Hydrodynamic limit of the Boltzmann-Monge-Ampere system

TL;DR

This paper studies the transition from a kinetic plasma model to fluid dynamics, proving convergence to the Euler equations in a specific regime using entropy methods.

Contribution

It establishes the hydrodynamic limit of the Boltzmann-Monge-Ampere system in the quasineutral regime, a novel result in plasma physics modeling.

Findings

Proves convergence of the system to Euler equations

Uses the relative entropy method for analysis

Provides rigorous mathematical justification

Abstract

In this paper we investigate the hydrodynamic limit of the Boltzmann-Monge-Ampere system in the so-called quasineutral regime. We prove the convergence of the Boltzmann-Monge-Ampere system to the Euler equation by using the relative entropy method.

Equations

$${\partial }_t{f}^{\epsilon }+\xi .{\nabla }_x{f}^{\epsilon }+{\nabla }_x{\phi }^{\epsilon }.{\nabla }_{\xi }{f}^{\epsilon }=$$

$$\text{mbox}det\left({\mathbb{I}}_d+{\epsilon }^2{D}^2{\phi }^{\epsilon }\right)=$$

$${\rho }^{\epsilon }(t,x)={\int }_{{\mathbb{R}}^d}{f}^{\epsilon }(t,x,\xi )d\xi$$

$$Q(f^{\epsilon },f^{\epsilon })(t,x,\xi )=\int {\int }_{S_{+}^{d-1}\times {\mathbb{R}}^d}\left({\left(f^{ϵ}\right)}^{\prime }{\left(f_1^{\epsilon }\right)}^{\prime }-f^{\epsilon }{f}_1^{\epsilon }\right)b\left(\xi -{\xi }_1,\sigma \right)d\sigma d{\xi }_1,$$

$${\xi }^{\prime }=\frac{\xi +{\xi }_1}{2}+\frac{\xi -{\xi }_1}{2}\sigma ,{\xi }_1^{\prime }=\frac{\xi +{\xi }_1}{2}-\frac{\xi -{\xi }_1}{2}\sigma .$$

$$\text{mbox}det\left({\mathbb{I}}_d+{\epsilon }^2{D}^2{\phi }^{\epsilon }\right)=1+{\epsilon }^2△{\phi }^{\epsilon }+O\left({\epsilon }^4\right).$$

$${\partial }_t{f}^{\epsilon }+\xi .{\nabla }_x{f}^{\epsilon }+{\nabla }_x{\phi }^{\epsilon }.{\nabla }_{\xi }{f}^{\epsilon }$$

$${\epsilon }^2△{\phi }^{\epsilon }$$

$${\int }_{{\mathbb{R}}^d}Q(f^{\epsilon },f^{\epsilon })d\xi ={\int }_{{\mathbb{R}}^d}{\xi }_{i}Q(f^{\epsilon },f^{\epsilon })d\xi ={\int }_{{\mathbb{R}}^d}{\left|\xi \right|}^{2}Q(f^{\epsilon },f^{\epsilon })d\xi =0,\text{mbox}i=1,2,...,d.$$

$$\frac{1}{2}{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{T}}^d}{\left|\xi \right|}^2{f}^{\epsilon }\left(t,x,\xi \right)dxd\xi +\frac{\epsilon }{2}{\int }_{{\mathbb{T}}^d}{\left|\nabla {\varphi }^{\epsilon }\left(t,x\right)\right|}^{2}dx=E_0$$

$$J^{\epsilon }\left(t,x\right)={\int }_{{\mathbb{R}}^d}\xi f^{\epsilon }\left(t,x,\xi \right)d\xi .$$

$${\partial }_t{\rho }^{\epsilon }+\nabla .J^{\epsilon }=0$$

$${\partial }_t{J}^{\epsilon }+{\nabla }_x.{\int }_{{\mathbb{R}}^d}\left(\xi \otimes \xi \right)f^{\epsilon }d\xi +\nabla {\varphi }^{\epsilon }+\frac{\epsilon }{2}\nabla \left({\left|\nabla {\varphi }^{\epsilon }\right|}^2\right)-\epsilon \nabla .\left(\nabla {\varphi }^{\epsilon }\otimes \nabla {\varphi }^{\epsilon }\right)=0.$$

$$\nabla .u=0,t>0,x\in {\mathbb{T}}^d$$

$${\partial }_{t}u+\left(u.\nabla \right)u+\nabla p=0$$

$$u\left(0,x\right)=u_0\left(x\right)\in {\mathcal{H}}^s,$$

$${\int }_{{\mathbb{R}}^d}{f}_0^{\epsilon }\left(x,\xi \right)d\xi =1+o\left({\epsilon }^{\frac{1}{2}}\right),\text{ as }\epsilon \to 0,$$

