On the Einstein-Vlasov system: Stationary Solutions and Small Data Solutions with Charged and Massless Particles

The Vlasov matter model describes an ensemble of collisionless particles moving through space-time. These particles interact via the gravitational field which they create collectively. In the framework of General Relativity this gravitational field is described by space-time curvature. Mathematically the situation is captured by the Einstein-Vlasov system. If the particles are charged an electro-magnetic field is created as well and the Maxwell equations are coupled to the system in addition. In astrophysics Vlasov matter is widely used to describe galaxies, globular clusters or galaxy clusters. Also in cosmology or plasma physics the Vlasov matter model plays an important role.In this thesis a collection of results on the Einstein-Vlasov system is presented. The Papers I to IV are concerned with stationary solutions and the Papers V and VI contain stability results for Minkowski space-time (the trivial solution of Einstein\u27s field equations describing an empty, flat space-time), i.e.~global existence results for the time evolution problem with small initial data.In Paper I, spherically symmetric, static solutions of the Einstein-Vlasov system with massless particles are constructed. These solutions constitute very thin and highly dense shells of matter with a vacuum region at the center. One can think of these shells as highly energetic, bent light which keeps itself together through the strong gravitational field created by itself. In Paper II, charged particles are considered and the existence of spherically symmetric, static solutions of the Einstein-Vlasov-Maxwell system is proven. It is possible to obtain the large variety of different spherically symmetric, static solutions that are known in the uncharged case, as for example balls, shells and multi-shells. Paper III is concerned with isotropic solutions, i.e.~solutions where the momenta are equally distributed among the particles. In this case Vlasov matter resembles a perfect fluid in many respects. It is shown that a uniqueness result for perfect fluids can be applied to Vlasov matter. This implies that every isotropic, static solution is uniquely determined by the surface potential and in particular spherically symmetric, if its overall pressure is not too high. In Paper IV solutions are constructed where the momenta are not equally distributed among the particles. These solutions have preferred axes of rotation or even an overall angular momentum. This way axially symmetric (but not spherically symmetric), stationary solutions of the Einstein-Vlasov-Maxwell system are obtained.In Paper V, exploiting the convenient conformal invariance properties of massless Vlasov matter, this matter model is integrated into the framework of the conformal Einstein field equations. In this framework, via a conformal rescaling, the physical space-time, which might be a perturbation of Minkowski space-time or de Sitter space-time, is identified with a compact portion of the Einstein-cylinder (or perturbations thereof). This way global Cauchy problems are turned into local Cauchy problems for which methods to obtain local existence are available. A semi-global stability result for Minkowski space-time and a global stability result for de Sitter space-time is obtained this way. In Paper VI the stability of Minkowski space-time for perturbations with massless Vlasov matter is proved with a completely different method, the vector field method for relativistic transport equations. Thereby an asymptotic stability result with very weak assumptions on the initial data is obtained, in particular no compact support assumptions of any kind are necessary for the initial data

On the Interaction of Charged Particles with Plasma

The last several years have seen great activity in the study of the properties of the free electron gas or plasma, An electronic plasma is understood to mean an assembly of electrons which may be regarded as nearly free in their response to a disturbance. The assembly is assumed to be electrically neutral on the whole, due to a stationary positive charge background which will be assumed uniform in density. These electrons interact with each other via the longrange coulomb potential. Due to the nature of this interaction, the system exhibits a very interesting collective behavior which evinces itself in the existence of plasma oscillations

The Kinetic and Hydrodynamic Bohm Criterions for Plasma Sheath Formation

The purpose of this paper is to mathematically investigate the formation of a plasma sheath, and to analyze the Bohm criterions which are required for the formation. Bohm derived originally the (hydrodynamic) Bohm criterion from the Euler--Poisson system. Boyd and Thompson proposed the (kinetic) Bohm criterion from kinetic point of view, and then Riemann derived it from the Vlasov--Poisson system. In this paper, we prove the solvability of boundary value problems of the Vlasov--Poisson system. On the process, we see that the kinetic Bohm criterion is a necessary condition for the solvability. The argument gives a simpler derivation of the criterion. Furthermore, the hydrodynamic criterion can be derived from the kinetic criterion. It is of great interest to find the relation between the solutions of the Vlasov--Poisson and Euler--Poisson systems. To clarify the relation, we also study the hydrodynamic limit of solutions of the Vlasov--Poisson system.Comment: 24 pages, 2 figure

Classical and Quantum Mechanical Models of Many-Particle Systems

The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and “to a lesser extent” numerical aspects of such equations. A highlight of this meeting was a mini-course on the recent mathematical theory of Landau damping

