Faculty of Sciences of Tunis, University of Tunis, El Manar, Tunisia
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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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)
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
ρε(t, x) ≥ 0 denote respectively the vacuum electric permittivity and
Q(fε, fε) is the Boltzman collision integral. This integral operates only on the ξ−argument of the distribution fε and is given by
where the terms defines, respectively the values
with given in terms of
The aim of this work is to investigate the hydrodynamic limit of the (BMA) system with optimal transport techniques.
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
In Hsiao et al.  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  and Loeper  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.
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.
• 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:
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 .
For simplicity, we set
So that, and the (BMAp) (p stands for periodic) system takes the following form
The energy is given by
It has been shown |2| that the energy is conserved.
The Euler equation for incompressible fluids reads
One can find in Loeper  more details for this kind of equations.
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
The constant C depends only on
Proof of the Theorem 3
Later, in the section, we need the following Lemma
Let be Lipschitz continuous such that then for all R>0 has one
From the BMA we have
It follows by integrating by party that
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
We deduce then the following inequality
Still using 4,
So once can transform (2.1) as
And by Gronwall’s inequality  yields
Finally we conclude that
Which achieves the proof of the theorem.