$$\frac{1}{2}{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{T}}^d}{\left|\xi -u_0\left(x\right)\right|}^2{f}_0^{\epsilon }\left(x,\xi \right)dxd\xi +\frac{\epsilon }{2}{\int }_{{\mathbb{T}}^d}{\left|\nabla {\varphi }^2\left(0,x\right)\right|}^{2}dx\to 0\text{ as }\epsilon \to 0.$$

$$H^{\epsilon }\left(t\right)=\frac{1}{2}{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{T}}^d}{\left|\xi -u\left(x\right)\right|}^2{f}^{\epsilon }\left(t,x,\xi \right)dxd\xi +\frac{\epsilon }{2}{\int }_{{\mathbb{T}}^d}{\left|\nabla {\varphi }^2\left(t,x\right)\right|}^{2}dx.$$

$$\det \left(I+\epsilon D^2{\varphi }^{\epsilon }\right)=1+\epsilon △{\varphi }^{\epsilon }+{\epsilon }^2\det D^2{\varphi }^{\epsilon }.$$

$$\int \left({\rho }^{\epsilon }\left(t,x\right)-1\right)\eta \left(x\right)dx$$

$$=\int \left(\epsilon △{\varphi }^{\epsilon }+{\epsilon }^2\det D^2{\varphi }^{\epsilon }\right)\eta \left(x\right)dx.$$

$$\det D^2{\varphi }^{\epsilon }=\frac{1}{2}\text{tr}\left(\left(\text{cof}{D}^2{\varphi }^{\epsilon }\right)D^2{\varphi }^{\epsilon }\right)=\frac{1}{2}\text{div}\left(\left(\text{cof}{D}^2{\varphi }^{\epsilon }\right)\nabla {\varphi }^{\epsilon }\right),$$

$$\int \left({\rho }^{\epsilon }\left(t,x\right)-1\right)\eta \left(x\right)dx$$

$$=\epsilon \int \nabla {\varphi }^{\epsilon }\nabla \eta \left(x\right)dx-\frac{{\epsilon }^2}{2}\int \left(\text{cof}{D}^2{\varphi }^{\epsilon }\right)\nabla {\varphi }^{\epsilon }.\nabla \eta \left(x\right)dx.$$

$$\left|\int \left({\rho }^{\epsilon }\left(t,x\right)-1\right)\right|\eta \left(x\right)dx$$

$$+\frac{{\epsilon }^2}{2}{\Vert \text{cof}{D}^2{\varphi }^{\epsilon }\Vert }_{L^2}{\Vert \nabla {\varphi }^{\epsilon }\Vert }_{L^2}{\Vert \nabla \eta \Vert }_{L^2}.$$

$${\Vert \text{cof}{D}^2{\varphi }^{\epsilon }\Vert }_{L^2}\lesssim {\epsilon }^{-\frac{1}{2}},$$

$$\left|\int \left({\rho }^{\epsilon }\left(t,x\right)-1\right)\eta \left(x\right)dx\right|$$

$$\le C_0{\epsilon }^{\text{nicefrac}12}{\Vert \nabla \eta \Vert }_{L^2}+C.\epsilon {\Vert \nabla \eta \Vert }_{L^2}$$

$$\le {\epsilon }^{\text{nicefrac}12}\left(C_0+C{\epsilon }^{\text{nicefrac}12}\right){\Vert \nabla \eta \Vert }_{L^2}$$

$$\int \left|J^{\epsilon }\left(t,x\right)\right|dx\le {\left(\int \int {\left|\xi \right|}^2{f}^{\epsilon }\left(t,x,\xi \right)dxd\xi \right)}^{\text{nicefrac}12}{\left(\int \int f^{\epsilon }\left(t,x,\xi \right)dxd\xi \right)}^{\text{nicefrac}12}\le C.$$

$$K\left({\rho }^{\epsilon },J^{\epsilon }\right)$$

$$=\underset{b}{\sup }{\int }_0^T\left\{-\frac{1}{2}{\left|b\left(t,x\right)\right|}^2{\rho }^{\epsilon }\left(t,x\right)+b\left(t,x\right)J^{\epsilon }\left(t,x\right)\right\}z\left(t\right)dt.$$

$${\int }_0^{T}z\left(t\right)\left(\int {\left|J\left(t,x\right)\right|}^{2}dx\right)dt\le C{\int }_0^{T}z\left(t\right)dt,$$

