May 29, 31 and June 3, 2019
May 29 and 31, 2019: Room 338, No. 5 Science Building, Minhang Campus, Shanghai Jiao Tong University
June 3, 2019: Room 306, No. 5 Science Building, Minhang Campus, Shanghai Jiao Tong University
|May. 29||16:00 ~ 17:00|
|May. 31||16:00 ~ 17:00|
|June. 3||16:00 ~ 17:00|
Kinetic models have found wide applications in physics, engineering, biology, and social sciences. Very often they contain more refined information than macroscopic equations and can provide more accurate description of physical, biological, or social phenomena. Mathematically, kinetic equations possess rich structures for which many advanced tools have been applied or developed. In these lectures, we will show analytical results of various kinetic equations used in engineering and biology.
In this talk we show a recent uniqueness result for a gas-disk interaction system with diffusive boundary conditions. This system aims at giving a more realistic description of the drag force acting on a solid when it is moving in a gas. Unlike the simplest model in the first-year physics class where the drag force is assumed to be proportional to the velocity of the solid, here we use a coupled system of the motion of the solid and the gas. In the particular model that we study, the solid motion is governed by Newton’s Second Law and the gas by a pure transport equation. In earlier works, coupling of these equations has been used to successfully explain the algebraic rate of decay to equilibrium of the solid rather than the exponential rate predicted by the simplest model. Mathematically, existence of near-equilibrium solutions for this type of systems is fairly well understood. However, uniqueness of any type of solutions was an open problem. In this talk we will show the first rigorous (and elementary) proof of the uniqueness among solutions that are only required to be locally Lipschitz; in particular, it holds for solutions far from equilibrium states. This is joint work with Anton Iatcenko.
When simulating dense gases over bounded domains, it is more desirable to use macroscopic models instead of kinetic models since computationally the latter is much more expensive. However, it is known that macroscopic approximations are only correct in the interior of the domain. Near the boundary of the physical domain, due to the sharp transition from the kinetic to macroscopic solutions a thin layer forms. Kinetic equations within this layer are called the boundary layer (BL) equations. It is crucial to understand the behaviour of the BL equations since their end-states provide the correct boundary conditions for the interior macroscopic equations. In this talk, we will review some numerical and analytical results related to kinetic boundary layer equations. In the first part of the talk, we will show efficient numerical schemes for solving these equations. In the second part, we will address the validity of the classical half-space equations. This was motivated by the work of Guo-Wu where they showed that classical kinetic BL equations break down when the boundary has a nontrivial curvature. Instead, they derived an -modified half-space equation as the correct characterization of the boundary layer. We will show an analytical comparison between the classical and Guo-Wu’s systems. These are joint works with Qin Li and Jianfeng Lu.
In this talk we deviate from the traditional transport equations and show some recent results of non-classical kinetic equations. By non-classical, we refer to those density functions which depend on extra variables in addition to time, position, and velocity as in the usual kinetic equations. We show two examples in this talk: the first one is a radiative transfer equation for photon scattering in clouds, the extra variable of which denotes the path length travelled by photons between collisions. The second example is a transport equation modelling E. Coli chemotaxis, where the extra variable measures the internal methylation level of the bacteria. We will use these kinetic models to explain the appearance of fraction diffusion on the macroscopic level. We will also show how flux-limited Keller-Segel models can originate from these kinetic equations. These are joint works with Martin Frank, Benoit Perthame, Min Tang, and Shugo Yasuda.