Global Classical Solutions to the Relativistic Boltzmann Equation Without Angular Cut-off

We prove the unique existence and exponential decay of global in time classical solutions to the special relativistic Boltzmann equation without any angular cut-off assumptions with initial perturbations in some weighted Sobolev spaces. We consider perturbations of the relativistic Maxwellian equilibrium states. We work in the case of a spatially periodic box. We consider the general conditions on the collision kernel from Dudynski and Ekiel-Jezewska (Commun. Math. Phys. 115(4):607-629,1985). Additionally, we prove sharp constructive upper and coercive lower bounds for the linearized relativistic Boltzmann collision operator in terms of a geometric fractional Sobolev norm; this shows that a spectral gap exists and that this behavior is similar to that of the non-relativistic case as shown by Gressman and Strain (Journal of AMS 24(3), 771-847, 2011). We also derive the relativistic analogue of Carleman dual representation of Boltzmann collision operator. Lastly, we explicitly compute the Jacobian of a collision map (p, q) to (cp\u27 + (1-c)p, q) for a fixed c in (0, 1), and it is shown that the Jacobian is bounded above in p and q. This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence for the relativistic kinetic theory without the Grad\u27s angular cut-off assumption

Computing Fixed Point Floer Homology

In the summer of 2009, our group developed a computer program that computes Hochschild Homology, a topological invariant. While we must assume that the reader has at least encountered algebraic topology, in this paper we provide the mathematical background and motivation for our algorithm. After presenting a number of definitions, we will explain how the algorithm works. Specifically, we first define the Floer complex of two curves on surface; the resulting homology is invariant under isotopies. Then, we introduce the Fukaya category associated to a sequence of curves. Next, we define the Hochschild complex of the Fukaya category. And finally, we describe an algorithm for computing Hochschild Homology and provide some examples

On the temperature distribution of a body heated by radiation

In this paper, we study the temperature distribution of a body when the heat is transmitted only by radiation. The heat transmitted by convection and conduction is ignored. We consider the stationary radiative transfer equation in the local thermodynamic equilibrium. We prove that the stationary radiative transfer equation coupled with the non-local temperature equation is well-posed in a generic case when emission-absorption or scattering of interacting radiation is considered. The emission-absorption and the scattering coefficients are generic and can depend on the frequency of radiation or the local temperature. We also establish the entropy production formula of the system.Comment: 33 pages, 2 figure

Kinetic Models for Semiflexible Polymers in a Half-plane

Based on a general discrete model for a semiflexible polymer chain, we introduce a formal derivation of a kinetic equation for semiflexible polymers in the half-plane via a continuum limit. The resulting equation is the kinetic Fokker-Planck-type equation with the Laplace-Beltrami operator under the trapping boundary condition if one assumes the energy-minimizing transition at the boundary. We then study the well-posedness and the long-chain asymptotics of the solutions of the resulting equation. In particular, we prove that there exists a unique measure solution for the corresponding boundary value problem. In addition, we prove that the equation is hypoelliptic and the solutions are locally H\"older continuous near the singular set. Finally, we provide the asymptotic behaviors of the solutions for large polymer chains.Comment: 70 pages, 4 figure

Effectiveness of vaccination and quarantine policies to curb the spread of COVID-19

A pandemic, the worldwide spread of a disease, can threaten human beings from the social as well as biological perspectives and paralyze existing living habits. To stave off the more devastating disaster and return to a normal life, people make tremendous efforts at multiscale levels from individual to worldwide: paying attention to hand hygiene, developing social policies such as wearing masks, social distancing, quarantine, and inventing vaccines and remedy. Regarding the current severe pandemic, namely the coronavirus disease 2019, we explore the spreading-suppression effect when adopting the aforementioned efforts. Especially the quarantine and vaccination are considered since they are representative primary treatments for block spreading and prevention at the government level. We establish a compartment model consisting of susceptible (S), vaccination (V), exposed (E), infected (I), quarantined (Q), and recovered (R) compartments, called SVEIQR model. We look into the infected cases in Seoul and consider three kinds of vaccines, Pfizer, Moderna, and AstraZeneca. The values of the relevant parameters are obtained from empirical data from Seoul and clinical data for vaccines and estimated by Bayesian inference. After confirming that our SVEIQR model is plausible, we test the various scenarios by adjusting the associated parameters with the quarantine and vaccination policies around the current values. The quantitative result obtained from our model could suggest a guideline for policy making on effective vaccination and social policies.Comment: 8 pages, 5 figure

Compactness and existence theory for a general class of stationary radiative transfer equations

In this paper, we study the steady-states of a large class of stationary radiative transfer equations in a $C^1$ convex bounded domain. Namely, we consider the case in which both absorption-emission and scattering coefficients depend on the local temperature $T$ and the radiation frequency $\nu.$ The radiative transfer equation determines the temperature of the material at each point. The main difficulty in proving existence of solutions is to obtain compactness of the sequence of integrals along lines that appear in several exponential terms. We prove a new compactness result suitable to deal with such a non-local operator containing integrals on a line segment. On the other hand, to obtain the existence theory of the full equation with both absorption and scattering terms we combine the compactness result with the construction of suitable Green functions for a class of non-local equations