Searcharxiv⌕ Search

arXiv subjects

I-Kun Chen

Publications and source records attributed to I-Kun Chen.

12 recordsLinked to original sources
arXiv2026

On thermal transpiration and thermomolecular pressure difference

In this article, we demonstrate the phenomenon of thermal transpiration in a bounded convex domain. We employ the stationary Boltzmann equation with a cutoff potential. For boundary condition, we partition the boundary into diffuse reflection and incoming regions. We establish the existence of solution in a weighted $L^\infty$ space. Furthermore, we consider a convex domain with diffuse reflection boundary condition in the middle and incoming boundary condition at the two ends. We first consider Maxwellians with the same pressure but different temperatures at the two ends. We prove that the total flux $U(x)$ is directed toward the hot end. Furthermore, we derive an estimate for the total flux: \begin{align} U(x)\geq C\left(1-\frac{1}{\sqrt{T_2}}\right). \end{align} In addition, we show that when the pressures and temperatures on the two ends satisfy the relation \begin{align} \frac{P_1}{P_2}=\sqrt{\frac{T_1}{T_2}}, \end{align} the total flux of the solution is of order $\mathcal{O}(\frac{1}κ)$. This result is consistent with Knudsen's finding of thermomolecular pressure difference in 1909.

math.AP↗
arXiv2025

On the existence and regularity of weakly nonlinear stationary Boltzmann equations : a Fredholm alternative approach

The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations \cite{Evans}. In this article, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski \cite{CP} to solve the existence of the stationary Boltzmann equation in a slab domain. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished.

math.AP↗
arXiv2023

Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation

We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in $W^{1,p}$ spaces for $1 \leq p<2$ for small domains. In contrast, if we further assume the positivity of the Gaussian curvature on the boundary, we prove the existence of solutions in $W^{1, p}$ spaces for $1 \leq p < 3$ provided that the diameter of the domain is small enough. In both cases, we provide counterexamples in the hard sphere model; a bounded convex domain with a flat boundary for $p = 2$, and a small ball for $p = 3$.

math.AP↗
arXiv2018

Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography

We consider a boundary value problem of the stationary transport equation with the incoming boundary condition in two or three dimensional bounded convex domains. We discuss discontinuity of the solution to the boundary value problem arising from discontinuous incoming boundary data, which we call the boundary-induced discontinuity. In particular, we give two kinds of sufficient conditions on the incoming boundary data for the boundary-induced discontinuity. We propose a method to reconstruct attenuation coefficient from jumps in boundary measurements.

math.AP↗
arXiv2018

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.

math.AP↗
arXiv2017

Propagation of boundary-induced discontinuity in stationary radiative transfer

We consider the boundary value problem of the stationary transport equation in the slab domain of general dimensions. In this paper, we discuss the relation between discontinuity of the incoming boundary data and that of the solution to the stationary transport equation. We introduce two conditions posed on the boundary data so that discontinuity of the boundary data propagates along positive characteristic lines as that of the solution to the stationary transport equation. Our analysis does not depend on the celebrated velocity averaging lemma, which is different from previous works. We also introduce an example in two dimensional case which shows that piecewise continuity of the boundary data is not a sufficient condition for the main result.

math-ph↗
arXiv2016

Regularity of stationary solutions to the linearized Boltzmann equations

We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary solutions solutions are Hölder continuous with order $\frac1{2}^-$ away from the boundary provided the incoming data have the same regularity. The key idea is to partially transfer the regularity in velocity obtained by collision to space through transport and collision.

math.AP↗
arXiv2008

Spherical Averaged Endpoint Strichartz Estimates for The Two-dimensional Schrodinger Equations with Inverse Square Potential

The endpoint Strichartz estimates for two-dimensional Schrodinger equations were recovered by averaging the solutions in L^2 in the angular variable by Tao. For Schrodinger equations with defocusing inverse square potential, we proved that the homogeneous endpoint estimates hold under this setting. In particular, the original versions of endpoint estimates hold for radial data.

math.AP↗