L infinity resolvent bounds for steady Boltzmann's equation

TL;DR

This paper establishes lower bounds on the resolvent operator for the steady Boltzmann equation in weighted L1 spaces, revealing unbounded operator norms across all Lp spaces, and clarifying differences with previous L2 results.

Contribution

It extends resolvent bounds for the steady Boltzmann equation to weighted L1 spaces, showing unboundedness in all Lp spaces, thus resolving prior discrepancies between L2 and Lp analyses.

Findings

Resolvent operator norm is unbounded in all Lp spaces for 1 < p ≤ ∞.

Lower bounds on the resolvent operator are derived in weighted L1 spaces.

The results reconcile differences between L2 and Lp space behaviors for the Boltzmann equation.

Abstract

We derive lower bounds on the resolvent operator for the linearized steady Boltzmann equation over weighted L1 Banach spaces in velocity, comparable to those derived by Pogan & Zumbrun in an analogous weighted L2 Hilbert space setting. These show in particular that the operator norm of the resolvent kernel is unbounded in Lp(R) for all $1<p \leq \infty$, resolving an apparent discrepancy in behavior between the two settings suggested by previous work.