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Global Properties of Kinetic Plasmas.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Global Properties of Kinetic Plasmas./
Author:	Chaturvedi, Sanchit.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2023,
Description:	290 p.
Notes:	Source: Dissertations Abstracts International, Volume: 85-04, Section: B.
Contained By:	Dissertations Abstracts International85-04B.
Subject:	Theoretical physics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=30615108
ISBN:	9798380484824

Global Properties of Kinetic Plasmas.
Chaturvedi, Sanchit.

Global Properties of Kinetic Plasmas. - Ann Arbor : ProQuest Dissertations & Theses, 2023 - 290 p.

Source: Dissertations Abstracts International, Volume: 85-04, Section: B.

Thesis (Ph.D.)--Stanford University, 2023.

This item must not be sold to any third party vendors.

Kinetic theory models the evolution of gas particles influenced by both external forces-such as external magnetic or electric fields and external potentials, and self-consistent forces- including collisions and self-generated magnetic or electric fields. It has been successfully used to study plasma including both terrestrial and astrophysical. Kinetic theory coupled to Einstien's theory of general relativity allows one to understand how gases influence the fabric of space-time. The first theories that attempted to understand the world at an atomic level go back thousands of years to ancient Indian and Greek civilizations. The golden age of science in Arabia and renaissance period in Europe saw major strides in theory of atoms and gases. The precise understanding of the dynamics of billions of gas atoms is a nearly impossible problem and thus a groundbreaking development in theory of gases was the use of a probability distribution function which tracks the dynamics of the gas on "average" at a mesoscopic level. This was pioneered in works of Maxwell and Boltzmann. The equation that governs the dynamics of this probability density function is called the Boltzmann equation.In this thesis, I will consider various models that show up in collisional kinetic theory and the focus of study is the aforementioned probability density function f(t, x, v) where t ∈ R+ is the time, x ∈ Ω is the space and v ∈ R3 is the velocity. In general, Ω could be any bounded domain, whole space R3 or a three-torus T3. In this compilation of works, Ω will either be R3 or T3. Collisional equations (where all other forces are neglected) are of the following general form, where Q(f, f) is the nonlinear collisional term and for the purposes of this work it will either be the non-cutoff Boltzmann or Landau kernel. The non-cutoff Boltzmann kernel has two parameters, γ which tells us about the "hardness" of the potential and s ∈ (0, 1) which tells us about the strength of the angular singularity. Landau kernel can be thought of a singular limit of Boltzmann kernel as s → 1.For physical plasmas, collisions are weak but the self-consistent electric field term cannot be ignored and thus we end up with the following equation, where 0 < ν ≪ 1 and Q(f, f) is given by the Landau kernel. This equation is called the Vlasov-Poisson-Landau equation.In the next two chapters we will consider the Landau equation, then in the third we will work with Boltzmann equation and in the final chapter we will look at the Vlasov-Poisson-Landau equation.1.2 Local well-posedness of Boltzmann and Landau equationsFrom a PDE point of view, the first thing we would like to know is whether the equation even has a solution for a general class of initial data and whether that solution is unique. The existence theory for spatially homogenous case for Boltzmann with hard sphere model was considered by Carleman in 1933. The first local existence result for a spatially inhomogeneous collisional equation is due to Grad. In his seminal 1958 paper, Grad introduced the notion of cut-off Boltzmann where he ignores the angular singularity and initiated a systematic study of cut-off Boltzmann. From a PDE point of view, the first thing we would like to know is whether the equation even has a solution for a general class of initial data and whether that solution is unique. The existence theory for spatially homogenous case for Boltzmann with hard sphere model was considered by Carleman in 1933. The first local existence result for a spatially inhomogeneous collisional equation is due to Grad. In his seminal 1958 paper, Grad introduced the notion of cut-off Boltzmann where he ignores the angular singularity and initiated a systematic study of cut-off Boltzmann.Theorem 1.2.1 (Grad,1958). There exists a local solution for the cut-off Boltzmann equation on the whole space with the so called psuedo-Maxwellian potential and for smooth enough initial data.Since then the local existence theory for all potentials and for cut-off Boltzmann equation with boundary has been well understood.The first local existence result for a collisional model with long range interaction is due to Alexandre-Morimoto-Ukai-Xu-Tang:Theorem 1.2.2 (Alexandre-Morimoto-Ukai-Xu-Tang,2009). There exists a local solution for the Boltzmann equation with s ∈ (0, 1/2) and γ such that γ + 2s < 1 and for smooth enough initial dataNote that this does not cover the full range of singularity parameter s ∈ (0, 1). This is due to a derivative loss issue and in particular it is worst for Landau equation which can be thought of as the singular s → 1 limit. In fact, a local (in time) theory for Landau equation was not known till very recently and the first general result appeared in 2017 due to Henderson-Snelson-Tarfulea.Theorem 1.2.3 (Henderson-Snelson-Tarfulea, 2017). There exists a local solution for the Landau equation with γ ∈ [−3, 0) (called the soft potentials) and for smooth enough initial data.Since then Henderson-Snelson-Tarfulea have proved local existence results for the non cut-off Boltzmann equation as well.Theorem 1.2.4 (Henderson-Snelson-Tarfulea, 2020 and 2023). There exists a local solution for the Boltzmann equation with γ ∈ (−3, 0), s ∈ (0, 1) and for smooth enough initial data.

