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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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From Monge-Ampere-Boltzman to Euler Equations

Ben Belgacem Fethi*

Faculty of Sciences of Tunis, University of Tunis, El Manar, Tunisia

*Corresponding Author:
Ben Belgacem Fethi
Faculty of Sciences of Tunis
University of Tunis, Monastir
El Manar, Tunisia
Tel: +21698955739
E-mail: belgacem.fethi@gmail.com

Received date: June 22, 2016; Accepted date: February 21, 2017; Published date: February 28, 2017

Citation: Fethi BB (2017) From Monge-Ampere-Boltzman to Euler Equations. J Appl Computat Math 6: 341. doi: 10.4172/2168-9679.1000341

Copyright: © 2017 Fethi BB. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This paper concerns with the convergence of the Monge-Ampere-Boltzman system to the in compressible Euler Equations in the quasi-neutral regime.

Keywords

Vlasov-Monge-Ampère-Boltzman system; Euler equations of the incompressible fluid

Introduction

In this paper, we are interested in the hydrodynamical limit of the Boltzman-Monge-Ampere system (BMA)

equation   (1.1)

equation   (1.2)

where equation the electronic density at time t ≥ 0 point x ∈[0, 1]d= equation and with a velocity equation and Id is the identity matrix defined by

equation

The spatially periodic electric potential is coupled with ?ε through the nonlinear Monge-Ampere equation (1.2). The quantities ε > 0 and

ρε(t, x) ≥ 0 denote respectively the vacuum electric permittivity and

equation   (1.3)

Q(fε, fε) is the Boltzman collision integral. This integral operates only on the ξ−argument of the distribution fε and is given by

equation

where the terms equation defines, respectively the values

equation withequation given in terms of

equation by

equation

The aim of this work is to investigate the hydrodynamic limit of the (BMA) system with optimal transport techniques.

Note that

equation

The linearization of the determinant about the identity matrix gives

equation

Where equation represents the identity matrix.

So, one can see that the BMA system is considered as a fully nonlinear version of the Vlasov Poisson-Boltzman (VPB) system defined by

equation   (1.4)

equation   (1.5)

The analysis of the VPB system has been considered by many authors and many results can be found in a vast literature [1-10].

In Hsiao et al. [11] study the convergence of the VPB system to the Incompressible Euler Equations. Bernier and Grégoire show that weak solution of Vlasov-Monge-Ampère converge to a solution of the incompressible Euler equations when the parameter goes to 0, Brenier [12] and Loeper [13] for details. So, is a ligitim question to look for the convergence of a weak solution of BMA (of course if such solution exists) to a solution of the incompressible Euler equations when the parameter goes to 0.

The study of the existence and uniqueness of solution to the BMA system seems a difficult matter. Here we assume the existence and uniqueness of smooth solution to the BMA and we just look to the asymptotic analysis of this system.

Definitions and Recalls

Definition 1

For a fixed bounded convex open set W of equation and a positive measure on equation of total mass |W|, we note by F[,ρ] the unique up to a constant convex function on satisfying

equation

Its Legendre-Fenchel transform denoted equation the function satisfying (1.6) equationwe may write Φ (resp. Ψ) instead of Φ [Ω,] (resp. Ψ[Ω,ρ]) if no confusion is possible.

Remark 2

• Existence and uniqueness of Φ is due to the polar factorization theorem.

• By setting the change of variables y=∇Ψ(x), we get dy=det D2Ψ(x)dx. So (1.6) can be transformed to:

equation

equation   (1.6)

Which is a weak version of the Monge–Ampere equation

equation

∇ mapps supp (ρ) in Ω

We assume that BMA system has a renormalized solution in the sense of DiePerna and Lions [3].

For simplicity, we set

equation

equation

So that, equation and the (BMAp) (p stands for periodic) system takes the following form

equation   (1.7)

The energy is given by

equation

equation

It has been shown |2| that the energy is conserved.

The Euler equation for incompressible fluids reads

equation   (1.8)

One can find in Loeper [13] more details for this kind of equations.

Theorem 3

Let fε be a weak solution of (1.7) with finite energy, let (t,x) → v (t, x) be a smooth C2([0,T] × Td) solution of (1.8) for t∈[0,T], and p(t,x) the corresponding pressure, let

equation

Then

equation

The constant C depends only on

equation

Proof of the Theorem 3

Later, in the section, we need the following Lemma

Lemma 4:

Let equation be Lipschitz continuous such thatequation then for all R>0 has one

equation

We have

equation

equation

From the BMA we have

equation

The last term is equal to zero from the property of Boltzman Operator [1,3,5,7-9,11].

It follows by integrating by party that

equation

Thus

equation

Let us begin with the first term A. Use Holder inequality and that equation to decompose

equation

From the second term D, one has

equation

From the definition of Φ, we have

equation

Since equation is divergence free, once gets

equation

Consider now the last term D.

equation

But since equation is divergence fre we haveequation Thus form Lemma 4

equation

equation

Since it costs no generality to suppose that for all t∈[0,T],∫P(t,x) dx=0,

we get from the equation of conservation of mass

equation

By Lemma4 and setting equation we can deduct that

equation

Thus

equation

We deduce then the following inequality

equation   (2.1)

Still using 4,

equation

Thus

equation

So once can transform (2.1) as

equation

And by Gronwall’s inequality [11] yields

equation

Finally we conclude that

equation

Which achieves the proof of the theorem.

References

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