中科院数学与系统科学研究院
数学研究所
学术报告
偏微分方程研讨班
报告人：Prof. Renjun Duan(The Chinese University of Hong Kong)
题 目：Compressible fluid limit for smooth solutions to the Landau equation
时 间：2022.07.15（星期五）09:30-10:30
地 点：Zoom ID: 924 888 5804 Passcode: AMSS2022
https://us06web.zoom.us/j/9248885804?pwd=OXg2VDhBT0pxRkd1Z3RJS0NFMnRDdz09
摘 要：Although the compressible fluid limit of the Boltzmann equation with cutoff has been well studied, there are few analogous results in case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, essentially due to the velocity diffusion effect of collision operator such that pointwise bounds of solutions are hard to obtain without replying on Sobolev embeddings. In the talk, we are concerned with the compressible Euler limit of the Landau equation for Coulomb potentials in the whole space. We develop a method of high-order energy estimates involving the small Knudsen number. Applications to other physical situations where the self-consistent electromagnetic field is coupled will also be discussed. In particular, we establish the convergence of smooth solutions of the Vlasov-Maxwell-Boltzmann/Landau system to the ones of the Euler-Maxwell system in the compressible setting. These are joint works with Dongcheng Yang and Hongjun Yu.
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报告人：Dr. Sanchit Chaturvedi (Stanford University)
题 目：Stability of vacuum for the non-cut-off Boltzmann equation with moderately soft potentials
时 间：2022.07.15（星期五）14:00-15:00
地 点：Zoom ID: 924 888 5804 Passcode: AMSS2022
https://us06web.zoom.us/j/9248885804?pwd=OXg2VDhBT0pxRkd1Z3RJS0NFMnRDdz09
摘 要：The vector field method developed by Klainerman has been widely successful in the study of wave equations and general relativity. Recently, the vector field approach has been adapted to understand the dispersion due to the transport operator in both collisionless and collisional kinetic models. As a proof of concept, I will discuss the stability of vacuum for Boltzmann equation with moderately soft potentials. The nonlocality of the Boltzmann operator poses a lot of difficulty and forces us to use a purely energy based approach. This is in contrast to the paper by Luk (Stability of vacuum for the Landau equation with moderately soft potentials) on Landau equation in a similar setting where a maximum principle is both proved and needed.
