Electron. J. Diff. Equ., Vol. 2012 (2012), No. 157, pp. 1-41.

Stability of peak solutions of a non-linear transport equation on the circle

Edith Geigant, Michael Stoll

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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