
Kevin Zumbrun

Indiana University

Instantaneous smoothing and exponential decay of solutions of a degenerate evolution equation with applications to Boltzmann's equation
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 halfline of certain degenerate Hilbert spacevalued 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, selfadjoint, 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.