[Paper Review] The asymptotic behavior of globally smooth solutions of bipolar non-isentropic compressible Euler-Maxwell system for plasma

Shu Wang, Yue‐Hong Feng+1 more|Feb 1, 2012

Navier-Stokes equation solutions17 references20 citations

TL;DR

This paper establishes the global existence and optimal $L^q$ time decay rates for smooth solutions to the three-dimensional bipolar non-isentropic compressible Euler-Maxwell system in plasma. It shows that total densities, temperatures, and magnetic fields decay at rate $(1+t)^{-rac{3}{2}+rac{3}{2q}}$, while density and temperature differences decay faster at $(1+t)^{-2-rac{1}{q}}$, and velocities and electric fields decay at $(1+t)^{-rac{3}{2}+rac{1}{2q}}$, revealing distinct charge transport dynamics compared to unipolar or isentropic systems.

ABSTRACT

The bipolar 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 total densities, total temperatures and magnetic field of two carriers converge to the equilibrium states at the same rate $(1+t)^{-3/2+3q/2}$ in $L^q$ norm. But, both the difference of densities and the difference of temperatures of two carriers decay at the rate $(1+t)^{-2-\frac{1}{q}}$, and the velocity and electric field decay at the rate $(1+t)^{-3/2+\frac{1}{2q}}$. This phenomenon on the charge transport shows the essential difference between the non-isentropic unipolar Euler-Maxwell and the bipolar isentropic Euler-Maxwell system.

Motivation & Objective

• To establish the global existence of smooth solutions for the bipolar non-isentropic compressible Euler-Maxwell system in $\mathbb{R}^3$.
• To analyze the asymptotic behavior of these solutions as $t \to \infty$ in terms of $L^q$ decay rates.
• To quantify the decay rates of key physical quantities: densities, temperatures, velocities, electric and magnetic fields.
• To highlight the essential differences in decay dynamics between the non-isentropic bipolar system and unipolar or isentropic counterparts.

Proposed method

• The analysis is based on energy estimates and $L^q$-type decay estimates in Sobolev spaces, using the structure of the symmetric hyperbolic system.
• A priori estimates are derived in high-order Sobolev norms to control the solution's regularity and decay.
• The system is linearized around the equilibrium state $(n_{\mu} = 1, u_{\mu} = 0, \theta_{\mu} = 1)$, and small initial data are assumed to ensure global existence.
• Interpolation inequalities and $L^2$-$L^\infty$ estimates are used to derive decay rates in $L^q$ norms for $2 \leq q \leq \infty$.
• The decay rates are obtained by combining energy decay, pointwise estimates, and the use of weighted norms in time.
• The proof relies on the symmetrizable hyperbolic structure of the system and careful control of nonlinear terms involving velocity, magnetic field, and temperature gradients.

Experimental results

Research questions

• RQ1 What are the optimal $L^q$ decay rates for globally smooth solutions of the 3D bipolar non-isentropic Euler-Maxwell system?
• RQ2 How do the decay rates of total densities, temperatures, and magnetic fields compare to those of density and temperature differences?
• RQ3 What is the role of non-isentropic effects in altering the asymptotic behavior compared to isentropic or unipolar systems?
• RQ4 How do the velocities and electric fields decay, and how does this compare to the decay of other quantities?
• RQ5 What is the mechanism behind the faster decay of density and temperature differences compared to their sums?

Key findings

• The total densities, total temperatures, and magnetic field of the two carriers decay at the rate $(1+t)^{-rac{3}{2}+rac{3}{2q}}$ in $L^q$ norm for $2 \leq q \leq \infty$.
• The difference in densities and temperatures between the two carriers decays at the faster rate $(1+t)^{-2 - \frac{1}{q}}$ in $L^q$ norm.
• The velocity and electric field decay at the rate $(1+t)^{-\frac{3}{2} + \frac{1}{2q}}$ in $L^q$ norm.
• The decay of the sum of densities and temperatures is slower than that of their difference, indicating distinct physical mechanisms in collective versus relative dynamics.
• The results reveal a fundamental difference in charge transport behavior between the non-isentropic bipolar system and the unipolar or isentropic cases.
• The asymptotic decay rates are sharp and optimal under the given smallness assumptions on initial data in $L^1 \cap H^{13}$.