Full text

1. Introduction and main results

The goal of this article is to study the hydrodynamical limit of the Boltzman-Monge-Ampere system (BMA)

[TABLE]

where $\mathbb{I}_{d}$ is the identity matrix and

[TABLE]

and $f^{\varepsilon}(t,x,\xi)\geq 0$ is the electronic density at time $t\geq 0$ point $x\in\left[0,1\right]^{d}=\mathbb{T}^{d},$ and with a velocity $\xi\in\mathbb{R}^{d}.$ The periodic electric potential $\varphi^{\varepsilon}$ is coupled with $f^{\varepsilon}$ through the nonlinear Monge-Ampere equation (1.2). The quantities $\varepsilon>0$ and $Q(f^{\varepsilon},f^{\varepsilon})$ denote respectively the vacum electric permitivity and the Boltzman collision integral. This latter, is given by (see [3,9])

[TABLE]

where the terms $f_{1}^{\varepsilon},$ $\left(f^{\varepsilon}\right)^{\prime}$ and $\left(f_{1}^{\varepsilon}\right)^{\prime}$ defines, respectively the values $f^{\varepsilon}(t,x,\xi_{1}),$ $f^{\varepsilon}(t,x,\xi^{\prime})$ and $f^{\varepsilon}(t,x,\xi_{1}^{\prime})$ with $\xi^{\prime}$ and $\xi_{1}^{\prime}$ given in terms of $\xi$, $\xi_{1}\in\mathbb{R}^{d},$ and $\sigma\in S_{+}^{d-1}=\left\{\sigma\in S^{d-1}/\sigma.\xi\geq\sigma.\xi_{1}\right\}$ by

[TABLE]

By linearising the determinant about the identity matrix $\mathbb{I}_{d},$ one get

[TABLE]

It follows that the BMA system is a fully nonlinear version of the Vlasov-Poisson-Boltzman (VPB) system defined by

[TABLE]

This latter, has been interested many authors. In [5] DiPerna and Lions showed the existence of renormalized solution. Desvilletes and Dolbeault [7] are interested to the long-time behavior of the weak solutions of the VPB system for the initial boundary problem. In [10] Guo established the global existence of smooth solutions to the VPB system in periodic boundary condition case. For more references for this subject, Boltzmann equation or Vlasov–Poisson system, one can see [1-7, 9–14].

In [11] L. Hsiao and al. studied the convergence of the VPB system to the Incompressible Euler Equations. If one consider the case $Q(f^{\varepsilon},f^{\varepsilon})=0$, we obtain the Vlasov-Monge-Ampère(VMA). This problem, was been considered by Y. Bernier and Grégoire[2]. They showed that weak solution of VMA converge to a solution of the incompressible Euler equations when the parameter $\varepsilon$ goes to $0.$

This work aims to extend these efforts to study such systems.

First, Note that

[TABLE]

and the conservation of total energy

[TABLE]

where

[TABLE]

From the conservation of mass and momentum, it follows that

[TABLE]

and

[TABLE]

Let us consider the periodic boundary problem of Euler equations to the incompressible fluid

[TABLE]

where the function space $\mathcal{H}^{s}$ is given by $\mathcal{H}^{s}=\left\{u\in H^{s}\left(\mathbb{T}^{d}\right),\>\nabla.u=0\right\}.$

We have the following result.

Theorem**.**

Let $0<T<T^{*}$ and $u_{0}$ in $\mathcal{H}^{s}\left(s>1+\frac{d}{2}\right),$ $\mathbb{Z}^{d}$ periodic in $x$. Assume that $f_{0}^{\varepsilon}\left(x,\xi\right)\geq 0$ to be smooth, $\mathbb{Z}^{d}$ periodic in $x$, and $f_{0}^{\varepsilon}$ decays fast as $\xi\rightarrow\infty.$ In addition, we assume that

[TABLE]

in the strong sense of the space $H^{-1}\left(\mathbb{T}^{d}\right)$ and

[TABLE]

Let $f^{\varepsilon}$ be any nonnegative smooth solution of (1.1)-(1.2). Then, up to the extraction of a subsequence, the current $J^{\varepsilon}$ converges weakly to the unique solution $u\left(x,t\right)$ of the Euler equations (1.9)-(1.10)-(1.11). Moreover, the divergence free part of $f$ converges to $u$ in $L^{\infty}\left(\left[0,T\right],L^{2}\left(\mathbb{T}^{d}\right)\right).$

2. Proof of the theorem

First introduce the modulated energy functional

[TABLE]

