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Workshop on Recent Advances in PDEs (X)

Date

November 23, 2019

Venue

Room 306, No. 5 Science Building, Minhang Campus, Shanghai Jiao Tong University

Invited Speakers

Schedule

Time Speaker Title
08:50 - 09:00 Openning Remarks  
09:00 - 09:50 Ning Jiang Boltzmann equation: fluid limits and boundary layers
10:00 - 10:50 Ning Jiang Boltzmann equation: fluid limits and boundary layers
10:50 - 11:10 Group Photo & Tea Break  
11:10 - 12:00 Yi Wang Wave phenomena to the fluid-particle model
12:00 - 14:30 Lunch  
14:30 - 15:20 Yi Wang Wave phenomena to the fluid-particle model
15:20 - 15:40 Tea Break  
15:40 - 16:30 Shuangqian Liu The Vlasov-Nordstrom-Fokker-Planck system in the whole space
16:40 - 17:30 Shuangqian Liu The Vlasov-Nordstrom-Fokker-Planck system in the whole space

Program

Boltzmann equation: fluid limits and boundary layers

Ning Jiang, Wuhan University

Abstract:
In bounded domain, the fluid limits of Boltzmann equation is closely related to the boundary layers, both viscous fluid and kinetic boundary layers. Different boundary conditions of Boltzmann equation lead to different boundary conditions of fluid equations. In this talk, we review the progress in this field recent years. It is based on my joint work with 1, Masmoudi (Maxwell reflection boundary condition of Boltzmann to Navier-Stokes equations), 2, Aoki and Golse (Incoming boundary condition for Boltzmann to acoustic system), and 3, Xu Zhang (Incoming data to Navier-Stokes).


Wave phenomena to the fluid-particle model

Yi Wang, Institute of Applied Mathematics, AMSS, CAS

Abstract: Fluid-particle model is extensively used in many industries such as sprays, aerosols or sedimentation problems arising in medicine, chemical engineering, waste water treatment and Diesel engines etc. In the talk we are concerned with the wave phenomena to a fluid-particle model described by the three-dimensional Vlasov-Fokker-Planck equation coupled with the compressible Navier-Stokes or Euler equations (denoted by NS/E-VFP in abbreviation) through the relaxation drag frictions and Vlasov forces between the macroscopic and microscopic momentums. First, we prove the time-asymptotically nonlinear stability of the planar rarefaction wave for both 3D NS-VFP and E-VFP systems. Note that such a wave phenomena has never been observed for either pure Fokker-Planck equation or compressible Euler fluid with friction damping and the new wave phenomena here comes essentially from the fluid-particle interactions through the relaxation drag frictions and Vlasov forces. To prove the stability of planar rarefaction wave to 3D NS/E-VFP system, we need establish a new micro-macro decomposition around the local Maxwellian to the kinetic part of the NS/E-VFP system and this decomposition presents a unified proof framework for the time-asymptotic stability of basic wave patterns to NS/E-VFP system. Consequently, a new two-fluid model with one fluid equipped with the isothermal pressure and the degenerate viscosity coefficients depending on the corresponding density function linearly, which is exactly same as the well-known Saint-Venant system for the viscous shallow water model, is derived from the Chapman-Enskog expansion of the Vlasov-Fokker-Planck equation around the local Maxwellian. Finally, it is proved that as the fluid shear and bulk viscosities tend to zero, the solution to 3D NS-VFP system around the planar rarefaction wave converges to the corresponding solution to E-VFP system with the uniform convergence rate with respect to the viscosities.


The Vlasov-Nordstrom-Fokker-Planck system in the whole space

Shuangqian Liu, Central China Normal University and Jinan University

Abstract: The Vlasov-Nordstrom-Fokker-Planck system, which represents a relativistic generalization of the well-known Vlasov-Poisson-Fokker-Planck system in the gravitational case, describes the ensemble motion of collision particles interacting by means of their own self-generated gravitational forces. We prove the global existence and uniqueness of the solution of the corresponding Cauchy problem with small initial data. Moreover, based on the analysis for the linearized system and the significant Gagaliardo-Nirenberg-Sobolev inequalities, the optimal large-time decay rates of the system are also presented. This is a joint work with Prof. Renjun Duan and Prof. Tong Yang.


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