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Large friction limit of the almost pressureless Euler-Poisson system

Xin Liu

Abstract

The goal of this work is to investigate the almost pressureless Euler-Poisson (EP) system with repulsive force in the large friction limit. The leading order equations in the limit are shown to be the hyperbolic-elliptic Keller-Segel (KS) system of consumption type. Under suitable assumptions on the initial data, we establish the unique global-in-time solutions to both the EP system and the KS system by establishing the global stability in the large friction limit. In particular, no singularity forms in the asymptotic limit. Moreover, the time asymptotic behavior of the one-dimensional KS flow with vacuum is also discussed.

Large friction limit of the almost pressureless Euler-Poisson system

Abstract

The goal of this work is to investigate the almost pressureless Euler-Poisson (EP) system with repulsive force in the large friction limit. The leading order equations in the limit are shown to be the hyperbolic-elliptic Keller-Segel (KS) system of consumption type. Under suitable assumptions on the initial data, we establish the unique global-in-time solutions to both the EP system and the KS system by establishing the global stability in the large friction limit. In particular, no singularity forms in the asymptotic limit. Moreover, the time asymptotic behavior of the one-dimensional KS flow with vacuum is also discussed.
Paper Structure (18 sections, 3 theorems, 120 equations)

This paper contains 18 sections, 3 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. The systems under consideration and motivation
  3. Background and literatures
  4. Reformulation of system \ref{['sys:PLEP-limit']}
  5. Main results
  6. Spectrum analysis
  7. Uniform-in-$\varepsilon$ estimates
  8. The Keller-Segel map
  9. The $L^2$-estimate
  10. The $H^3$-estimate
  11. Local-in-time estimate
  12. Decay estimate of $\Vert \rho-M \Vert_{L^\infty}$ and $\Vert \nabla \rho \Vert_{L^4}$
  13. Global-in-time estimate
  14. Closing the a prior assumption \ref{['ass-prior:rho']}
  15. Large friction limit
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $0 < \alpha < 2$. Consider system sys:PLEP-perturbation equipped with initial data $(\rho_0, {\bf w}_0)$ satisfying initial:000--initial:002.

Theorems & Definitions (10)

  • Remark 1.1
  • Definition 1.2: Classical solution
  • Theorem 1.3: Uniform regularity
  • Remark 1.4
  • proof
  • Theorem 1.5: Large friction limit
  • proof
  • Corollary 1.6
  • proof
  • Remark 2.1