arXiv subjects

Gayoung An

Publications and source records attributed to Gayoung An.

5 recordsLinked to original sources

arXiv2025

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

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 ($0\leq γ\leq 2$) 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 $0<γ\leq 2$. (4) Finally, we show propagation of $L^\infty$ Gaussian and polynomial upper bounds when $b$ is constant and $0<γ\leq 1$.

math.AP↗

arXiv2025

Quantitative pointwise estimates of the cooling process for inelastic Boltzmann equation

In this paper, we study the homogeneous inelastic Boltzmann equation for hard spheres. We first prove that the solution $f(t,v)$ is bounded pointwise from above by $C_{f_0}\langle t \rangle^3$ and establish that the cooling time is infinite $T_c = +\infty$ under the condition $f_0 \in L^1_2 \cap L^{\infty}_{s}$ for $s > 2$. Away from zero velocity, we further prove that $f(t,v)\leq C_{f_0, |v|} \langle t \rangle$ for $v \neq 0$ at any time $t > 0$. This time-dependent pointwise upper bound is natural in the cooling process, as we expect the density near $v = 0$ to grow rapidly. We also establish an upper bound that depends on the coefficient of normal restitution constant, $α\in (0,1]$. This upper bound becomes constant when $α= 1$, restoring the known upper bound for elastic collisions [8]. Consequently, through these results, we obtain Maxwellian upper bounds on the solutions at each time.

math.AP↗

arXiv2025

Optimal $C^{\frac{1}{2}}$ regularity of the Boltzmann equation in non-convex domains

Regularity of the Boltzmann equation, particularly in the presence of physical boundary conditions, heavily relies on the geometry of the boundaries. In the case of non-convex domains with specular reflection boundary conditions, the problem remained outstanding until recently due to the severe singularity of billiard trajectories near the grazing set, where the trajectory map is not differentiable. This challenge was addressed in [32], where $C^{\frac{1}{2}-}_{x,v}$ H\"{o}lder regularity was proven. In this paper, we introduce a novel dynamical singular regime integration methodology to establish the optimal $C^{\frac{1}{2}}_{x,v}$ regularity for the Boltzmann equation past a convex obstacle.

math.AP↗

arXiv2024

The Mixed Convex-Concave effect on the regularity of a Boltzmann solution

The geometric properties of domains are well-known to be crucial factors influencing the regularity of Boltzmann boundary problems. In the presence of non-convex physical boundaries, highly singular behaviors are investigated including the propagation of discontinuities \cite{Kim11} and Hölder regularity \cite{CD2022}. In this paper, we focus on studying the $C^{0,\frac{1}{4}-}$ Hölder regularity of the Boltzmann equation within concentric cylinders where both convex and concave features are present on the boundaries. Our findings extend the previous results of \cite{CD2022}, which primarily addressed concavity while considering the exterior of convex objects.

math.AP↗

arXiv2022

High-velocity tails of the inelastic and the multi-species mixture Boltzmann equations

We study high-velocity tails of some homogeneous Boltzmann equations on $v \in \mathbb{R}_{v}^d$. First, we consider spatially homogeneous inelastic Boltzmann equation with noncutoff collision kernel, in the case of moderately soft potentials. We also study spatially homogeneous mixture Boltzmann equations : for both noncutoff collision kernel with moderately soft potentials and cutoff collision kernel with hard potentials. In the case of noncutoff inelastic Boltzmann, we obtain \[ f(t,v) \geq a(t) e^{-b(t)|v|^p}, \quad 2 < p < 6.213 \] by extending Cancellation lemma and spreading lemma and assuming $f\in C^{\infty}$. For the Mixture type Boltzmann equations, we prove Maxwellian $p=2$.

math.AP↗