Edith Geigant, Michael Stoll
Abstract:
We study solutions of a transport-diffusion equation on the circle.
The velocity of turning is given by a non-local term that models attraction
and repulsion between elongated particles.
Having mentioned basics like invariances, instability criteria and non-existence
of time-periodic solutions, we prove that
the constant steady state is stable at large diffusion.
We show that without diffusion localized initial distributions and attraction lead
to formation of several peaks.
For peak-like steady states two kinds of peak stability are analyzed:
first spatially discretized with respect to the relative position of the peaks,
then stability with respect to non-localized perturbations.
We prove that more than two peaks may be stable up to translation and slight
rearrangements of the peaks.
Our fast numerical scheme which is based on the Fourier-transformed system allows
to study the long-time behaviour of the equation.
Numerical examples show backward bifurcation, mixed-mode solutions, peaks with
unequal distances, coexistence of one-peak and two-peak solutions and peak
formation in a case of purely repulsive interaction.
Submitted July 16, 2012. Published September 7, 2012.
Math Subject Classifications: 35R09, 35B35, 35Q92, 45K05, 92B05.
Key Words: Transport equation on the circle; peak solutions;
local and global stability; numerical algorithms and simulations.
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Edith Geigant Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany email: Edith.Geigant@uni-bayreuth.de | |
Michael Stoll Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany email: Michael.Stoll@uni-bayreuth.de |
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