Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries

Zbigniew Banach; Wieslaw Larecki

doi:10.3934/krm.2017035

Article Contents

2017, Volume 10, Issue 4: 879-900. Doi: 10.3934/krm.2017035

This issue Previous Article Next Article Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries

Zbigniew Banach^, and
Wieslaw Larecki

Polish Academy of Sciences, Institute of Fundamental Technological Research, Department of Theory of Continuous Media, Pawinskiego 5B, 02-106 Warsaw, Poland

* Corresponding author: Zbigniew Banach
* Corresponding author: Zbigniew Banach

Received: May 31, 2016

Revised: October 31, 2016

Published: November 2017

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Abstract

The mixed three-moment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of one-dimensional flows. The conditions for realizability of the mixed three moments chosen as the energy density and two partial heat fluxes are established. The domain of admissible values of those moments is determined and the existence of the solution to the optimization problem is proved. Here, the standard approaches related to either the truncated Hausdorff or Markov moment problems do not apply because the non-negative fermionic distribution function, denoted $f$, must satisfy the inequality $f≤q 1$ and, at the same time, there are three different intervals of integration in the integral formulae defining the mixed moments. The hydrodynamic equations are obtained in the form of the symmetric hyperbolic system for the Lagrange multipliers of the optimization problem with constraints. The potentials generating this system are explicitly determined as dilogarithm and trilogarithm functions of the Lagrange multipliers. The invertibility of the relation between moments and Lagrange multipliers is proved. However, the inverse relation cannot be determined in a closed analytic form. Using the $H$-theorem for the radiative transfer equation, it is shown that the derived system of hydrodynamic radiation equations has as a consequence an additional balance law with a non-negative source term.

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Mathematics Subject Classification: 85A25, 82C40, 94A17, 46N10, 35L40, 35Q93.

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Figure 1. The set $\Omega$ and its location in $\mathbb{R}^3$. In the 'horizontal' directions, we use the notation $\mbox{'energy'}$ for $\varepsilon$ and $\mbox{'heat'}$ for $q_{ +}$. The tick marks along the 'vertical' direction represent the values of $q_{ -}$

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