Andrzej Raczyński

Stability property of the two-dimensional Keller–Segel model

The paper contains results concerning the stability properties of solutions to a simplified Keller–Segel problem modelling chemotaxis in the whole space R2. First, the existence of solutions to considered system is proved, as well as for this system with the elliptic equation replaced by its parabolic version with time derivative multiplied by a positive parameter e. Secondly, the stability of the model is shown, namely the convergence of solutions of parabolic–parabolic system to a solution of parabolic–elliptic one, when a suitable parameter in the system under consideration converges towards zero.

Sticky particles on the line under harmonic interactions

We consider the hydrodynamical limit of a system of particles of finite size on the line which interact with each other via an harmonic force. The particles are assumed to stick to each other upon collision, to form compound particles whose mass and size is the sum of masses and sizes of the particles before collision, and whose velocity after collision is determined by the conservation of linear momentum at the collision time. The main result reads: under reasonable conditions on the initial data there exists a unique hydrodynamical limit, and this limit is a weak solution of an associated system of gas dynamics equations.