Group analysis of the one-dimentional kinetic equation and the problem of closing the moment system

Abstract: The report is devoted to a problem that goes back to the works of Maxwell and Clausius, the relationship between the kinetic equations of the particles of the medium and the macroscopic characteristics of the medium. In the modern form, the question is how to obtain the equations of a continuum media from the kinetic equations. The fundamental problem is the following: integration of the kinetic equation with power-law weights over velocities gives an infinite system of equations, the first of which are very similar to the equations of a continuous medium. But the system of equations of a continuous medium is finite. This means that the infinite system must be truncated and closed. The problem consists of two questions: where to truncate and what ratio use to close. The report will present an approach based on group methods. The idea is to calculate the symmetry group of the kinetic equation, transfer its action to macroscopic quantities, find invariants already in terms of macroscopic quantities, and use them to construct a closure. This was successfully implemented in the one-dimensional case, the details will be presented in the report.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods