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.Comment: 15 pages, no figure

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

In this dissertation, I investigate the two-dimensional Schrodinger equation with repulsive inverse square potential. I prove a version of the homogeneous endpoint Strichartz estimate, in which I replace the supremum norm on space by a norm that takes $L^2$ average in angular variable first and then supremum norm on radial variable

Boundary Singularity for Thermal Transpiration Problem of the Linearized Boltzmann Equation

We study the boundary singularity of the fluid velocity for the thermal transpiration problem in the kinetic theory. Logarithmic singularity has been demonstrated through the asymptotic and computational analysis. The goal of this paper is to confirm this logarithmic singularity through exact analysis. We use an iterated scheme, with the “gain” part of the collision operator as a source. The iterated scheme is appropriate for large Knudsen numbers considered here and yields an explicit leading term

Singularity of the Velocity Distribution Function in Molecular Velocity Space

We study the boundary singularity of the solutions to the Boltzmann equation in the kinetic theory. The solution has a jump discontinuity in the microscopic velocity ζ on the boundary and a secondary singularity of logarithmic type around the velocity tangential to the boundary, ζn∼0-, where ζn is the component of molecular velocity normal to the boundary, pointing to the gas. We demonstrate this secondary singularity by obtaining an asymptotic formula for the derivative of the solution on the boundary with respect to ζnn that diverges logarithmically when ζn∼0-. Our study is for the thermal transpiration problem between two plates for the hard sphere gases with sufficiently large Knudsen number and with the diffuse reflection boundary condition. The solution is constructed and its singularity is studied by an iteration procedure

Temperature-Sensitive Nanocapsules for Controlled Drug Release Caused by Magnetically Triggered Structural Disruption

Self-assembled nanocapsules containing a hydrophilic core and a crosslinked yet thermosensitive shell have been successfully prepared using poly(ethylene-oxide)-poly(propylene-oxide)-poly(ethylene-oxide) block copolymers, 4-nitrophenyl chloroformate, gelatin, and 1-ethyl-3-(3- dimethylaminopropyl) carbodiimide. The core is further rendered magnetic by incorporating iron oxide nanoparticles via internal precipitation to enable externally controlled actuation under magnetic induction. The spherical nanocapsules exhibit a hydrophilic-to-hydrophobic transition at a characteristic but tunable temperature reaching 40ºC, triggering a size contraction and shrinkage of the core. The core content experiences very little leakage at 25ºC, has a half life about 5 h at 45ºC, but bursts out within a few minutes under magnetic heating due to iron oxide coarsening and core/shell disruption. Such burst-like response may be utilized for controlled drug release as illustrated here using a model drug Vitamin B12

On the Existence of H1H^1 solutions for Stationary Linearized Boltzmann Equations in a Small Convex Domain

In this article, we investigate the incoming boundary value problem for the stationary linearized Boltzmann equations in $\Omega \subseteq \mathbb{R}^{3}$. For a $C^2$ bounded domain with boundary of positive Gaussian curvature, the existence theory is established in $H^{1}(\Omega \times \mathbb{R}^{3})$ provided that the diameter of the domain $\Omega$ is small enough.Comment: 19 pages, 1 figur

Graphene controlled Brewster angle device for ultra broadband terahertz modulation

Terahertz modulators with high tunability of both intensity and phase are essential for effective control of electromagnetic properties. Due to the underlying physics behind existing approaches there is still a lack of broadband devices able to achieve deep modulation. Here, we demonstrate the effect of tunable Brewster angle controlled by graphene, and develop a highly-tunable solid-state graphene/quartz modulator based on this mechanism. The Brewster angle of the device can be tuned by varying the conductivity of the graphene through an electrical gate. In this way, we achieve near perfect intensity modulation with spectrally flat modulation depth of 99.3 to 99.9 percent and phase tunability of up to 140 degree in the frequency range from 0.5 to 1.6 THz. Different from using electromagnetic resonance effects (for example, metamaterials), this principle ensures that our device can operate in ultra-broadband. Thus it is an effective principle for terahertz modulation