The goal of this thesis is to apply the theory of multilinear weighted Fourier estimates to nonlinear dispersive equations in order to tackle problems in regularity, well-posedness, and pointwise convergence of solutions. Dispersion of waves is a ubiquitous physical phenomenon that arises, among others, from problems in shallow-water propagation, nonlinear optics, quantum mechanics, and plasma physics. A natural tool for understanding the related physics is to study waves/signals simultaneously from both physical and spectral perspectives. Specifically, we will treat nonlinearities as multilinear operator perturbations, which (by the method of spacetime Fourier transforms), exhibit smoothing properties in norms defined to reflect the dispersive natures of the solutions. Our model equation is the quantum Zakharov system, which can be viewed as a variation on the cubic nonlinear Schrödinger equation (NLS). We investigate the model in various contexts (adiabatic limits, nonlinear Schrödinger limits, semi-classical limits). We additionally study a variation of Carleson's Fourier convergence problem in the context of pointwise convergence of the full Schrödinger operator with non-zero potential.

The method of steepest descent is used to solve partial differential equations of mixed type. In the main hypothesis for this paper, H, L, and S are Hilbert spaces, T: H -> L and B: H -> S are functions with locally Lipshitz Fréchet derivatives where T represents a differential equation and B represents a boundary condition. Define ∅(u) = 1/2 II T(u) II^2. Steepest descent is applied to the functional ∅. A new smoothing technique is developed and applied to Tricomi type equations (which are of mixed type). Finally, the graphical outputs on some test boundary conditions are presented in the table of illustrations.

The effective degree model has been applied to both SIR and SIS type diseases (those which confer permanent immunity and those which do not, respectively) with great success. The original model considers a large system of ODEs to keep track of the number of infected and susceptible neighbours of an individual. In this thesis, we use a generating function approach on the SIR effective degree model to transform the system of ODEs into a single PDE. This has the advantage of allowing the con- sideration of infinite networks. We derive existence and uniqueness of solutions to the PDE. Furthermore, we show that the linear stability of the PDE is governed by the same disease threshold derived by the ODE model, and we also show the nonlinear instability of the PDE agrees with the same disease threshold. / Graduate

<div>This thesis mainly focuses on the PDE theories that arise from the study of hydrodynamics of nematic liquid crystals. </div><div><br></div><div>In Chapter 1, we give a brief introduction of the Ericksen--Leslie director theory and Beris--Edwards <i>Q</i>-tensor theory to the PDE modeling of dynamic continuum description of nematic liquid crystals. In the isothermal case, we derive the simplified Ericksen--Leslie equations with general targets via the energy variation approach. Following this, we introduce a simplified, non-isothermal Ericksen--Leslie system and justify its thermodynamic consistency. </div><div><br></div><div>In Chapter 2, we study the weak compactness property of solutions to the Ginzburg--Landau approximation of the simplified Ericksen--Leslie system. In 2-D, we apply the Pohozaev type argument to show a kind of concentration cancellation occurs in the weak sequence of Ginzburg--Landau system. Furthermore, we establish the same compactness for non-isothermal equations with approximated director fields staying on the upper semi-sphere in 3-D. These compactness results imply the global existence of weak solutions to the limit equations as the small parameter tends to zero. </div><div><br></div><div>In Chapter 3, we establish the global existence of a suitable weak solution to the co-rotational Beris–Edwards system for both the Landau–De Gennes and Ball–Majumdar bulk potentials in 3-D, and then study its partial regularity by proving that the 1-D parabolic Hausdorff measure of the singular set is 0.</div><div><br></div><div>In Chapter 4, motivated by the study of un-corotational Beris--Edwards system, we construct a suitable weak solution to the full Ericksen--Leslie system with Ginzburg--Landau potential in 3-D, and we show it enjoys a (slightly weaker) partial regularity, which asserts that it is smooth away from a closed set of parabolic Hausdorff dimension at most 15/7.</div>

