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arXiv · 2511.02273

On the Boltzmann-Fermi-Dirac Equation for Hard Potential: Global Existence and Uniqueness, Gaussian Lower Bound, and Moment Estimates

Gayoung An·Sungbin Park
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Abstract

In this paper, we study the global existence and uniqueness, Gaussian lower bound, and moment estimates in the spatially homogeneous Boltzmann equation for Fermi-Dirac particles for hard potential ($0\leq \gamma\leq 2$) with angular cutoff $b$. Our results extend classical results to the Boltzmann-Fermi-Dirac setting. In detail, (1) we show existence, uniqueness, and $L^1_2$ stability of global-in-time solutions of the Boltzmann-Fermi-Dirac equation. (2) Assuming the solution is not a saturated equilibrium, we prove creation of a Gaussian lower bound for the solution. (3) We prove creation and propagation of $L^1$ polynomial and exponential moments of the solution under additional assumptions on the angular kernel $b$ and $0<\gamma\leq 2$. (4) Finally, we show propagation of $L^\infty$ Gaussian and polynomial upper bounds when $b$ is constant and $0<\gamma\leq 1$.

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Gayoung An, Sungbin Park. 2025-11-04. On the Boltzmann-Fermi-Dirac Equation for Hard Potential: Global Existence and Uniqueness, Gaussian Lower Bound, and Moment Estimates. https://arxiv.org/abs/2511.02273

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