Department of Mathematics - Faculty Publications
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Item Notes on Ascona Converence: Nonlinear Hamiltonian PDEs July 1 - 6, 2012(2012-09-25) Richards, GeordieThis document contains the abstracts of talks given at a conference in Ascona 2012, followed by handwritten notes of the talks taken by Geordie Richards.Item Maximal-in-time behavior of deterministic and stochastic dispersive partial differential equations(2012-09-19) Geordie, RichardsItem Critical Sobolev Inequalities and Applications to Navier-Stokes Equations(2012-02-10) Wang, YunIn this talk, some critical Sobolev inequalities are introduced. These inequalities are generalizations of Brezis-Gallouet-Wainger inequality. We apply such inequalities to the two-dimensional non-homogeneous incompressible Navier-Stokes problem and prove global existence of strong solutions.Item Computing Steady State Solutions of a Rotating Thin Film Equation(2011-10-25) Ginsberg, Daniel ; Simpson, GideonThis script demonstrates the use of an iterative spectral algorithm for computing steady state solutions of the thin film equation on a rotating cylinder. This is intended to complement the manuscript "Analytical and Numerical Results on the Positivity of Steady State Solutions of a Thin Film Equation" by D. Ginsberg & G. Simpson. See http://arxiv.org/abs/1101.3261 for details of the algorithm.Item Anderson localization triggered by spin disorder(2011-10-23) Egli, DanielThe phenomenon of Anderson localization is studied for a class of one-particle Schrödinger operators with random Zeeman interactions. These operators arise as follows: Static spins are placed randomly on the sites of a simple cubic lattice according to a site percolation process with density x and coupled to one another ferromagnetically. Scattering of an electron in a conduction band at these spins is described by a random Zeeman interaction term that originates from indirect exchange. It is shown rigorously that, for positive values of x below the percolation threshold, the spectrum of the one-electron Schrödinger operator near the band edges is dense pure-point, and the corresponding eigenfunctions are exponentially localized. Localization near the band edges persists in a weak external magnetic field, H, but disappears gradually, as H is increased. Our results lead us to predict the phenomenon of colossal (negative) magnetoresistance and the existence of a Mott transition, as H and/or x are increased. Our analysis is motivated directly by experimental results concerning the magnetic alloy 𝖤𝗎x𝖢𝖺1−x𝖡6.Item Breakdown Criteria for Nonvacuum Einstein Equations(2011-10-13) Shao, ArickWe extend a recent breakdown/continuation result of S. Klainerman and I. Rodnianski for the Einstein-vacuum equations to the Einstein-scalar and the Einstein-Maxwell equations. Roughly, the main theorem states that if an existing local solution of these equations satisfy certain uniform bounds for the second fundamental form, lapse, and matter field, then it can be further extended in time. This can also be reformulated as conditions that must be satisfied when such a solution blows up. In particular, in these nonvacuum settings, we encounter additional difficulties resulting from the nontrivial Ricci curvature and from the coupling between the Einstein and the matter field equations.Item Damped Driven Hamiltonian PDE(2011-07-16) Sergei, KuksinThere are the slides for a presentation given by Sergei Kuksin the the Analysis and Applied Mathematics Seminar in the Department of Mathematics at the University of Toronto. The talk was delivered on 2011-07-15.Item Computer Codes for Collapsing Vortex Soliton Blowups for $L^2$-Critical NLS(2011-06-14T02:29:18Z) Simpson, Gideon ; Zwiers, IanThis software accompanies "Collapsing Vortex Soliton Blowups for $L^2$-Critical NLS" by Simpson & Zwiers. Using these codes, we provide a proof of stability for the log-log blow up rate of vortex soliton solutions to 2D NLS.Item 2006 MAT1638HS Fluid Dynamics Course Notes(2011-03-01T16:39:57Z) James, CollianderThese are hand-written notes from a graduate course in fluid dynamics taught by J. Colliander in Winter 2006.Item Computer Codes for Embedded Eigenvalues and the Nonlinear Schrodinger Equation(2011-02-11T23:08:38Z) Asad, Reza ; Simpson, GideonThis software accompanies "Embedded Eigenvalues and the Nonlinear Schrodinger Equation" by Asad & Simpson. Using these codes, we provide a proof that when certain nonlinear Schrodinger equations are linearized about soliton solutions, the resultant linear operators have no purely imaginary eigenvalues or endpoint resonances.Item Some problems concerning a Quasilinear Schrödinger Equation(2011-02-05T20:12:59Z) Alessandro, SelvitellaIn this seminar we will talk about different issues concerning a Quasilinear Schrodinger Equation. In particular we will discuss a joint work with Prof. Louis Jeanjean about uniqueness and nondegeneracy of the ground state. We will also give an outline of what is known, what will be known soon (hopefully...) and what is going to be not known (for a while at least) about the Cauchy Problem for this equation.Item 2006 PDE Course Lecture Notes(2011-01-11T15:38:57Z) James, CollianderThese are hand-written lecture notes for the second semester of a year long graduate course in partial differential equations. The course number was MAT1061HS.Item Mean Field Limits for Interacting Bose Gases and the Cauchy Problem for Gross-Pitaevskii Hierarchies(2011-01-08T14:20:58Z) Chen, Thomas ; Pavlovic, NatasaThis talk surveys some recent results, all based on joint work with Natasa Pavlovic, related to the dynamics of Bose gases, and the Cauchy problem for Gross-Pitaevskii (GP) hierarchies. A GP hierarchy is an infinite system of coupled partial differential equations describing an interacting Bose gas in a mean field limit. First, we describe how the quintic nonlinear Schrodinger equation is derived from an N-body Schrodinger system with 3-body interactions and an associated GP hierarchy.Then, the local well-posedness theory for more general GP hierarchies is addressed, for focusing, defocusing, cubic and quintic interactions. In particular, the occurrence of blowup solutions is discussed (joint work with N. Pavlovic and N. Tzirakis). Furthermore, we present new conserved energy functionals which we apply to extend local to global well-posedness.Item BOOTSTRAPPED MORAWETZ ESTIMATES AND RESONANT DECOMPOSITION FOR LOW REGULARITY GLOBAL SOLUTIONS OF CUBIC NLS ON \(R^2\)(AIM Sciences, 2011-03) Colliander, James ; Roy, TristanWe prove global well-posedness for the \(L^2\)-critical cubic defocusing nonlinear Schr odinger equation on \(R^2\) with data \(u_0 \in H^s( R^2)\) for \(s > \frac{1}{3}\) . The proof combines a priori Morawetz estimates obtained and the improved almost conservation law obtained in earlier works. There are two technical di culties. The firrst one is to estimate the variation of the improved almost conservation law on intervals given in terms of Strichartz spaces rather than in terms of \(X^{s,b}\) spaces. The second one is to control the error of the a priori Morawetz estimates on an arbitrary large time interval, which is performed by a bootstrap via a double layer in time decomposition.Item Computer Codes for the Spectral Analysis for Matrix Hamiltonian Operators(2010-11-11T20:39:53Z) Simpson, Gideon ; Marzuola, Jeremy L.This software accompanies "Spectral Analysis for Matrix Hamiltonian Operators" by Marzuola & Simpson. Using these codes, we provide a proof that when the cubic nonlinear Schrodinger equation in 3D is linearized about a soliton, the resultant operator has no purely imaginary eigenvalues or resonances.Item Polar homology and holomorphic bundles(Royal Society, 2001) Khesin, B.; Rosly, A.