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Green's function and Large time behavior for the 1-D compressible Euler-Maxwell system

Boyu Liang, Mingying Zhong

TL;DR

This work analyzes the 1D compressible Euler-Maxwell system with relaxation by combining spectral-Green's function analysis and nonlinear energy methods. The authors decouple the linearized dynamics into a longitudinal $( ho,u_1,E_1)$ block and a transverse $(u_r,E_r,B_r)$ block, derive explicit Green's function representations and sharp decay in Fourier space, and show exponential decay for the longitudinal part while the transverse part exhibits diffusive, polynomial decay. They then prove global existence for small initial data via a robust energy framework and establish precise time-decay rates for the nonlinear system using Duhamel's principle and the Green's-function bounds, including lower bounds under low-frequency assumptions. Overall, the paper provides a rigorous characterization of the large-time behavior of solutions to the 1D Euler-Maxwell system with relaxation, highlighting the difference between exponential and polynomial decay regimes and offering a methodological template for related kinetic-electromagnetic models.

Abstract

We study Green's function and the large time behavior of the one-dimensional Euler-Maxwell System with relaxation. Firstly, we construct the Green's function of linearized system and obtain the optimal time decay rates of its solutions. And then, we obtain the global existence and the optimal time decay rates of solutions to the nonlinear system by using Green's function and a suitable energy estimate.

Green's function and Large time behavior for the 1-D compressible Euler-Maxwell system

TL;DR

This work analyzes the 1D compressible Euler-Maxwell system with relaxation by combining spectral-Green's function analysis and nonlinear energy methods. The authors decouple the linearized dynamics into a longitudinal block and a transverse block, derive explicit Green's function representations and sharp decay in Fourier space, and show exponential decay for the longitudinal part while the transverse part exhibits diffusive, polynomial decay. They then prove global existence for small initial data via a robust energy framework and establish precise time-decay rates for the nonlinear system using Duhamel's principle and the Green's-function bounds, including lower bounds under low-frequency assumptions. Overall, the paper provides a rigorous characterization of the large-time behavior of solutions to the 1D Euler-Maxwell system with relaxation, highlighting the difference between exponential and polynomial decay regimes and offering a methodological template for related kinetic-electromagnetic models.

Abstract

We study Green's function and the large time behavior of the one-dimensional Euler-Maxwell System with relaxation. Firstly, we construct the Green's function of linearized system and obtain the optimal time decay rates of its solutions. And then, we obtain the global existence and the optimal time decay rates of solutions to the nonlinear system by using Green's function and a suitable energy estimate.
Paper Structure (7 sections, 11 theorems, 143 equations)

This paper contains 7 sections, 11 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Green's Function
  3. Spectrum Analysis
  4. Decay estimate of Green's Function
  5. The Original Nonlinear Problem
  6. Energy Estimate
  7. Time Decay Rates of Nonlinear System

Key Result

Theorem 1.1

Let $N \ge 2$ and compatable holds. There exists a positive constant $\delta$ such that if then, the Cauchy problem re-em_1--re-initial has a unique global solution $(\rho,u_1,u_r,E_1,E_r,B_r)$ satisfying | (ρ,u_1,u_r,E_1,E_r,B_r) |_H^N^2 + ∫_0^t ( | (ρ,u_1,u_r)|_H^N^2 + | (E_1,E_r)|_H^N-1^2 + | ∂_x B_r|_H^N-2^2) dτ C| (ρ_0,u_10,u_r0,E_10,E_r0,B_r0)|_H^N^2, t 0.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • proof
  • proof
  • Definition 2.1
  • proof
  • proof
  • proof
  • Proposition 2.6
  • ...and 24 more