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We establish an instantaneous smoothing property for decaying solutions onthe half-line $(0,+\infty)$ of certain degenerate Hilbert space-valuedevolution equations arising in kinetic theory, including in particular thesteady Boltzmann equation. Our results answer the two main open problems posedby Pogan and Zumbrun in their treatment of $H^1$ stable manifolds of suchequations, showing that $L^2_{loc}$ solutions that remain sufficiently small in$L^\infty$ (i) decay exponentially, and (ii) are $C^\infty$ for $t>0$, hencelie eventually in the $H^1$ stable manifold constructed by Pogan and Zumbrun

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