Medical, Pharma, Engineering, Science, Technology and Business

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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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

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

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

(1.1)

(1.2)

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

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

*ρ ^{ε}(t, x) *≥ 0 denote respectively the vacuum electric permittivity and

(1.3)

*Q(f ^{ε}, f^{ε})* is the Boltzman collision integral. This integral operates only on the ξ−argument of the distribution

where the terms defines, respectively the values

with given in terms of

by

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

Note that

The linearization of the determinant about the identity matrix gives

Where 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

(1.4)

(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.

**Definition 1**

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

Its Legendre-Fenchel transform denoted the function satisfying (1.6) we 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:

(1.6)

Which is a weak version of the Monge–Ampere 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

So that, and the (BMA_{p}) (*p* stands for periodic) system takes the following form

(1.7)

The energy is given by

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

The Euler equation for incompressible fluids reads

(1.8)

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

**Theorem 3**

Let *f ^{ε}* be a weak solution of (1.7) with finite energy, let (t,x) →

Then

The constant C depends only on

**Proof of the Theorem 3**

Later, in the section, we need the following Lemma

**Lemma 4:**

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

We have

From the BMA we have

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

Thus

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

From the second term *D*, one has

From the definition of Φ, we have

Since is divergence free, once gets

Consider now the last term *D*.

But since is divergence fre we have Thus form Lemma 4

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

By Lemma4 and setting we can deduct that

Thus

We deduce then the following inequality

(2.1)

Still using 4,

Thus

So once can transform (2.1) as

And by Gronwall’s inequality [11] yields

Finally we conclude that

Which achieves the proof of the theorem.

- Cercignani C (1988) The Boltzmann equation. The Boltzmann Equation and Its Applications. Springer New York, pp: 40-103.
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