Global existence conditions for a nonlocal problem arising in statistical mechanics

Tijdschriftartikel

Duijn, van, C.J., Guerra Benavente, I.A. & Peletier, M.A. (2004). Global existence conditions for a nonlocal problem arising in statistical mechanics. Advances in Differential Equations, 9(1-2), 133-158. In Scopus Cited 7 times. Lees meer: Medialink/Full text

Abstract

 

We consider the evolution of the density and temperature of

11 three-dimensional cloud of self-interacting particles. This phenomenon

is modeled by a parabolic equation for the density distribution combining

temperature-dependent diffusion and convection driven by the gradient

of the gravitational potential. This equation is coupled with Poisson's

equation for the potential generated by the density distribution. The

system preserves mass by imposing a zero-flux boundary condition. Finally

the temperature is fixed by energy conservation; that is, the sum of

kinetic energy (temperature) and gravitational energy remains constant

in time. This model is thermodynamically consistent, obeying the first

and the second laws of thermodynamics. We prove local existence and

uniqueness of weak solutions for the system, using a Schauder fixed-point

theorem. In addition, we give sufficient conditions for global-in-time existence

and blow-up for radially symmetric solutions. We do this using

a comparison principle for an equation for the accumulated radial mass.