# 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

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 $(d/dt)A x= -x + G(x)$, where $G(0)=0$, $|dG|\leq \gamma<1$, 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 $H^1$ stable manifolds of such equations, showing that $L^2_{loc}$ solutions that merely remain sufficiently small in $L^\infty$ (i) decay exponentially, and (ii) are $C^1$ for $t > 0$, hence lie eventually in the $H^1$ 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.