25 research outputs found
On the non-integrability of the Popowicz peakon system
We consider a coupled system of Hamiltonian partial differential equations
introduced by Popowicz, which has the appearance of a two-field coupling
between the Camassa-Holm and Degasperis-Procesi equations. The latter equations
are both known to be integrable, and admit peaked soliton (peakon) solutions
with discontinuous derivatives at the peaks. A combination of a reciprocal
transformation with Painlev\'e analysis provides strong evidence that the
Popowicz system is non-integrable. Nevertheless, we are able to construct exact
travelling wave solutions in terms of an elliptic integral, together with a
degenerate travelling wave corresponding to a single peakon. We also describe
the dynamics of N-peakon solutions, which is given in terms of an Hamiltonian
system on a phase space of dimension 3N.Comment: 8 pages, AIMS class file. Proceedings of AIMS conference on Dynamical
Systems, Differential Equations and Applications, Arlington, Texas, 200
Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models
We consider the linear dissipative Boltzmann equation describing inelastic
interactions of particles with a fixed background. For the simplified model of
Maxwell molecules first, we give a complete spectral analysis, and deduce from
it the optimal rate of exponential convergence to equilibrium. Moreover we show
the convergence to the heat equation in the diffusive limit and compute
explicitely the diffusivity. Then for the physical model of hard spheres we use
a suitable entropy functional for which we prove explicit inequality between
the relative entropy and the production of entropy to get exponential
convergence to equilibrium with explicit rate. The proof is based on
inequalities between the entropy production functional for hard spheres and
Maxwell molecules. Mathematical proof of the convergence to some heat equation
in the diffusive limit is also given. From the last two points we deduce the
first explicit estimates on the diffusive coefficient in the Fick's law for
(inelastic hard-spheres) dissipative gases.Comment: 25 page