The main part of the thesis is set to review and extend the theory of the so called Strichartztype estimates. We present a new viewpoint on the subject according to which our primary goal is the study of the (endpoint) inhomogeneous Strichartz estimates. This is based on our result that the class of all homogeneous Strichartz estimates (understood in the wider sense of homogeneous estimates for data which might be outside the energy class) are equivalent to certain types of endpoint inhomogeneous Strichartz estimates. We present our arguments in the abstract setting but make explicit derivations for the most important dispersive equations like the Schr¨odinger , wave, Dirac, Klein-Gordon and their generalizations. Thus some of the explicit estimates appear for the first time although their proofs might be based on ideas that are known in other special contexts. We present also several new advancements on well-known open problems related to the Strichartz estimates. One problem we pay a special attention is the endpoint homogeneous Strichartz estimate for the kinetic transport equation (and its generalization to estimates with vector-valued norms.) For example, this problem was considered by Keel and Tao [30], but at the time the authors were not able to resolve it. We also fall short of resolving that problem but instead we prove a weaker version of it that can be useful for applications. Moreover, we also make a conjecture and give a counterexample related to that problem which might be useful for its potential resolution. Related to the latter is the fact that we now primarily use complex interpolation in the proof of the homogeneous and the inhomogeneous Strichartz estimates, which produces more natural norms in the vector-valued and the abstract setting compared to the real method of interpolation employed in earlier works. Another important direction of the thesis is to study the range of validity of the Strichartz estimates for the kinetic transport equation which requires a separate and more delicate approach due to its vector-valued dispersive inequality and a special invariance property. We produce an almost optimal range of estimates for that equation. It is an interesting fact that the failure of certain endpoint estimates with L∞ or L1-space norms can be shown on characteristics of Besicovitch sets. With regard to applications of these estimates we demonstrate for the first time in the context of a nonlinear kinetic system (the Othmer-Dunbar-Alt kinetic model of bacterial chemotaxis) that its global well-posedness for small data can be achieved via Strichartz estimates for the kinetic transport equation. Another new development in the thesis is connected to the question of the global regularity of the Dirac-Klein-Gordon system in space dimensions above one for large initial data. That question was instigated in the 1970’s by Chadam and Glassey [12, 13, 22] and although a great number of mathematicians have made contributions in the past 30 years, we, together with the independent recent preprint by Gr¨unrock and Pecher [24], present the first global result for large data. In particular, we prove that in two space dimensions the system has spherically symmetric solutions for all time if the initial data is spherically symmetric and lies in a certain regularity class. Our result is achieved via new inhomogeneous Strichartz estimates for spherically symmetric functions that we prove in the abstract setting and in particular for the wave equation. We make a number of other lesser improvements and generalizations in relation to the Strichartz estimates that shall be presented in the main body of this text.

We present a new iterative approach to solving neutral-particle transport problems. The scheme divides the transport solution into its particular and homogeneous or “source-free” components. The particular problem is solved directly, while the homogeneous problem is found iteratively. To organize the iterative inversion of the homogeneous components, we exploit the structures of the so called Case-modes that compose it. The asymptotic Case-modes, those that vary slowly in space and angle, are assigned to a diffusion solver. The remaining transient Case-modes, those with large spatial gradients, are assigned to a transport solver. The scheme iterates on the contribution from each solver until the particular plus homogeneous solution converges. The iterative method is implemented successfully in slab geometry with isotropic scattering and one energy group. The convergence rate of the method is only weakly dependent on the scattering ratio of the problem. Instead, the rate of convergence depends strongly on the material thickness of the slab, with thick slabs converging in few iterations. The transient solution is obtained by applying a One Cell Inversion scheme instead of a Source Iteration based scheme. Thus, the transient unknowns are calculated with little coordination between them. This independence among unknowns makes our scheme ideally suited for transport calculations on parallel architectures. The slab geometry iterative scheme is adapted to XY geometry. Unfortunately, this attempt to extend the slab geometry iterative scheme to multiple dimensions has not been successful. The exact filtering scheme needed to discriminate asymptotic and transient modes has not been obtained and attempts to approximate this filtering process resulted in a divergent iterative scheme. However, the development of this iterative scheme yield valuable analysis tools to understand the Case-mode structure of any spatial discretization under arbitrary material properties.

