high regularity. By a pure energy method, we also derive the optimal time decay rates when . We find a phenomenon that still decays even if the initial data does not possess smallness. Notably, the low-frequency smallness assumption is removed in the case with .
Paper Structure (7 sections, 12 theorems, 160 equations)

This paper contains 7 sections, 12 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Notations and main results
  4. Motivations and main ideas
  5. Preliminaries
  6. Global regularity and sharp decay rates of small solutions to CNS equations
  7. Global regularity and sharp decay rates of large solutions to CNS equations

Key Result

Theorem 1.1

Let $\frac{1}{2}<\beta<1$, $s\ge\beta$ or $\beta=\frac{1}{2}$, $s>\frac{1}{2}$. 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 that for all $t>0$, there holds

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • ...and 13 more