abstract: Numerous works have addressed the control of multi-robot systems for coverage, mapping, navigation, and task allocation problems. In addition to classical microscopic approaches to multi-robot problems, which model the actions and decisions of individual robots, lately, there has been a focus on macroscopic or Eulerian approaches. In these approaches, the population of robots is represented as a continuum that evolves according to a mean-field model, which is directly designed such that the corresponding robot control policies produce target collective behaviours. This dissertation presents a control-theoretic analysis of three types of mean-field models proposed in the literature for modelling and control of large-scale multi-agent systems, including robotic swarms. These mean-field models are Kolmogorov forward equations of stochastic processes, and their analysis is motivated by the fact that as the number of agents tends to infinity, the empirical measure associated with the agents converges to the solution of these models. Hence, the problem of transporting a swarm of agents from one distribution to another can be posed as a control problem for the forward equation of the process that determines the time evolution of the swarm density. First, this thesis considers the case in which the agents' states evolve on a finite state space according to a continuous-time Markov chain (CTMC), and the forward equation is an ordinary differential equation (ODE). Defining the agents' task transition rates as the control parameters, the finite-time controllability, asymptotic controllability, and stabilization of the forward equation are investigated. Second, the controllability and stabilization problem for systems of advection-diffusion-reaction partial differential equations (PDEs) is studied in the case where the control parameters include the agents' velocity as well as transition rates. Third, this thesis considers a controllability and optimal control problem for the forward equation in the more general case where the agent dynamics are given by a nonlinear discrete-time control system. Beyond these theoretical results, this thesis also considers numerical optimal transport for control-affine systems. It is shown that finite-volume approximations of the associated PDEs lead to well-posed transport problems on graphs as long as the control system is controllable everywhere. / Dissertation/Thesis / Doctoral Dissertation Mechanical Engineering 2019

We study energy-driven nonlocal pattern forming systems with opposing interactions. Selections are drawn from the area of Quantum Physics, and nonlocalities are present via Coulombian type interactions. More precisely, we study Thomas-Fermi-Dirac-Von Weizsacker (TFDW) type models, which are mass-constrained variational problems. The TFDW model is a physical model describing ground state electron configurations of many-body systems. First, we consider minimization problems of the TFDW type, both for general external potentials and for perturbations of the Newtonian potential satisfying mild conditions. We describe the structure of minimizing sequences, and obtain a more precise characterization of patterns in minimizing sequences for the TFDW functionals regularized by long-range perturbations. Second, we consider the TFDW model and the Liquid Drop Model with external potential, a model proposed by Gamow in the context of nuclear structure. It has been observed that the TFDW model and the Liquid Drop Model exhibit many of the same properties, especially in regard to the existence and nonexistence of minimizers. We show that, under a "sharp interface'' scaling of the coefficients, the TFDW energy with constrained mass Gamma-converges to the Liquid Drop model, for a general class of external potentials. Finally, we present some consequences for global minimizers of each model. / Thesis / Doctor of Philosophy (PhD)

This thesis studies the traveling wavefront created by the autocatalytic cubic chemical reaction A + 2B → 3B involving two chemical species A and B, where A is the reactant and B is the auto-catalyst. The diffusion coefficients for A and B are given by DA and DB. These coefficients differ as a result of the chemical species having different size and/or weight. Theoretical results show there exist bounds, v* and v*, depending on DB/DA, where for speeds v ≥ v*, a traveling wave solution exists, while for speeds v < v*, a solution does not exist. Moreover, if DB ≤ DA, and v* and v* are similar to one another and in the order of DB/DA when it is small. On the other hand, when DA ≤ DB there exists a minimum speed vmin, such that there is a traveling wave solution if the speed v > vmin. The determination of vmin is very important in determining the dynamics of general solutions. To fill in the gap of the theoretical study, we use numerical methods to determine vmin for various cases. The numerical algorithm used is the fourth-order Runge-Kutta method (RK4).

The aim of this paper is to show how the spine of a PDE surface can be generated and how it can be used to efficiently parameterise a PDE surface. For the purpose of the work presented here an approximate analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. Furthermore, it is shown that a parameterisation can be introduced on the spine enabling intuitive manipulation of PDE surfaces.

No / Geometric modelling using Partial Differential Equations (PDEs) has been gradually recognised due to its smooth instinct, as well as the ability to generate a variety of geometric shapes by intuitively manipulating a relatively small set of PDE boundary curves. In this paper we explore and demonstrate the feasibility of the PDE method in facial geometry parameterisation. The geometry of a generic face is approximated by evaluating spectral solutions to a group of fourth order elliptic PDEs. Our PDE-based parameterisation scheme can produce and animate a high-resolution 3D face with a relatively small number of parameters. By taking advantage of parametric representation, the PDE method can use one fixed animation scheme to manipulate the facial geometry in varying Levels of Detail (LODs), without any further process.

We propose the use of Partial Differential Equations (PDEs) for shape modelling within visual cyberworlds. PDEs, especially those that are elliptic in nature, enable surface modelling to be defined as boundary-value problems. Here we show how the PDE based on the Biharmonic equation subject to suitable boundary conditions can be used for shape modelling within visual cyberworlds. We discuss an analytic solution formulation for the Biharmonic equation which allows us to define a function based geometry whereby the resulting geometry can be visualised efficiently at arbitrary levels of shape resolutions. In particular, we discuss how function based PDE surfaces can be readily integrated within VRML and X3D environments