This work presents a method to quantitatively calculate the error induced through application of approximate boundary conditions in quantum charge transport simulations based on the Wigner transport equation (WTE). Except for the special case of homogeneous material, there exists no methodology for the calculation of exact boundary conditions. Consequently, boundary conditions are customarily approximated by equilibrium or near-equilibrium distributions known to be correct in the classical limit. This practice can, however, exert deleterious impact on the accuracy of numerical calculations and can even lead to unphysical results. The Yoder group has recently developed a series expansion for exact boundary conditions which, when truncated, can be used to calculate boundary conditions of successively greater accuracy through consideration of successively higher order terms, the computational penalty for which is however not to be underestimated. This thesis focuses on the calculation and analysis of the second order term of the series expansion. A method is demonstrated to calculate the term for any general device structure in one spatial dimension. In addition, numerical analysis is undertaken to directly compare the first and second order terms. Finally a method to incorporate the first order term into simulation is formulated.

This is Part B. / <p> Another approach is adopted for deriving the moments equations in spherical geometry using a spherical harmonics expansion of the neutron transport equation over a variable range of the direction cosine. Because of complications and uncertainties in establishing boundary conditions for the equations, only the zero'th order equations are solved, in an idealized situation, in order that a feel for equations and boundary conditions may be obtained.</p> <p> The equations are compared to equations given in a paper 'Directionally Discontinuous Harmonic Solutions of the Neutron Transport Equation in Spherical Geometry', by A. A. Harms and E. A. Attia. Analytical solutions for the zero'th order equations are given for equations developed there and to the equations developed in this paper. Numerical values are presented to give an idea of what accuracies might be expected. It is hoped that similar techniques can be used to solve the higher order equations analytically, and that appropriate boundary conditions can be found.</p> / Thesis / Master of Engineering (MEngr)

This dissertation is devoted to investigating the time dependent neutron transport equations with reflecting boundary conditions. Two typical geometries --- slab geometry and spherical geometry --- are considered in the setting of <I>L^p</I> including <I>L^1</I>. Some aspects of the spectral properties of the transport operator <I>A</I> and the strongly continuous semigroup <I>T(t)</I> generated by <I>A</I> are studied. It is shown under fairly general assumptions that the accumulation points of { m Pas}(A):=sigma (A) cap { lambda :{ m Re}lambda > -lambda^{ast} }, if they exist, could only appear on the line { m Re}lambda =-lambda^{ast}, where lambda^{ast} is the essential infimum of the total collision frequency. The spectrum of <I>T(t)</I> outside the disk {lambda : |lambda| leq exp (-lambda^{ast} t)} consists of isolated eigenvalues of <I>T(t)</I> with finite algebraic multiplicity, and the accumulation points of sigma (T(t)) igcap{ lambda : |lambda| > exp (-lambda^{ast} t)}, if they exist, could only appear on the circle {lambda :|lambda| =exp (-lambda^{ast} t)}. Consequently, the asymptotic behavior of the time dependent solution is obtained. / Ph. D.

The Boltzmann Transport Equation (BTE) has been the keystone of the kinetic theory, which is at the center of Statistical Mechanics bridging the gap between the atomic structures and the continuum-like behaviors. The existence of solutions has been a great mathematical challenge and still remains elusive. As a grazing limit of the Boltzmann operator, the Fokker-Planck-Landau (FPL) operator is of primary importance for collisional plasmas. We have worked on the following three different projects regarding the most important kinetic models, the BTE and the FPL Equations. (1). A Discontinuous Galerkin Solver for Nonlinear BTE. We propose a deterministic numerical solver based on Discontinuous Galerkin (DG) methods, which has been rarely studied. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. To save the tremendous computational cost with increasing order of polynomials and number of mesh nodes, as well as to resolve loss of conservations due to domain truncations, the following combined procedures are applied. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum and/or energy), with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. We applied the property of ``shifting symmetries" in the weight matrix, which consists in finding a minimal set of basis matrices that can exactly reconstruct the complete family of collision weight matrix. This procedure, together with showing the sparsity of the weight matrix, reduces the computation and storage of the collision matrix from O(N3) down to O(N^2). (2). Spectral Gap for Linearized Boltzmann Operator. Spectral gaps provide information on the relaxation to equilibrium. This is a pioneer field currently unexplored form the computational viewpoint. This work, for the first time, provides numerical evidence on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a ``collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function the Rayleigh quotient of the "collision matrix" and with constraints the conservation laws. A conservation correction then applies. We also study the convergence of the approximate Rayleigh quotient to the real spectral gap. (3). A Conservative Scheme for Approximating Collisional Plasmas. We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equations coupled with Poisson equations. The original problem is splitted into two subproblems: collisonless Vlasov problem and collisonal homogeneous Fokker-Planck-Landau problem. They are handled with different numerical schemes. The former is approximated using Runge-Kutta Discontinuous Galerkin (RKDG) scheme with a piecewise polynomial basis subspace covering all collision invariants; while the latter is solved by a conservative spectral method. To link the two different computing grids, a special conservation routine is also developed. All the projects are implemented with hybrid MPI and OpenMP. Numerical results and applications are provided. / text

