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Wed Sep 30

PDE Seminar

3:35pm - https://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi
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 Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting 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 Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting 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 Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting 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 Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting 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
Singularity formation in incompressible fluids and related models
Jiajie Chen , Caltech

In this talk, we will discuss the self-similar singularity formation in the Hou-Luo (HL) model for the 3D asymmetric Euler equations with boundary. Several observations obtained in the analysis of the HL model have been used to study other equations. We will also talk about some features of the singularity formation in the 2D Boussinesq and 3D asymmetric Euler equations with $C^{\alpha}$ velocity and boundary that have connections to the Hou-Luo’s computation for the potential 3D Euler singularity. Some of the results are joint with Tom Hou and De Huang. Join Zoom Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting ID: 917 6595 9125Passcode: VinH16 

Wed Feb 17

PDE Seminar

3:35pm - via Zoom
Global Existence for the 3D Muskat problem
Stephen Cameron, NYU-Courant

The Muskat problem studies the evolution of the interface between two incompressible, immiscible fluids in a porous media. In the case that the fluids have equal viscosity and the interface is graphical, this system reduces to a single nonlinear, nonlocal parabolic equation for the parametrization. Even in this stable regime, wave turning can occur leading to finite time blowup for the slope of the interface. Before that blowup though, we prove that an imperfect comparison principle still holds. Using this, we are able to show that solutions exist for all time so long as either the initial slope is not too large, or the slope stays bounded for a sufficiently long time. Join Zoom Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting ID: 917 6595 9125Passcode: VinH16

Wed Mar 03

PDE Seminar

3:35pm - via Zoom
Boundary unique continuation of Dini domains
Zihui Zhao, University of Chicago
Wed Mar 03

PDE Seminar

3:35pm - via Zoom
Boundary unique continuation of Dini domains
Zihui Zhao, University of Chicago

Let u be a harmonic function in \Omega \subset \mathbb{R}^d. It is known that in the interior, the singular set \mathcal{S}(u) = \{u=|\nabla u|=0 \} is (d-2)-dimensional, and moreover \mathcal{S}(u) is (d-2)-rectifiable and its Minkowski content is bounded (depending on the frequency of u). We prove the analogue at the boundary for C^1-Dini domains: If the harmonic function u vanishes on an open subset E of the boundary, then near E the singular set \mathcal{S}(u) \cap \overline{\Omega} is (d-2)-rectifiable and has bounded Minkowski content. Dini domain is the optimal domain for which \nabla u is continuous towards the boundary, and in particular every C^{1,\alpha} domain is Dini. The main difficulty is the lack of monotonicity formula for boundary and interior points of a Dini domain. This is joint work with Carlos Kenig.Join Zoom Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting ID: 917 6595 9125Passcode: VinH16

Wed Mar 10

PDE Seminar

3:35pm - via Zoom
Universal dynamics of pulled fronts
Montie Avery, UMN

The formation of structure in spatially extended systems is often mediated by an invasion process, in which a pointwise stable state invades a pointwise unstable state. A fundamental goal is then to predict the speed of this invasion. The marginal stability conjecture postulates that, absent a mechanism through which the nonlinearity enhances propagation, the invasion speed is predicted by marginal linear stability of the pointwise unstable background state in a suitable norm. We introduce a set of largely model-independent conceptual assumptions under which we establish nonlinear propagation at the linear spreading speed, thereby resolving the marginal stability conjecture in the general case of stationary invasion. Our assumptions hold for open classes of parabolic equations, including higher order equations without comparison principles, while previous results rely on special structure of the equation and the presence of a comparison principle. Our result also establishes universality of the logarithmic in time delay in the position of the front, compared with propagation strictly at the linear speed, as predicted in generality by Ebert and van Saarloos and first established in the special case of the Fisher-KPP equation by Bramson. Our proof describes the invasion process through the interaction of a Gaussian leading edge with the pulled front in the wake. Technically, we rely on sharp linear decay estimates to control errors from this matching procedure and corrections from the initial data.