Hamiltonian and Godunov Structures of the Grad Hierarchy

TL;DR

This paper explores the Hamiltonian, Godunov, and gradient structures within the Grad hierarchy of the Boltzmann equation, linking mechanics, thermodynamics, and extended hydrodynamics.

Contribution

It extends the Hamiltonian and Godunov structures to the Grad hierarchy, including finite hierarchies, and analyzes their interrelations.

Findings

Identified Hamiltonian and Godunov structures in the infinite Grad hierarchy.

Demonstrated these structures in finite extended hydrodynamic equations.

Explored relationships between Hamiltonian and Godunov structures within Grad's hierarchies.

Abstract

The time evolution governed by the Boltzmann kinetic equation is compatible with mechanics and thermodynamics. The former compatibility is mathematically expressed in the Hamiltonian and Godunov structures, the latter in the structure of gradient dynamics guaranteeing the growth of entropy and consequently the approach to equilibrium. We carry all three structures to the Grad reformulation of the Boltzmann equation (to the Grad hierarchy). First, we recognize the structures in the infinite Grad hierarchy and then in several examples of finite hierarchies representing extended hydrodynamic equations. In the context of Grad's hierarchies we also investigate relations between Hamiltonian and Godunov structures.