Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

Xiang Bai, Changhui Tan, Liutang Xue

Abstract

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter $α\in(0,1)$. This highlights the distinct feature of the alignment operator.

Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

Abstract

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter . This highlights the distinct feature of the alignment operator.
Paper Structure (12 sections, 20 theorems, 213 equations, 1 figure)

This paper contains 12 sections, 20 theorems, 213 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Global well-posedness
  4. Spectral analysis of the linearized system
  5. A priori estimates
  6. Proof of Theorem \ref{['thm:global']}
  7. Asymptotic behavior
  8. Green's matrix for the linearized system
  9. Rough upper bounds
  10. Refined upper bounds for the velocity field
  11. The lower bound
  12. The expression of Green's matrix

Key Result

Theorem 1.1

Let $s> \frac{N}{2}+1$. Consider the Euler-alignment system eq.EAS with pressure eq:pressure, alignment interaction eq:alignment with $0<\alpha\leqslant 1$, and initial data $(\rho_0-1,u_0)\in H^s(\mathbb{R}^N)$. There exists a small constant $\varepsilon>0$ such that if then the Euler-alignment system eq.EAS has a global unique strong solution $(\rho,u)$ such that

Figures (1)

  • Figure 1: The optimal decay rates.

Theorems & Definitions (36)

  • Theorem 1.1: Global well-posedness
  • Remark 1.1
  • Theorem 1.2: Asymptotic behavior
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Commutator estimates
  • Lemma 2.4: Composition estimates
  • proof : Proof of Lemma \ref{['Lem:composite']}
  • ...and 26 more