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

Abstract

<p style='text-indent:20px;'>We establish an instantaneous smoothing property for decaying solutions on the half-line <inline-formula><tex-math id="M1">\begin{document}$ (0, +\infty) $\end{document}</tex-math></inline-formula> of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of <inline-formula><tex-math id="M2">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula> stable manifolds of such equations, showing that <inline-formula><tex-math id="M3">\begin{document}$ L^2_{loc} $\end{document}</tex-math></inline-formula> solutions that remain sufficiently small in <inline-formula><tex-math id="M4">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> (i) decay exponentially, and (ii) are <inline-formula><tex-math id="M5">\begin{document}$ C^\infty $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ t&gt;0 $\end{document}</tex-math></inline-formula>, hence lie eventually in the <inline-formula><tex-math id="M7">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula> stable manifold constructed by Pogan and Zumbrun.</p>