Shock phenomena for the one-dimensional Boltzmann equation and related kinetic problems

In this thesis, we present a work on the derivation of fluid models from microscopic dynamics, inspired by Hilbert’s sixth problem, on the justification of continuum equations from atomistic interactions. In particular, we focus on approximations of two-dimensional incompressible flow by finitely many interacting point vortices. Additionally, we study approximations of compressible gas dynamics by the Boltzmann equation, which models gases using a statistical description of molecular interactions, in which particles interact by elastic collisions. Furthermore, we develop the stability theory of the Boltzmann equation in one dimension for general data. In Chapter 1, we present a summary of the works included in this thesis. The first part of this thesis addresses point-vortex approximations of the incompressible Navier-Stokes equations in two dimensions. In Chapter 2, we present a work deriving convergence of a stochastic vortex system with Brownian noise to solutions of the Navier- Stokes equations in the many particle mean-field limit, where vortices are assumed to be of mixed-sign. The main method of this chapter is to prove propagation of chaos of the stochastic differential equation by comparison to the mean-field equation and relative entropy methods. More precisely, we prove quantitative estimates on the growth in time of correlations between vortices, and we show that these vanish in the many particle limit, proving our result. The second part of this thesis concerns the dynamics of collisional gases depending on one spatial dimension. In Chapter 3, we prove a large-data global well-posedness theory for the Boltzmann equation in one spatial dimension, with periodic boundary conditions and on the line. We prove this result for truncated hard-sphere interactions, for which local well-posedness results and propagation of derivatives and moments are elementary. Global well-posedness then follows using a Lyapunov functional well-adapted to one-dimensional transport problems, whose derivative exhibits a gain of regularity which is sufficient to control the nonlinearity, by a nonlinear Gronwall-type estimate. This estimate also gives decay estimates for finite mass and finite energy initial data on the line. In Chapter 4, we construct shock profile solutions for the Boltzmann equation with long-range interactions, near Navier-Stokes viscous shocks. We prove uniqueness up to translation, and show by a Chapman-Enskog argument that small amplitude Boltzmann shocks are well approximated by the Navier-Stokes equation. Our construction follows by a linearization procedure in the Chapman-Enskog scheme and a fixed-point argument. Translation invariance of the problem gives a linearized operator with a one-dimensional kernel, which we track using the stability theory of viscous shocks. Existence of solutions then follows by linearized energy estimates and by a Galerkin scheme, using stable/unstable dimensionality arguments. To overcome a loss of moment from the discretization procedure, we replace the collision operator Q with a regularized operator Q_κ, and take limits κ → 0 using functional-analytic continuity arguments.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• Wynter, Dominic Louis

Advisor dc:contributor.advisor

• Mouhot, Clement

• Fluid Dynamics
• Kinetic Theory
• Mathematical Physics
• Partial Differential Equations

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/be0986a5-e4ba-4c05-b340-8192a11b889c/download
• https://creativecommons.org/licenses/by/4.0/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.112066

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/373767