Global Well-Posedness and Large Data Estimates for the 1D Boltzmann Equation

Abstract: We prove quantitative growth estimates for large data solutions to the 1D Boltzmann equation, for a collision kernel with angular cutoff and relative velocity cutoff. We present proofs for the global well-posedness results presented in the note of Biryuk, Craig, and Panferov, in which global solutions for this equation are shown to exist for large data, with density bounded for all time. We show that these solutions propagate moments in $v$, and derivatives in $x$ and $v$. Our main contribution is to develop new, sharp integral inequality estimates, which allow us to prove exponential growth bounds in $L^\infty_x L<sup>1_v$ for large data, and to prove dissipation in $L^\infty_x L<sup>1_v$ for finite energy data on the line.