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Wed Sep 30

PDE Seminar

3:35pm -
On the Cauchy problem for the Hall magnetohydrodynamics
Sungjin Oh, University of California Berkeley 

 In this talk, I will describe a recent series of work with I.-J. Jeong on the incompressible Hall MHD equation without resistivity. This PDE, first investigated by the applied mathematician M. J. Lighthill, is a one-fluid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate the ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. Our ill posedness mechanism is sharp, in that it remains true under fractional dissipation of any subcritical order. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which not only includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field) but also possibly large perturbations thereof. Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character. With such a viewpoint, the key ill- and well-posedness mechanisms can be understood in terms of the properties of the bi-characteristic flow associated with the appropriate principal symbol.Join Zoom Meeting ID: 917 6595 9125Passcode: VinH16  

Wed Oct 07

PDE Seminar

3:35pm - Via Zoom
Two problems related to the boundary layer in fluids
Siming He, Duke University 

  In this talk, I will present two works related to boundary layers in thefluid. The first result concerns the 2D Navier-Stokes equations linearized around the Couette flow in the periodic channel in the vanishing viscosity limit. We split the vorticity evolution into thefree-evolution (without a boundary) and a boundary corrector that is exponentially localized. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of both thevelocity and the boundary layer's vorticity. We also observe that both velocity and vorticity satisfy the expected enhanced dissipation. This is joint work with Jacob Bedrossian. The second work is related to aboundary layer model designed to understand the Hou-Luo Scenario associated with the 3D Euler equation's blow-up. We show that there exists initial data which yields blow-up in the model. This is joint workwith Alexander Kiselev.Join Zoom Meeting ID: 917 6595 9125Passcode: VinH16

Wed Oct 14

PDE Seminar

3:35pm - Via Zoom
Radon measures and lipschitz graphs
Lisa Naples, Macalaster College

In geometric measure theory there is interest in understanding measures by studying interactions with particular collections of sets. Here, we will discuss a recent characterization of Radon measures on R^n which are carried by the collection of m-Lipschitz graphs. That is, we will provide necessary and sufficient conditions for a Radon measure under which there exist countably many Lipschitz graphs that capture almost all of the mass. Our characterization will involve only countably many evaluations of the measure. This is joint work with Matthew Badger.Join Zoom Meeting ID: 917 6595 9125Passcode: VinH16  

Wed Oct 21

PDE Seminar

3:35pm - Via Zoom
Quantitative stability for minimizing Yamabe metrics
Robin Neumeyer, Northwestern University

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.

We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

Wed Oct 28

PDE Seminar

3:35pm - Via Zoom
Stationary Euler flows near the Kolmogorov and Poiseuille flows
Michele Coti Zelati, Imperial College London

We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow. This is in contrast with both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel, for which we can show that the only stationary states near them must be shears. This has surprising consequences in the context of inviscid damping in 2D Euler and enhanced dissipation in Navier-Stokes.Join Zoom Meeting ID: 917 6595 9125Passcode: VinH16

Wed Nov 04

PDE Seminar

3:35pm - Via Zoom
The N-membrane problem
Hui Yu, Columbia University

The N-membrane problem is the study of shapes of elastic membranes being pushed against each other. The main questions are the
regularity of the functions modeling the membranes, and the regularity
of the contact regions between consecutive membranes.

These are classical questions in free boundary problems. However, very
little is known when N is larger than 2. In this case, there are
multiple free boundaries that cross each other, and most known
techniques fail to apply.

In this talk, we discuss, for general N, the optimal regularity of the
solutions in arbitrary dimensions, and a classification of blow-up
solutions in 2D. Then we focus on the regularity of the free
boundaries when N=3.

This talk is based on two recent joint works with Ovidiu Savin
(Columbia University).

Wed Nov 11

PDE Seminar

3:35pm - Via Zoom
A Harnack inequality for weak solutions of non-elliptic equations
Max Goering, University of Washington-Seattle

We'll introduce a broad class of PDEs which arise from the Calculus of Variations. After producing specific examples of some PDEs that fall within this class, we will outline a Moser Iteration based argument to derive a harnack inequality for weak solutions. This demonstrates that for 0th order regularity, the aspect of "ellipticity" which is useful is the fixed homogeneity. This raises the question of whether or not some notion of convexity can be used to replace ellipticity and still recover 1st order regularity of solutions.

Wed Nov 18

PDE Seminar

3:35pm - Via Zoom
PDE Seminar
Jiajie Chen , Caltech

Join Zoom Meeting ID: 917 6595 9125Passcode: VinH16