# Regularity for the Boltzmann Equation Conditional to Pressure and Moment Bounds

## TL;DR

This paper shows that solutions to the Boltzmann equation are smooth and decay under certain physical constraints, with results also applying to the Landau equation.

## Contribution

New uniform regularity and decay estimates for the Boltzmann equation without cut-off, conditional on macroscopic bounds.

## Key findings

- Solutions to the Boltzmann equation have uniform L∞ bounds under macroscopic constraints for hard potentials.
- C∞ estimates and derivative decay follow from the L∞ bounds.
- Results extend to the Landau equation as the limit s approaches 1.

## Abstract

We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in \documentclass[12pt]{minimal}
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\begin{document}$$L^\infty $$\end{document}L∞ in the case of hard potentials. As a consequence, we derive \documentclass[12pt]{minimal}
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\begin{document}$$C^{\infty }$$\end{document}C∞ estimates and decay estimates for all derivatives, conditional to these macroscopic bounds. Our \documentclass[12pt]{minimal}
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\begin{document}$$L^\infty $$\end{document}L∞ estimates are uniform in the limit \documentclass[12pt]{minimal}
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\begin{document}$$s \nearrow 1$$\end{document}s↗1 and hence we recover the same results also for the Landau equation.