A Kinetic Concentration criterion for Large-data Boltzmann Dynamics
Abstract
We establish a conditional theory of genuinely non-perturbative global dynamics for the cutoff Boltzmann equation with hard potentials on the three-dimensional torus. We prove that any global mild solution satisfying supt≥0∫ℝ3W(v)supx∈𝕋3F(t,x,v) dv<∞, for a suitable stretched-exponential weight W, enjoys uniform weighted L∞ control, is unique in the dissipative mild class, and converges exponentially to the global Maxwellian. In particular, these conclusions hold for arbitrary-amplitude solutions admitting a uniform stretched-exponential velocity upper bound. The key mechanism is a *tail-to-coercivity principle*: uniform velocity-tail control generates positive lower bounds for the local density and nonlinear collision frequency, without imposing macroscopic coercivity a priori. A double-Duhamel positivity argument also generates a Gaussian lower bound at arbitrarily short positive times, allowing initial data with local vacuum. We establish a polynomially weighted counterpart under an additional spatially uniform lower velocity profile, obtaining analogous weighted L∞ bounds, uniqueness, and exponential relaxation. Our results provide a low-regularity alternative to conditional theories requiring smoothness and pointwise macroscopic bounds. They reveal how a single velocity-tail condition controls large-data Boltzmann dynamics and suggest a route toward problems with physical boundary conditions.
Related
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BibTeX
@misc{he2026kinetic,
title = {{A Kinetic Concentration criterion for Large-data Boltzmann Dynamics}},
author = {Lingbing He and Jin-Cheng Jiang and Jong-In Kim and Donghyun Lee and Sungbin Park},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12093},
eprint = {2610.12093},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12093},
}
RIS
TY - UNPB
TI - A Kinetic Concentration criterion for Large-data Boltzmann Dynamics
AU - Lingbing He
AU - Jin-Cheng Jiang
AU - Jong-In Kim
AU - Donghyun Lee
AU - Sungbin Park
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12093
UR - https://arxiv.org/abs/2610.12093
AB - We establish a conditional theory of genuinely non-perturbative global dynamics for the cutoff Boltzmann equation with hard potentials on the three-dimensional torus. We prove that any global mild solution satisfying sup_t≥0∫_ℝ^3W(v)sup_x∈𝕋^3F(t,x,v) dv<∞, for a suitable stretched-exponential weight W, enjoys uniform weighted L^∞ control, is unique in the dissipative mild class, and converges exponentially to the global Maxwellian. In particular, these conclusions hold for arbitrary-amplitude solutions admitting a uniform stretched-exponential velocity upper bound. The key mechanism is a *tail-to-coercivity principle*: uniform velocity-tail control generates positive lower bounds for the local density and nonlinear collision frequency, without imposing macroscopic coercivity a priori. A double-Duhamel positivity argument also generates a Gaussian lower bound at arbitrarily short positive times, allowing initial data with local vacuum. We establish a polynomially weighted counterpart under an additional spatially uniform lower velocity profile, obtaining analogous weighted L^∞ bounds, uniqueness, and exponential relaxation. Our results provide a low-regularity alternative to conditional theories requiring smoothness and pointwise macroscopic bounds. They reveal how a single velocity-tail condition controls large-data Boltzmann dynamics and suggest a route toward problems with physical boundary conditions.
ER -