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1	Standing Ring Blowup Solutions for the Cubic Nonlinear Schrodinger Equation Zwiers, Ian 05 December 2012 (has links) The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension. Nonlinear Schrodinger Equation Blowup and Singularity Formation 0405
2	Standing Ring Blowup Solutions for the Cubic Nonlinear Schrodinger Equation Zwiers, Ian 05 December 2012 (has links) The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension. Nonlinear Schrodinger Equation Blowup and Singularity Formation 0405
3	Mathematical Modeling of Charged Liquid Droplets: Numerical Simulation and Stability Analysis Vantzos, Orestis 05 1900 (has links) The goal of this thesis is to study of the evolution of 3D electrically charged liquid droplets of fluid evolving under the influence of surface tension and electrostatic forces. In the first part of the thesis, an appropriate mathematical model of the problem is introduced and the linear stability analysis is developed by perturbing a sphere with spherical harmonics. In the second part, the numerical solution of the problem is described with the use of the boundary elements method (BEM) on an adaptive mesh of triangular elements. The numerical method is validated by comparison with exact solutions. Finally, various numerical results are presented. These include neck formation in droplets, the evolution of surfaces with holes, singularity formation on droplets with various symmetries and numerical evidence that oblate spheroids are unstable. Fluid mechanics -- Mathematical models. boundary elements method mathematical modeling viscous flow stability analysis singularity formation charged droplets
4	Asymptotic Analysis of Models for Geometric Motions Gavin Ainsley Glenn (17958005) 13 February 2024 (has links) <p dir="ltr">In Chapter 1, we introduce geometric motions from the general perspective of gradient flows. Here we develop the basic framework in which to pose the two main results of this thesis.</p><p dir="ltr">In Chapter 2, we examine the pinch-off phenomenon for a tubular surface moving by surface diffusion. We prove the existence of a one parameter family of pinching profiles obeying a long wavelength approximation of the dynamics.</p><p dir="ltr">In Chapter 3, we study a diffusion-based numerical scheme for curve shortening flow. We prove that the scheme is one time-step consistent.</p> Numerical analysis Partial differential equations partial differential equations (PDE) motion by mean curvature Mean Curvature Flow surface diffusion models gradient flows diffusion generated motion codimension-2 pinch-off dynamics consistency estimates heat equation axisymmetric surface diffusion Geometric PDEs one-parameter family singularity formation Numerical analysis. convergence rate Differential equations. Ordinary Differential Equations (ODE)

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