ISBN: 9798380484824Subjects--Topical Terms:

2144760
Theoretical physics.

Global Properties of Kinetic Plasmas.
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502 $a Thesis (Ph.D.)--Stanford University, 2023.
506 $a This item must not be sold to any third party vendors.
520 $a Kinetic theory models the evolution of gas particles influenced by both external forces-such as external magnetic or electric fields and external potentials, and self-consistent forces- including collisions and self-generated magnetic or electric fields. It has been successfully used to study plasma including both terrestrial and astrophysical. Kinetic theory coupled to Einstien's theory of general relativity allows one to understand how gases influence the fabric of space-time. The first theories that attempted to understand the world at an atomic level go back thousands of years to ancient Indian and Greek civilizations. The golden age of science in Arabia and renaissance period in Europe saw major strides in theory of atoms and gases. The precise understanding of the dynamics of billions of gas atoms is a nearly impossible problem and thus a groundbreaking development in theory of gases was the use of a probability distribution function which tracks the dynamics of the gas on "average" at a mesoscopic level. This was pioneered in works of Maxwell and Boltzmann. The equation that governs the dynamics of this probability density function is called the Boltzmann equation.In this thesis, I will consider various models that show up in collisional kinetic theory and the focus of study is the aforementioned probability density function f(t, x, v) where t ∈ R+ is the time, x ∈ Ω is the space and v ∈ R3 is the velocity. In general, Ω could be any bounded domain, whole space R3 or a three-torus T3. In this compilation of works, Ω will either be R3 or T3. Collisional equations (where all other forces are neglected) are of the following general form, where Q(f, f) is the nonlinear collisional term and for the purposes of this work it will either be the non-cutoff Boltzmann or Landau kernel. The non-cutoff Boltzmann kernel has two parameters, γ which tells us about the "hardness" of the potential and s ∈ (0, 1) which tells us about the strength of the angular singularity. Landau kernel can be thought of a singular limit of Boltzmann kernel as s → 1.For physical plasmas, collisions are weak but the self-consistent electric field term cannot be ignored and thus we end up with the following equation, where 0 < ν ≪ 1 and Q(f, f) is given by the Landau kernel. This equation is called the Vlasov-Poisson-Landau equation.In the next two chapters we will consider the Landau equation, then in the third we will work with Boltzmann equation and in the final chapter we will look at the Vlasov-Poisson-Landau equation.1.2 Local well-posedness of Boltzmann and Landau equationsFrom a PDE point of view, the first thing we would like to know is whether the equation even has a solution for a general class of initial data and whether that solution is unique. The existence theory for spatially homogenous case for Boltzmann with hard sphere model was considered by Carleman in 1933. The first local existence result for a spatially inhomogeneous collisional equation is due to Grad. In his seminal 1958 paper, Grad introduced the notion of cut-off Boltzmann where he ignores the angular singularity and initiated a systematic study of cut-off Boltzmann. From a PDE point of view, the first thing we would like to know is whether the equation even has a solution for a general class of initial data and whether that solution is unique. The existence theory for spatially homogenous case for Boltzmann with hard sphere model was considered by Carleman in 1933. The first local existence result for a spatially inhomogeneous collisional equation is due to Grad. In his seminal 1958 paper, Grad introduced the notion of cut-off Boltzmann where he ignores the angular singularity and initiated a systematic study of cut-off Boltzmann.Theorem 1.2.1 (Grad,1958). There exists a local solution for the cut-off Boltzmann equation on the whole space with the so called psuedo-Maxwellian potential and for smooth enough initial data.Since then the local existence theory for all potentials and for cut-off Boltzmann equation with boundary has been well understood.The first local existence result for a collisional model with long range interaction is due to Alexandre-Morimoto-Ukai-Xu-Tang:Theorem 1.2.2 (Alexandre-Morimoto-Ukai-Xu-Tang,2009). There exists a local solution for the Boltzmann equation with s ∈ (0, 1/2) and γ such that γ + 2s < 1 and for smooth enough initial dataNote that this does not cover the full range of singularity parameter s ∈ (0, 1). This is due to a derivative loss issue and in particular it is worst for Landau equation which can be thought of as the singular s → 1 limit. In fact, a local (in time) theory for Landau equation was not known till very recently and the first general result appeared in 2017 due to Henderson-Snelson-Tarfulea.Theorem 1.2.3 (Henderson-Snelson-Tarfulea, 2017). There exists a local solution for the Landau equation with γ ∈ [−3, 0) (called the soft potentials) and for smooth enough initial data.Since then Henderson-Snelson-Tarfulea have proved local existence results for the non cut-off Boltzmann equation as well.Theorem 1.2.4 (Henderson-Snelson-Tarfulea, 2020 and 2023). There exists a local solution for the Boltzmann equation with γ ∈ (−3, 0), s ∈ (0, 1) and for smooth enough initial data.
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