PDE-Applied Math Seminar

Organizer: Antoine Mellet
When: Thursdays @ 3:30pm
Where: Math 3206

Archives: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018

  • Liquid drops on Rough surfaces

    Speaker: Inwon Kim (UCLA) -

    When: Thu, September 14, 2017 - 3:30pm
    Where: Kirwan Hall 3206
  • Data-based stochastic model reduction for chaotic systems

    Speaker: Fei Lu (John Hopkins University) - http://www.math.jhu.edu/~feilu/

    When: Thu, October 12, 2017 - 3:30pm
    Where: Kirwan Hall 3206

    View Abstract

    Abstract: The need to develop reduced nonlinear statistical-dynamical models from time series of partial observations of complex systems arises in many applications such as geophysics, biology and engineering. The challenges come mainly from memory effects due to the nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete data.

    We address these challenges by introducing a discrete-time stochastic parametrization framework, in which we infer nonlinear autoregression moving average (NARMA) type models to take the memory effects into account. We show by examples that the NARMA type stochastic reduced models that can capture the key statistical and dynamical properties, and therefore can improve the performance of ensemble prediction in data assimilation. The examples include the Lorenz 96 system (which is a simplified model of the atmosphere) and the Kuramoto-Sivashinsky equation of spatiotemporally chaotic dynamics.

  • A free boundary problem with facets

    Speaker: Will Feldman (University of Chicago) -

    When: Thu, October 19, 2017 - 3:30pm
    Where: Kirwan Hall 3206

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    Abstract: I will discuss a variational problem on the lattice analogous to the Alt-Caffarelli problem. The scaling limit is a free boundary problem for the Laplacian with a discontinuous constraint on the normal derivative at the boundary. The discontinuities cause the formation of facets in the free boundary. The problem is related to models for contact angle hysteresis of liquid drops studied by Caffarelli-Lee and Caffarelli-Mellet.
  • Probabilistic scattering for the 4D energy-critical defocusing nonlinear wave equation

    Speaker: Jonas Luehrmann (Johns Hopkins University) - http://www.math.jhu.edu/~luehrmann/

    When: Thu, October 26, 2017 - 3:30pm
    Where: Kirwan Hall 3206

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    Abstract: We consider the Cauchy problem for the energy-critical defocusing nonlinear wave equation on R^4. It is known that for initial data at energy regularity, the solutions exist globally in time and scatter to free waves. However, the problem is ill-posed for initial data at super-critical regularity, i.e. for regularities below the energy regularity. In this talk we study the super-critical data regime for this Cauchy problem from a probabilistic point of view, using a randomization procedure that is based on a unit-scale decomposition of frequency space. We will present an almost sure global existence and scattering result for randomized radially symmetric initial data of super-critical regularity. The main novelties of our proof are the introduction of an approximate Morawetz estimate to the random data setting and new large deviation estimates for the free wave evolution of randomized radially symmetric data.
    This is joint work with Ben Dodson and Dana Mendelson.

  • On L^p approximations of Landau equation solutions in the Coulomb case

    Speaker: Sona Akopian (Brown University) - https://www.brown.edu/academics/applied-mathematics/sona-akopian

    When: Thu, November 2, 2017 - 3:30pm
    Where: Kirwan Hall 3206

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    Abstract: We examine a class of Boltzmann equations with an abstract collision kernel in the form of a singular mass concentrated at very low collision angles and relative velocities between interacting particles. Similarly to the classical Boltzmann operator, this particular collision operator also converges to the collision term in the Landau equation as the characterizing parameter \epsilon tends to zero. We will address the existence of L^p solutions to this family of Boltzmann equations and discuss their approximations of solutions to the Landau equation as \epsilon vanishes.
  • Degenerate disperisve equations and compactons

    Speaker: Benjamin Harrop-Griffiths (New York University ) - https://math.nyu.edu/~griffiths/

    When: Thu, November 9, 2017 - 3:30pm
    Where: Kirwan Hall 3206

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    Abstract: We consider a family of Hamiltonian toy models for degenerate dispersion that admit compactly supported solitons or “compactons”. We discuss their variational properties and stability. This is joint work with Pierre Germain and Jeremy Marzuola.
  • Data-based stochastic model reduction for chaotic systems

    Speaker: Fei Lu (John Hopkins University) - http://www.math.jhu.edu/~feilu/

    When: Thu, November 30, 2017 - 3:30pm
    Where: Kirwan Hall 3206

    View Abstract

    Abstract: The need to develop reduced nonlinear statistical-dynamical models from time series of partial observations of complex systems arises in many applications such as geophysics, biology and engineering. The challenges come mainly from memory effects due to the nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete data.

