Tadeusz Nadzieja

The

We study the properties and the large time asymptotics of radially symmetric solutions of a chemotaxis system in a disc of R 2 when the parameter is either critical and equal to 8… or subcritical.

8\pi

We study properties of solutions of the system ut =r (#ru +ur’); ’ = u; E = M# + 1 Z u’dx: This system was proposed by Chavanis, Sommeria and Robert for description of evolution of density of a system of gravitating particles. In physical interpretation u(x;t), ’(x;t) are the density and the gravitational potential, respectively. The temperature #(t) is uniform in the domain , where the problem is considered. M is the total mass and E is the energy of the particles. We are interested in the existence of global solutions, blow-up phenomena and stationary solutions of our system. The results have been obtained jointly with Piotr Biler (Uniwersytet Wroc lawski) and Ignacio Guerra (Universidad de Chile).

-problem for radially symmetric solutions of a chemotaxis model in a disc

We review Streater’s energy-transport models which describe the temporal evolution of the density and temperature of a cloud of gravitating particles, coupled to a mean field Poisson equation. In particular we consider the existence of stationary solutions in a bounded domain with given energy and mass. We discuss the influence of the dimension and geometry of the domain on existence results.

Global and exploding solutions in a model of self-gravitating systems

We propose some nonlinear parabolic-elliptic systems modelling the evolution of the density of particles (charged or massive) interacting with themselves, and coupled to a temperature field. These models are thermodynamically consistent, i.e., they obey the first and the second laws of thermodynamics. We study their steady states and the asymptotic behaviour for large time.

Steady States for Streater’s Energy-Transport Models of Self-Gravitating Particles

1. IntroductionThispaperisdevotedtoastudyofaparticularmodelproblemofelectrodi⁄usionofions. The intriguing phenomenon of electrodi⁄usion appears to be of importance invarious areas of science: electrolysis, semiconductor theory, biological systems (poly-electrolytes, membrane transport). The book [21] is a very good comprehensiveintroduction to mathematical aspects of these problems.The history of basic PDEOs in electrolytes theory is quite long. W. Nernst andM. Planck formulated at the end of the nineteenth century a system of para-bolic

SELF-INTERACTION OF BROWNIAN PARTICLES COUPLED WITH THERMODYNAMIC PROCESSES

Energy-transport models introduced by R. F. Streater describe the evolution of the density and temperature of a cloud of self-gravitating particles. We study the existence of steady states with prescribed mass and energy for these models.

A Singular Problem in Electrolytes Theory

Abstract A nonlocal parabolic equation describing the evolution of a cloud of particles is studied.

Structure of steady states for Streater's energy-transport models of gravitating particles

The aim of this note is to present a simple proof of the existence of solutions of nonlocal equations appearing in the theory of self-interacting Brownian particles. A result on the nonexistence of solutions is also proved.

Nonlocal parabolic problems in statistical mechanics

Abstract We give the conditions for a flow generated by a smooth vector field X which guarantee that every smooth vectorfield Y in some C0-neighborhood of X defines a flow with positively Lagrange stable trajectories.