This work evaluates whether the branch of Reynolds Averaging in Computational Fluid Dynamics can be used to, based on real field measurements, find the source of the measured gas in question. The method to do this is via the adjoint to the Reynolds Averaged Scalar Transport equation, explained and derived herein. Since the Inverse is only as good as the main equation, forward runs are made to evaluate the turbulence model. Reynolds Averaged Navier Stokes is solved in a domain containing 4 cubes in a 2x2 grid, generating a velocity field for said domain. The turbulence model in question is a union of two modifications to the standard two equation k-ε model in order to capture blunt body turbulence but also to model the atmospheric boundary layer. This field is then inserted into the Reynolds Averaged Scalar Transport equation and the simulation is compared to data from the Environmental Flow wind tunnel in Surrey. Finally the adjoint scalar transport is solved, both for synthetic data that was generated in the forward run, but also for the data from EnFlo. It was discovered that the turbulent Schmidt number plays a major role in capturing the dispersed gas, three different Schmidt numbers were tested, the standard 0.7, the unconventional 0.3 and a height dependent Schmidt number. The widely accepted value of 0.7 did not capture the dispersion at all and gave a huge model error. As such the adjoint scalar transport was solved for 0.3 and a height dependent Schmidt number. The interaction between measurements, the real source strength (which is not used in the adjoint equation, but needed to find the source) and the location of the source is intricate indeed. Over estimation and under estimation of the forward model may cancel out in order to find the correct source, with the correct strength. It is found that Reynolds Averaged Computational fluid dynamics may prove useful in source term estimation. / Detta arbete utvärderar hurvida Reynolds medelvärdesmodellering inom flödessimuleringar kan användas till att finna källan till en viss gas baserat på verkliga mätningar ute i fält. Metoden går ut på att använda den adjungerade ekvationen till Reynolds tidsmedlade skalära transportekvationen, beskriven och härledd häri. Då bakåtmodellen bygger på framåtmodellen, måste såleds framåtmodellen utvärderas först. Navier-Stokes ekvationer med en turbulensmodell löses i en domän, innehållandes 4 kuber i en 2x2 orientering, för vilken en hastighetsprofil erhålles. Turbulensmodellen som användes är en union av två olika k-ε modeller, där den ena fångar turbulens runt tröga objekt och den andra som modellerar atmosfäriska gränsskiktet. Detta fält används sedan i framåtmodellen av skalära transportekvationen, som sedan jämförs med körningar från EnFlo windtunneln i Surrey. Slutligen testkörs även den adjungerade ekvationen, både för syntetiskt data genererat i framåtkörningen men även för data från EnFlo tunneln. Då det visade sig att det turbulenta Schmidttalet spelar stor roll inom spridning i det atmosfäriska gränsskiktet, gjordes testkörningar med tre olika Schmidttal, det normala 0.7, det väldigt låga talet 0.3 samt ett höjdberoende Schmidttal. Det visade sig att det vanligtvis använda talet 0.7 inte alls lyckas fånga spridningen tillfredställande och gav ett stort modellfel. Därför löstes den adjungerade ekvationen för 0.3 samt för ett höjdberoende Schmidttal. Interaktionen mellan mätningar, den riktiga källstyrkan (som är okänd i den adjungerade ekvationen) samt källpositionen är onekligen intrikat. Över- samt underestimationer av framåtmodellen kan ta ut varandra i bakåtmodellen för att finna rätt källa, med rätt källstyrka. Det ter sig som Reynolds turbulensmodellering mycket möjligt kan användas inom källtermsuppskattning.