    We address these challenges by introducing a discrete-time stochastic parametrization framework, in which we infer nonlinear autoregression moving average (NARMA) type models to take the memory effects into account. We show by examples that the NARMA type stochastic reduced models that can capture the key statistical and dynamical properties, and therefore can improve the performance of ensemble prediction in data assimilation. The examples include the Lorenz 96 system (which is a simplified model of the atmosphere) and the Kuramoto-Sivashinsky equation of spatiotemporally chaotic dynamics.

  • Global strong solution with latent singularity: application of gradient flow theory to thin film equations in epitaxial growth

    Speaker: Yuan Gao (Hong Kong University of Science and Technology) -

    When: Thu, December 7, 2017 - 3:30pm
    Where: Kirwan Hall 3206

    View Abstract

    Abstract: We consider a class of step flow models from mesoscopic view and their continuum limit to 4th order degenerate parabolic equations. Using the regularized method we obtain a global weak solution to the slope equation, which is sign-preserved almost everywhere. However, in order to study the global strong solution with latent singularity, which occurs whenever the solution approaches zero, we formulate the problem as the gradient flow of a suitably-defined convex functional in a non-reflexive Banach space and establish a framework to handle a class of degenerate parabolic equations, including exponential model for epitaxial growth, described by gradient flow in metric space.
  • Elasticity and curvature: the elastic energy of non-Euclidean thin bodies

    Speaker: Cy Maor (University of Toronto) - http://www.math.toronto.edu/cmaor/

    When: Thu, January 25, 2018 - 3:30pm
    Where: Kirwan Hall 3206

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    Abstract: Non-Euclidean, or incompatible elasticity is an elastic theory for bodies that do not have a reference, stress-free configuration. It applies to many systems, in which the elastic body undergoes inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a geometric calculus of variations question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension.
    Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them.
    In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results on the relations between their elastic behavior and their curvature.
    Based on joint work with Asaf Shachar.
  • Sticky particles and the Euler-Poisson equations

    Speaker: Ryan Hynd (U-Penn) - https://web.sas.upenn.edu/rhynd/

    When: Thu, March 1, 2018 - 3:30pm
    Where: Kirwan Hall 3206

    View Abstract

    Abstract: We will consider the dynamics of a finite number of particles that interact pairwise and undergo perfectly inelastic collisions.
    Such physical systems conserve mass and momentum and satisfy the Euler-Poisson equations. In one spatial dimension, we will show
    how to derive an extra entropy estimate which allows us to characterize the limit as the number of particles tends to infinity.
  • Alexandrov Theorem Revisited

    Speaker: Matias Delgadino (Imperial College - London) -

    When: Thu, March 15, 2018 - 3:30pm
    Where: Kirwan Hall 3206

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    Abstract: We will introduce the notion of sets of finite perimeter and we will show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional. This is joint work with Francesco Maggi.
  • Approximation diffusion and fluid limits for random kinetic equations [Appl Math Colloquium]

    Speaker: Arnaud Debussche (ENS Rennes) - http://w3.bretagne.ens-cachan.fr/math/people/arnaud.debussche/

    When: Tue, May 1, 2018 - 3:30pm
    Where: Kirwan Hall 3206

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    Abstract: In this talk, I consider kinetic equations containing random terms. The kinetic models contain a small parameter and it is wellknown that, after parabolic rescaling, when this parameter goes to zero the limit problem is a diffusion equation in the PDE sense, ie a parabolic equation of second order. A smooth noise is added, accounting for external perturbation. It scales also with the small parameter. It is expected that the limit equation is then a stochastic parabolic equation where the noise is in Stratonovitch form.
    Our aim is to justify in this way several SPDEs commonly used. We first treat linear equations with multiplicative noise. Then show how to extend the methods to some nonlinear equations or to the more physical case of a random forcing term. The method is to combine the classical perturbed test function method with PDE argument.
  • Stochastic nonlinear Schroedinger equations [Aziz Lecture]

    Speaker: Arnaud Debussche (ENS Rennes) - http://w3.bretagne.ens-cachan.fr/math/people/arnaud.debussche/

    When: Fri, May 4, 2018 - 3:15pm
    Where: Kirwan Hall 3206

    View Abstract

    Abstract: The nonlinear Schroedinger equation is a prototype model to describe propagation of waves in dispersive media. It arises in several modelisation and noise appears naturally. It may represent the noise due to amplifiers or random dispersion in the fiber. In this talk I will present some aspects of well-posedness and influence on blow-up phenomena for the stochastic nonlinear Schroedinger.