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

We analyze the pattern forming ability and pattern stability for a one-dimensional non-linear transport-diffusion equation on the circle. We show that the trivial steady state is stable when diffusion is sufficiently strong. In the limit for vanishing diffusion, linear combinations of delta peaks can be stationary solutions; we study their stability properties. Finally, we present numerical examples exhibiting a variety of behaviors.