First Page

Advances in Differential Equations
Volume 9, Numbers 5-6, May/June 2004, Pages 563–586
ON AN EVOLUTION SYSTEM DESCRIBING
SELF-GRAVITATING FERMI–DIRAC PARTICLES
Piotr Biler
Instytut Matematyczny, Uniwersytet Wroc!lawski
pl. Grunwaldzki 2/4, 50–384 Wroc!law, Poland
Philippe Laurençot
Mathématiques pour l’Industrie et Physique, CNRS UMR 5640
Université Paul Sabatier – Toulouse 3, 118 route de Narbonne
F–31062 Toulouse cedex 4, France
Tadeusz Nadzieja
Instytut Matematyki, Uniwersytet Zielonogórski
ul. Szafrana 4a, 65–516 Zielona Góra, Poland
(Submitted by: Herbert Amann)
Abstract. The global-in-time existence of solutions for a system describing the interaction of gravitationally attracting particles that obey
the Fermi–Dirac statistics is proved. Stationary solutions of that system
are also studied.
1. Introduction and derivation of the system
Our aim in this paper is to study a nonlinear, nonlocal, parabolic system
with nonlinear diffusion describing the evolution of a cloud of self-gravitating
particles that obey the Fermi–Dirac statistics. This model has been introduced in [14] on the basis of considerations of kinetic equations, and has
been studied in [13, 10].
Unlike the models of interacting particles where the particles are subject
to linear Brownian diffusion (see, e.g., [7, 4, 5, 6, 23, 24]), the assumption
that the density 0 ≤ f = f (x, v, t) of particles at the point (x, t) ∈ Ω ×
R+ , Ω ⊂ Rd , moving at the velocity v ∈ Rd is bounded by, say η0 > 0,
leads to mathematically completely different models. They involve nonlinear
diffusion resembling that for fast-diffusing gases (at large densities).
The plan of this paper is following: after discussing the derivation of
the system of partial differential equations in Section 1 and properties of
Accepted for publication: January 2004.
AMS Subject Classifications: 35Q, 35K60, 35B40, 82C21.
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