In this thesis, we investigate the regularity and approximation of a hyperbolic-elliptic coupled problem. In particular, we consider the Poisson and the transport equation where both are assigned nonhomogeneous Dirichlet boundary conditions. The coupling of the two problems is executed as follows. The right hand side function of the Poisson equation is the solution ρ of the transport equation whereas the gradient field E = −∇u, with u being solution of the Poisson problem, is the convective field for the transport equation. The analysis is done throughout on a nonconvex, not simply connected domain that is supposed to be homeomorph to an annular domain. In the first part of this thesis, we will focus on the existence and uniqueness of a classical solution to this highly nonlinear problem using the framework of Hölder continuous functions. Herein, we distinguish between a time dependent and time independent formulation. In both cases, we investigate the streamline functions defined by the convective field E. These are used in the time dependent case to derive an operator equation whose fixed point is the streamline function to the gradient of the classical solution u. In the time independent setting, we formulate explicitly the solution operators L for the Poisson and T for the transport equation and show with a fixed point argument the existence and uniqueness of a classical solution (u,ρ). The second part of this thesis deals with the approximation of the coupled problem in Sobolev spaces. First, we show that the nonlinear transport equation can be formulated equivalently as variational inequality and analyse its Galerkin finite element discretization. Due to the nonlinearity of the coupled problem, it is necessary to use iterative solvers. We will introduce the staggered algorithm which is an iterative method solving alternating the Poisson and transport equation until convergence is obtained. Assuming that LοT is a contraction in the Sobolev space H1(Ω), we will investigate the convergence of the discrete staggered algorithm and obtain an error estimate. Subsequently, we present numerical results in two and three dimensions. Beside the staggered algorithm, we will introduce other iterative solvers that are based on linearizing the coupled problem by Newton’s method. We illustrate that all iterative solvers converge satisfactorily to the solution (u, ρ).

Heat dissipation is a critical bottleneck for microelectronic device performance and longevity. At micrometer and nanometer length scales heat carriers scatter at the boundaries of the material reducing its thermal conductivity. Additionally, thermal boundary conductance across dissimilar material interfaces becomes a dominant factor due to the increase in surface area relative to the volume of device layers. Therefore, techniques for monitoring spatially varying temperature profiles, and methods to improve thermal performance are critical to future device design and optimization. The first half of this thesis focused on frequency domain thermoreflectance (FDTR) to measure thermal transport in nanometer-thick polymer films and across an organic-inorganic interface. Hybrid structures of organic and inorganic materials are widely used in devices such as batteries, solar cells, transistors, and flexible electronics. The Langmuir-Blodgett (LB) technique was used to fabricate nanometer-thick polymer films ranging from 2 - 30 nm. FDTR was then used to experimentally determine the thermal boundary conductance between the polymer and solid substrates. The second half of the thesis focused on developing a fundamental understanding of thermal transport in wide-bandgap (WBG) materials, such as GaN, and ultrawide-bandgap (UWBG) materials, such as diamond, to improve thermal dissipation in power electronic devices. Improvements in WBG materials and device technologies have slowed as thermal properties limit their performance. UWBG materials can provide a dramatic leap in power electronics technologies while temporarily sidestepping the problems associated with their WBG cousins. However, for power electronic devices based on WBG- and UWBG-materials to reach their full potential the thermal dissipation issues in these hard-driven devices must be understood and solved. FDTR provides a comprehensive pathway towards fully understanding the physics governing phonon transport in WBG- and UWBG-based devices. By leveraging FDTR imaging and measuring samples as a function of temperature, defect concentration, and thickness, in conjunction with transport models, a well-founded understanding of the dominant thermal-carrier scattering mechanisms in these devices was achieved. With this knowledge we developed pathways for their mitigation.

This thesis investigates how magnetographic data may be used to study the longterm behaviour of the magnetic field distribution across the surface of the sun.