Uniform regularity of the isentropic Navier-Stokes-Maxwell system

1.   Introduction

In this paper, we consider the following isentropic compressible Navier-Stokes-Maxwell system ^[1,2]:

Here, $\rho$ is the electron density, $u$ is the velocity, $E$ and $b$ represent electronic and magnetic fields respectively. The pressure is $p: = a\rho^{\gamma}$ with constants $a > 0$ and $\gamma > 1$. $j$ is the electric current expressed by Ohm's law. The force term $j\times B$ in the Navier-Stokes equations comes from Lorentz force under a quasi-neutrality assumption of the net charge carried by the fluid. The third equation is the Ampère-Maxwell equation for an electric field E and the fourth equation is Faraday's law. The viscosity coefficients $\mu$ and $\lambda$ of the fluid satisfy $\mu > 0$ and $\lambda+\frac{2\mu}{3}\geq0$. $\epsilon$ is the dielectric constant. The Navier-Stokes-Maxwell system is a plasma physical model that describes the motion of charged particles in electromagnetic field, which can be derived from the Vlasov-Maxwell-Boltzmann system. It includes many classical models. For example, when $j = 0$, (1.1) and (1.2) reduce to the well-known isentropic compressible Navier-Stokes system, Gong-Li-Liu-Zhang ^[3] and Huang^[4] showed the local well-posedness of strong solutions.

When $\inf\rho_{0} > 0$, the problem of Navier-Stokes-Maxwell system has attracted much attention. Jiang-Li ^[5,6,7] studied the vanishing limit of dielectric constant $\epsilon_{1}$. Fan-Li-Nakamura ^[8,9,10] considered the vanishing limits of dielectric constant $\epsilon_{1}$ or the Mach number $\epsilon_{2}$. Chen-Li-Zhang ^[11] and Mi-Gao ^[12] established the long-time asymptotic behavior of the smooth solutions.

Before stating our main results, we recall the local existence of smooth solutions to the problem (1.1)–(1.6). Since the system (1.1)–(1.6) is a parabolic-hyperbolic one, the results in ^[13] imply that

Proposition 1.1. (^[13]). Let $\rho_{0}, u_{0}, E_{0}, b_{0}\in H^3$ and $\frac{1}{C_{0}}\leq\rho_{0}$, for a positive constant $C_{0}$. Then the problem $(1.1)–(1.6)$ has a unique smooth solution $(\rho, u, E, b)$ satisfying

for some $0 < T\leq\infty$.

The aim of this paper is to prove uniform regularity estimates in $(\lambda, \mu, \epsilon)$. We will prove the following

Theorem 1.1. Let $0 < \mu < 1, 0 < \lambda+\mu < 1, 0 < \epsilon < 1, 0 < \frac{1}{C_{0}}\leq\rho_{0}, \rho_{0}, u_{0}, E_{0}, b_{0}\in H^3(\mathbb{T}^3)$. Let $(\rho, u, E, b)$ be the unique local smooth solutions to the problem $(1.1)–(1.6)$. Then

hold true for some positive constants $C$ and $T_{0}\ (\leq T)$ independent of $\lambda, \mu$ and $\epsilon > 0$.

We define

We can prove

Theorem 1.2. For any $t\in[0, 1]$, it holds that

for some nondecreasing continuous functions $C_{0}(\cdot)$ and $C(\cdot)$.

It follows from (1.10) that

In the following proofs, we will use the bilinear commutator and product estimates due to Kato-Ponce ^[14]:

with $s > 0$ and $\frac1p = \frac{1}{p_{1}}+\frac{1}{q_{1}} = \frac{1}{p_{2}}+\frac{1}{q_{2}}$.

We only need to show Theorem 1.2.

2.   Proof of Theorem 1.2

First, multiplying (1.1) by $\rho^{q-1}$ and integrating the resulting equation yields

from which it follows that

thus, one can have

and

by taking $q\to +\infty$.

It follows from (1.1) that

Multiplying (2.3) by $\left(\frac{1}{\rho}\right)^{q-1}$, and integrating the resulting equation yields that

or equivalently,

from which it follows that

and

by taking $q\to+\infty$. It follows from (2.2) and (2.4) that

It is easy to verify that

which implies

Multiplying (1.3) by and $E$, (1.4) by $b$, integrating with respect to $x$ respectively, and summing up the results, then it follows that

which implies

Applying $D^3$ to (1.3) and (1.4), multiplying by $D^3E$ and $D^3b$, respectively, and summing up the results, one can observe that

which yields

Differentiating (1.3) and (1.4) with respect to t, multiplying by $E_{t}$ and $b_{t}$, respectively, and summing up the results, one can deduce that

which implies

It is clear that

and hence

It is obvious that

Similarly, applying $D^{3}$ to (2.11), multiplying the equation by $D^{3}p$ and integrating with respect to $x$, then it follows from (2.11), (1.12) and (1.13) that

where we have used the estimate ^[15]:

It is obvious that

Applying $D^2$ to (1.2), multiplying the resulting equality by $D^2 \partial_tu$ and integrating with respect to $x$, by (1.12)–(1.13) and some direct calculations, we obtain

which gives rises to

Applying $D^3$ to (1.2), multiplying the resulting equation by $D^3u$ and integrating with respect to $x$, and it follows from (1.1), (1.12) and (1.13) that

Summing (2.12) and (2.16) up, one can deduce that

Noting that the last term of LHS of (2.17) is zero, it follows from (2.15) that

On the other hand, it follows from (1.2) that

By the aid of the following estimate ^[15]:

with $\rho = f(p): = \left(\frac{p}{a}\right)^\frac1\gamma$, and

we have

Combining (2.4)–(2.10), (2.18), (2.19) with (2.21), we conclude that (1.10) holds true.

This completes the proof.

Acknowledgments

The authors would like to thank the anonymous reviewers and the editor-in-chief for their comments to improve the paper. Qingkun Xiao is partially supported by NSFC (No. 11801270).

Conflict of interest

We declare that we have no conflict of interest.