Short- and long-time behavior in evolution equations: the role of the hypocoercivity index

Abstract

The index of hypocoercivity is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it quantifies `how degenerate` a hypocoercive evolution equation is, both for ODEs and for evolutions equations in a Hilbert space. We show that this index characterizes the polynomial decay of the propagator norm for short time and illustrate these concepts for the Lorentz kinetic equation on a torus. This talk is based on joint work with F. Achleitner, E. Carlen, E. Nigsch, and V. Mehrmann.