Instantaneous smoothing and exponential decay of solutions of a degenerate evolution equation with applications to Boltzmann's equation

Abstract: 

We establish an instantaneous smoothing property for decaying solutions on the half-line of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation, namely (d/dt)Ax=−x+G(x)(�/��)��=−�+�(�), where G(0)=0�(0)=0, |dG|≤γ<1|��|≤�<1, and A� is bounded, self-adjoint, but singular. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of H1�1 stable manifolds of such equations, showing that L2loc����2 solutions that merely remain sufficiently small in L∞�∞ (i) decay exponentially, and (ii) are C1�1 for t>0�>0, hence lie eventually in the H1�1 stable manifold constructed by Pogan and Zumbrun. Surprisingly, it is small velocities (leading to singularity of A�) rather than large that present the main difficulty for Boltzmanns' equation in this context.

Joint work with Fedor Nazarov.