In the squel we need the following two Lemmas

Lemma 1**.**

Under the hypothesis of the above theorem ,we have up to the extraction of a sequence, $\rho^{\varepsilon}$ converges to 1 in $C^{0}\left(\left[0,T\right],\mathcal{D}^{\prime}\left(\mathbb{T}^{d}\right)\right),$ the current $J^{\varepsilon}$ converges to $J$ in $L^{\infty}\left(\left[0,T\right],\mathcal{D}^{\prime}\left(\mathbb{T}^{d}\right)\right),$ $J\in L^{\infty}\left(\left[0,T\right],L^{2}\left(\mathbb{T}^{d}\right)\right),$ and the divergence free parts of $J^{\varepsilon}$ converges to $J$ in $C^{0}\left(\left[0,T\right],\mathcal{D}^{\prime}\left(\mathbb{T}^{d}\right)\right).$

Proof.

we take $d=2,$ and we notice that

[TABLE]

We first show that $\rho^{\varepsilon}\rightarrow 1$ in $C^{0}\left(\left[0,T\right],\mathcal{D}^{\prime}\left(\mathbb{T}^{d}\right)\right).$ In fact, for $\eta\in C_{0}^{\infty}\left(\mathbb{T}^{d}\right),$ we get

[TABLE]

But

[TABLE]

it follows by integrating by parts that

[TABLE]

Thus, by the Hölder inequality one has

[TABLE]

Recall that from regularity result of Monge-Ampère equation we have [8]

[TABLE]

So, by the conservation of the energy, one deduce that

[TABLE]

∎

By the total energy equality (1.6) we have

[TABLE]

Thus $J^{\varepsilon}$ is bounded in $L^{\infty}\left(\left[0,T\right],L^{1}\left(\mathbb{T}^{d}\right)\right).$ Up to extracting a subsequence, we can assume that $J^{\varepsilon}$ has a limit $J$ in the sens of (Radon) measures on $\left[0,T\right]\times\nicefrac{{\mathbb{R}^{d}}}{{\mathbb{Z}^{d}}}=\mathbb{T}^{d}.$ Let us define as in [11], for each non-negative function $z\left(t\right)\in C^{0}\left(\left[0,T\right]\right),$ the convex functional of a (Radon) measure

[TABLE]

where $b$ belongs to the space of all continuous functions from $\left[0,T\right]\times\mathbb{T}^{d}$ to $\mathbb{R}^{d}.$ From (2.1) and since the functional $K$ is lower semi-continuous with respect to the convergence of measure, it follows that

[TABLE]

which means that $J\in L^{\infty}\left(\left[0,T\right],L^{2}\left(\mathbb{T}^{d}\right)\right).$

From (1.7) and (1.8) one write

[TABLE]

thus

[TABLE]

For $\eta\in C_{0}^{\infty}\left(\mathbb{T}^{d}\right),$ we have

[TABLE]

thus $J$ is divergence free in $x$ in the sense of distribution.

By (1.8), we deduce that $\partial_{t}J$ is bounded in $L^{\infty}\left(\left[0,T\right],D^{\prime}\left(\mathbb{T}^{d}\right)\right).$ So, we obtain that up to the exraction of a subsequance, $J\in C^{0}\left(\left[0,T\right],L^{2}\left(\mathbb{T}^{d}\right)-w\right).$

In the same way , we can show that the divergence -free part of $J^{\varepsilon}$ converges to $J$ in $C^{0}\left(\left[0,T\right],D^{\prime}\left(\mathbb{T}^{d}\right)\right).$ Since $J^{\varepsilon}$ converges to $J$, it remains to show that $J=u$ in $L^{\infty}\left(\left[0,T\right],L^{2}\left(\mathbb{T}^{d}\right)\right).$ For this, it suffies to use the next Lemma.

Lemma 2**.**

[11]Let $u$ be the unique solution of the Euler equations (1.9)-(1.10) with initial datum and $u_{0}$ and the hypotheses of theorem 1 hold. Then, for any $t\in\left(0,T\right],$ $H^{\varepsilon}\left(t\right)\rightarrow 0$ as $\varepsilon\rightarrow 0.$

To end the proof of the Theorem, we define a new functional

[TABLE]

With $b$ belongs to the space of all continuous functions from $\mathbb{T}^{d}$ to $\mathbb{R}^{d}.$ By the Cauchy-Shwarz inequality, one get

[TABLE]

Since $\rho^{\varepsilon}\rightarrow 1,$ $J^{\varepsilon}\rightarrow J$ and from the convexity of the functional defined by (2.2), we obtain

[TABLE]

This finish the proof of Theorem 1.2.