Plasma Dynamics

Contains research objectives and reports on three research projects.Contract AF19(604)-4551 with Air Force Cambridge Research CenterAtomic Energy Commission under Contract AT(30-1)-1842Air Force Cambridge Research Center under Contract AF19(604)-5992National Science Foundation under Grant G-9330WADD Contract AF33(616)-7624 with Flight Accessories Laboratory, Wright-Patterson Air Force Base, Ohi

Nonlinear laser absorption under high-energy-density conditions

Tato disertační práce zkoumá interakce vysokointenzitního laseru s plazmaty pro účely "shock ignition" při fúzi inerciálním sbližování. Zaměřuje se na kontrolu nelineární absorpce laseru a studium interakcí laseru s pórovitými materiály nízké hustoty. Výzkum klade důraz na důležitost potlačení stimulovaného Brillouinova rozptylu (SBS) a poskytuje poznatky o účinných metodách potlačení SBS. Teoretická analýza a numerické simulace ukazují efektivní absorpci laserové energie a urychlování iontů v expandujících plazmových dutinách. Práce přispívá k pochopení interakcí vysokointenzitního laseru s materiály se strukturou a navrhuje vývoj podřízeného modelu pro ohřev pěny. Získané poznatky otevírají nové perspektivy pro studium interakce laseru s plazmatem a poukazují na potřebu dalšího výzkumu potlačení SBS a pěnových terčů.This thesis explores high-intensity laser interactions with plasmas for shock ignition in inertial confinement fusion. It focuses on controlling nonlinear laser absorption and studying laser interactions with low-density porous materials. The research emphasizes the importance of mitigating stimulated Brillouin scattering (SBS) and provides insights into effective SBS suppression methods. Theoretical analysis and numerical simulations demonstrate efficient laser energy absorption and ion acceleration in expanding plasma cavities. The thesis also contributes to understanding high-intensity laser interactions with structured materials and suggests the development of a sub-grid model for foam heating. The findings open new perspectives for laser-plasma interaction studies and highlight the need for further research in SBS mitigation and foam targets

A numerical and analytical study of kinetic models for particle-wave interaction in plasmas

This dissertation presents a study of particle-wave interaction in plasmas. It focuses on a kinetic model called quasilinear theory, which is a reduction of VlasovMaxwell (or Vlasov-Poisson) system in the weak turbulence regime. The quantized waves in plasmas, known as plasmons, are absorbed or emitted by charged particles. Meanwhile, the particles change their states due to such emission/absorption process, therefore resulting in a nonlinear kinetic system for the pdf (probability density function) of particles and plasmons. The research presented here unfolds in two main topics: structure-preserving numerical solvers, and solvability of the kinetic model. On the first topic, we are interested in numerical simulation of non-uniform magnetized plasmas, which involves two processes: particle-wave interaction and wave propagation (plasmon advection). For particle-wave interaction in homogeneous magnetized plasmas, we propose a finite element scheme that preserves all the conservation laws. Firstly, an unconditionally conservative weak form is constructed. By “unconditional” we mean that conservation is independent of the transition probabilities. Then we design a discretization that preserves such unconditional conservation property, and discuss the conditions for positivity and stability. We present numerical examples with a “bump on tail” initial configuration, showing that the particle-wave interaction results in a strong anisotropic diffusion of the particles. We generalize the strategy to obtain a conservative DG (discontinuous Galerkin) scheme. The evolution of plasmon pdf is governed by a Liouville equation with additional reaction term caused by particle-wave interaction, where the dominant Poisson bracket term necessitates trajectorial average. Hence, we propose a Galerkin approach for trajectorial average in dynamical systems. The weak form of averaged equation is derived, and the concept of trajectory bundle is introduced. To compute and store the trajectory bundles, we propose a novel algorithm, named connection-proportion algorithm, which transforms a continuous topological problem into a discrete graph theory problem. The conservative DG scheme, combined with our trajectorial average method, renders a structure-preserving solver for particle-wave interaction in non-uniform magnetized plasmas. We demonstrate that discrete weak form with/without average differs only in the choice of test/trial spaces. The complexity of each procedure is analyzed. Finally, a numerical example for a non-uniform magnetized plasma in an infinitely long symmetric cylinder is presented. It is verified that the connectionproportion algorithm allows to distinguish different trajectory bundles, and the proposed DG scheme rigorously preserves all the conservation laws. On the second topic, the existence of global weak solution to quasilinear theory for electrostatic plasmas is proved. In the one-dimensional case, both the particle pdf and the plasmon pdf can be expressed with the same auxiliary function. The auxiliary function itself, is the solution of a porous medium equation with nonlinear source terms, defined on an unbounded domain. The solvability is then proved in two steps: Firstly, the equation on finite cut-off domain with Dirichlet’s boundary condition is solved. Next, the solution, extended by zero outside the cut-off domain, turns out to be a solution to the same equation on the unbounded domain.Computational Science, Engineering, and Mathematic