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Current Colloquia & Seminars

  1. Model problems in fluid dynamics: Instability via degenerate dispersion for generalized surface quasi-geostrophic models with singular velocities

    Location: MSRI: Online/Virtual
    Speakers: Sung-Jin Oh (University of California, Berkeley)

    To participate in this seminar, please register here: https://www.msri.org/seminars/25657

    Abstract: The primary purpose of this talk is to elucidate an instability mechanism, which will be referred to as degenerate dispersion, that leads to illposedness of the Cauchy problem in arbitrarily high-regularity Sobolev spaces for a number of nonlinear PDEs of hydrodynamics and magnetohydrodynamics (MHD) that respect conservation of energy. Due to the conservation structure, the instability mechanism is necessarily different from that of, say, the reverse heat equation; rather, it is a mechanism by which energy gets concentrated into small scales at an arbitrarily fast rate due to the degeneration of the dispersion relation. 

    In this talk, I will focus on generalized surface quasi-geostrophic (gSQG) models with singular velocities. I will give a heuristic description of the phenomenon via geometric-optical ideas (or classical-quantum correspondence), and then discuss the mathematical tools recently developed to capture this phenomenon in the nonlinear setting. This talk is based on joint works with Dongho Chae and In-Jee Jeong.

    Updated on Mar 04, 2021 05:43 PM PST
  2. Free surface flows in fluid dynamics (UCB Chancellor Professor Course)

    Location: MSRI: Online/Virtual
    Speakers: Thomas Alazard (Ecole Normale Supérieure Paris-Saclay; Centre National de la Recherche Scientifique (CNRS))

    To participate in this seminar, please register here: https://www.msri.org/seminars/25657

     

    A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

    These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

    Updated on Mar 01, 2021 10:45 AM PST