Well-posedness of a Gas-disk Interaction System

Peer reviewed: 
No, item is not peer reviewed.
Scholarly level: 
Graduate student (Masters)
Date created: 
Weiran Sun
Science: Department of Mathematics
Integro-differential equations
Kinetic equations
Gas-body interaction
Drag force
System with memory

This thesis concerns a gas-disk interaction system: the disk is immersed in a gas and acted on by a drag force and an external force. The evolution of the system is described by a coupled system of integro-differential equations. More specifically, we use a pure kinetic transport equation to model the gas and a Newton’s Second Law ODE to model the disk. The two are coupled via the drag force exerted on the disk by the gas and the boundary condition for the gas colliding with the disk.

Systems of this type have been extensively studied in the literature, both analytically and numerically. To the best of our knowledge, existing works focus on existence of nearequilibrium solutions and their long-time behaviour. However, uniqueness of solutions has not been investigated previously. In the first part of the thesis we will give the first rigorous proof of existence and uniqueness of solutions for general initial data and external forcing.

The most important physical feature of this system is its inherent recursivity: particles can collide with the disk time and time again. Recognizing this structure and introducing recursivity into the equations by the means of gas decomposition is the key to obtaining the well-posedness result.

In the second part of the thesis we will present a simple numerical method for computing the trajectory of the disk using the aforementioned gas decomposition. We will contrast it with methods used previously, and also use it to show that considering only one or two precollisions for the gas particles is sufficient to accurately compute the density distribution of the gas and the velocity of the disk.

Thesis type: 
(Thesis) M.Sc.
Document type: 
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