[Paper Review] Global existence and asymptotic decay of solutions to the Non-isentropic Euler-Maxwell system

Yue‐Hong Feng, Shu Wang+1 more|arXiv (Cornell University)|Feb 1, 2012

Navier-Stokes equation solutions8 references4 citations

TL;DR

This paper establishes the global existence and optimal $L^q$ time decay rates for smooth solutions to the three-dimensional non-isentropic Euler-Maxwell system near equilibrium. Using refined energy methods and Fourier multiplier techniques, it proves that density and temperature decay as $(1+t)^{-11/4}$, while velocity and magnetic fields decay as $(1+t)^{-3/2 + 3/(2q)}$, with electric field decaying as $(1+t)^{-2 + 3/(2q)}$, under small initial data assumptions in high-order Sobolev and $L^1$ norms.

ABSTRACT

The non-isentropic compressible Euler-Maxwell system is investigated in $R^3$ in the present paper, and the $L^q$ time decay rate for the global smooth solution is established. It is shown that the density and temperature of electron converge to the equilibrium states at the same rate $(1+t)^{-11/4}$ in $L^q$ norm.

Motivation & Objective

• To establish the global existence of smooth solutions to the non-isentropic Euler-Maxwell system in $\mathbb{R}^3$ near the equilibrium state $(1,0,1,0,0)$.
• To derive sharp time decay estimates for the $L^q$ norms of density, temperature, velocity, electric field, and magnetic field as $t \to \infty$.
• To analyze the asymptotic behavior of the system under small initial perturbations in Sobolev and $L^1$ spaces.
• To extend decay rate analysis to the non-isentropic case, which involves coupled energy and momentum equations, differing from the isentropic setting.

Proposed method

• Employing a refined energy method in Sobolev spaces to derive global a priori estimates for high-order derivatives of the solution.
• Utilizing a symmetrizer technique to handle the non-isentropic coupling in the Euler equations, enabling energy estimates for density, velocity, and temperature.
• Applying Fourier multiplier techniques to analyze the linearized system, particularly focusing on the third-order dissipative wave equation governing the error functions $\rho$, $\Theta$, and $\nabla \cdot u$.
• Establishing $L^p$-$L^q$ time decay estimates for the linearized operator by analyzing the $L^\infty$ behavior of Fourier transforms and solving the resulting third-order wave system.
• Using $L^2$-$L^\infty$ interpolation and nonlinear energy estimates to close the bootstrap argument and derive optimal decay rates.
• Verifying compatibility conditions and smallness assumptions in $H^s$ and $L^1$ norms to ensure global existence and decay.

Experimental results

Research questions

• RQ1 What is the optimal time decay rate for the $L^q$ norms of the density and temperature in the non-isentropic Euler-Maxwell system?
• RQ2 How does the presence of temperature dynamics affect the decay behavior compared to the isentropic case?
• RQ3 Can global smooth solutions exist for the non-isentropic Euler-Maxwell system under small initial perturbations in $H^s \cap L^1$?
• RQ4 What is the role of the third-order dissipative wave structure in the decay of the error functions $\rho$, $\Theta$, and $\nabla \cdot u$?
• RQ5 How do the decay rates of the electric and magnetic fields differ from those of the fluid variables due to the Maxwell subsystem's symmetric structure?

Key findings

• The density $n(t)$ and temperature $\theta(t)$ converge to their equilibrium states at the rate $(1+t)^{-11/4}$ in $L^q$ norm for $2 \leq q \leq \infty$.
• The velocity $u(t)$ and magnetic field $B(t)$ decay as $(1+t)^{-3/2 + 3/(2q)}$ in $L^q$ norm.
• The electric field $E(t)$ decays as $(1+t)^{-2 + 3/(2q)}$ in $L^q$ norm.
• The decay rate for $\rho = n-1$ and $\Theta = \theta - 1$ is slower than that of the velocity and fields, reflecting the influence of the third-order dissipative wave equation.
• The $L^\infty$ estimates for $\rho$ and $\Theta$ are bounded by $C(1+t)^{-11/4}$, confirming the sharpness of the $L^q$ decay result.
• The analysis reveals a distinct decay behavior for the magnetic field $B$, which depends on temperature $\Theta$, differing from the isentropic case.