Abstract
We prove an existence and uniqueness of solution to the Maxwell- Boltzmann coupled system globally in time. We first of all describe the background space-time and the unknown functions, making some concerning the potentials of gravitation $a$ and $b$, which determine the gravitational field $g$, the distribution function $f$ which is unknown and is subject to the Boltzmann equation, as well as the collision kernel $\sigma$ which appears in the collision operator. We after clarify the choice of the function spaces and we establish step by step, using Sobolev theorems, all the essential energy estimations leading to the global existence theorem. The method used for the investigation of the global existence combine the Galerkin method which is applied in a particular separable Hilbert space which is a Sobolev space with weight, and the standard theory on the first order differential systems. We then give at the end, the physical significance of our work.
Highlights
In this paper we consider the coupled MAXWELL- BOLTZMANN System which is one of the basic systems of the kinetic theory.The relativistic Boltzmann equation rules the dynamics of a kind of particles subject to mutual collisions, by determining their distribution function, which is a non-negative real-valued function of both the position and the momentum of the particles
We first of all describe the background space-time and the unknown functions, making some hypotheses concerning the potentials of gravitation a and b, which determine the gravitational field g, the distribution function f which is unknown and is subject to the Boltzmann equation, as well as the collision kernel σ which appears in the collision operator
The method used for the investigation of the global existence combine the Galerkin method which is applied in a particular separable Hilbert space which is a Sobolev space with weight, and the standard theory on the first order differential systems
Summary
In this paper we consider the coupled MAXWELL- BOLTZMANN System which is one of the basic systems of the kinetic theory. The relativistic Boltzmann equation rules the dynamics of a kind of particles subject to mutual collisions, by determining their distribution function, which is a non-negative real-valued function of both the position and the momentum of the particles. We consider the case where the electromagnetic field F is generated, through the Maxwell equations by the Maxwell current defined by the distribution function f of the colliding particles, a charge density e, and a future pointing unit vector u, tangent at any point to the temporal axis. - firstly to prove the global existence in time and uniqueness of solution to the coupled Maxwell-Boltzmann system, clarifying things in the method used by Mucha, explaining the choice made for the function spaces, demonstrating completely the main theorems;.
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