Advances in Differential Equations

Global existence conditions for a nonlocal problem arising in statistical mechanics

I. A. Guerra, M. A. Peletier, and C. J. van Duijn

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We consider the evolution of the density and temperature of a 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.

Article information

Adv. Differential Equations Volume 9, Number 1-2 (2004), 133-158.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q72
Secondary: 35B40: Asymptotic behavior of solutions 35K60: Nonlinear initial value problems for linear parabolic equations 82C21: Dynamic continuum models (systems of particles, etc.)


van Duijn, C. J.; Guerra, I. A.; Peletier, M. A. Global existence conditions for a nonlocal problem arising in statistical mechanics. Adv. Differential Equations 9 (2004), no. 1-2, 133--158.

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