98 research outputs found
Infinite energy solutions to the homogeneous Boltzmann equation
The goal of this work is to present an approach to the homogeneous Boltzmann
equation for Maxwellian molecules with a physical collision kernel which allows
us to construct unique solutions to the initial value problem in a space of
probability measures defined via the Fourier transform. In that space, the
second moment of a measure is not assumed to be finite, so infinite energy
solutions are not {\it a priori} excluded from our considerations. Moreover, we
study the large time asymptotics of solutions and, in a particular case, we
give an elementary proof of the asymptotic stability of self-similar solutions
obtained by A.V. Bobylev and C. Cercignani [J. Stat. Phys. {\bf 106} (2002),
1039--1071]
Asymptotic stability of Landau solutions to Navier-Stokes system
It is known that the three dimensional Navier-Stokes system for an
incompressible fluid in the whole space has a one parameter family of explicit
stationary solutions, which are axisymmetric and homogeneous of degree -1. We
show that these solutions are asymptotically stable under any
-perturbation
Blow-up versus global existence of solutions to aggregation equations
A class of nonlinear viscous transport equations describing aggregation
phenomena in biology is considered. Optimal conditions on an interaction
potential are obtained which lead either to the existence or to the
nonexistence of global-in-time solutions
- …