From Monge-Ampere-Boltzman to Euler Equations

This paper concerns with the convergence of the Monge-Ampere-Boltzman system to the in compressible Euler Equations in the quasi-neutral regime. Citation: Fethi BB (2017) From Monge-Ampere-Boltzman to Euler Equations. J Appl Computat Math 6: 341. doi: 10.4172/2168-9679.1000341


Introduction and Main Results
In this paper, we are interested in the hydrodynamical limit of the Boltzman-Monge-Ampere system (BMA) where f ε (t, x,ξ) ≥ 0 the electronic density at time t ≥ 0 point x ∈[0, 1] d = d, and with a velocity ξ ∈  d , and Id is the identity matrix defined by The spatially periodic electric potential is coupled with ϕ 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 ρ ε (t, x)= d f ε (t, x,ξ)dξ. (1. 3) Q(f ε , f ε ) is the Boltzman collision integral. This integral operates only on the ξ−argument of the distribution f ε and is given by The linearization of the determinant about the identity matrix gives det( d +ε 2 D 2 ϕ ε )=1+ε 2 ∆ϕ ε +O(ε 4 ).
Where  d 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 The analysis of the VPB system has been considered by many authors and many results can be found in a vast literature [1][2][3][4][5][6][7][8][9][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.

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: Which is a weak version of the Monge-Ampere equation We assume that BMA system has a renormalized solution in the sense of DiePerna and Lions [3].
For simplicity, we set , and the (BMA p ) (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 [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) 
It follows by integrating by party that

Let us begin with the first term A. Use Holder inequality and that
.
From the second term D, one has ( ) From the definition of Φ, we have