No Gelation and Global Existence for a Boltzmann Equation with Regularly Varying Mass-Exchange Rates

Abstract

We develop a no-gelation mechanism that yields global solutions to the spatially homogeneous Boltzmann equation with mass exchange for Grad-cutoff hard potentials $B=E^{\gamma }b(\xi ),0<\gamma <1$ and regularly varying mass-exchange rates. Without assuming any higher mass moment, we construct a convex superlinear weight assembled from dyadic hinges. Production of the weighted moment by collisions between large particles of comparable mass is absorbed by dissipation through collisions with a uniformly populated reservoir of bounded-mass particles. Regular variation makes the signed increments in these two collision configurations comparable, while mass conservation and $\gamma <1$ yield the vanishing factor $L^{\gamma -1}$. This yields a uniform moment bound of the mass-cutoff approximations on every finite time interval and rules out finite-time mass escape to infinity. Consequently, for every nonnegative initial density with finite physical moments, we obtain a global integral weak solution.

arXiv comment40 pages, 2 figures, 1 table