Monthly Archives: August 2018

分科会「運動論方程式，流体力学とその周辺」（第6回）

日時：	平成30年8月24日（金）　15:00-17:00
場所：	京都大学桂キャンパスCクラスタ総合研究棟III（C3棟）　3階　b3n03室（航空宇宙工学専攻会議室）
講演1：	Entropy methods and cross-diffusion systems: derivation and entropy structure
	Prof. Ansgar Jüngel (Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria)
要旨1：	Nature is dominated by systems composed of many individuals, belonging to various species, with a collective behavior. Instead of calculating the trajectories of all individuals, it is computationally much simpler to describe the dynamics of the individuals on a macroscopic level by averaged quantities such as population densities. This leads to systems of highly nonlinear partial differential equations with cross diffusion, which may reveal surprising effects such as uphill diffusion and diffusion-induced instabilities. In this talk, we detail some approaches on the derivation of cross-diffusion equations from kinetic, fluiddynamical, and stochastic models. Relations to thermodynamic principles and the results of Kawashima and Shizuta are detailed. The entropy structure can also be found in nonstandard applications like van-der-Waals fluids, population dynamics, and exotic financial derivatives. It allows for a mathematical existence theory and stable numerical approximations with guaranteed lower and upper bounds.
講演2：	Linear Boltzmann equation and fractional diffusion
	Prof. François Golse (Centre de mathématiques Laurent Schwartz, Ecole Polytechnique, France)
要旨2：	Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient σ. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient α. Moreover, assume that there is a tem- perature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where σ → +∞ and 1 – α ~ C/σ, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of a kinetic model which is based on the harmonic extension definition of √-Δ. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (“heavy tails”) or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273-280].