## Seminar Categories

This page lists seminar series that have events scheduled between two months ago and twelve months from now and have speaker information available.

## Current Series

Wed Sep 30 |
## PDE Seminar3:35pm - https://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNiOn 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 Seminar3:35pm - Via ZoomTwo 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 Seminar3:35pm - Via ZoomRadon 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 Seminar3:35pm - Via ZoomQuantitative 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 showin a quantitative sensethat 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 Seminar3:35pm - Via ZoomStationary 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 Seminar3:35pm - Via ZoomThe 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 These are classical questions in free boundary problems. However, very In this talk, we discuss, for general N, the optimal regularity of the This talk is based on two recent joint works with Ovidiu Savin |

Wed Nov 11 |
## PDE Seminar3:35pm - Via ZoomA 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 Seminar3:35pm - Via ZoomSingularity 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-Luos 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 Seminar3:35pm - via ZoomGlobal 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 Seminar3:35pm - via ZoomBoundary unique continuation of Dini domains Zihui Zhao, University of Chicago |

Wed Mar 03 |
## PDE Seminar3:35pm - via ZoomBoundary 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 Seminar3:35pm - via ZoomUniversal 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 |

Wed Mar 31 |
## PDE Seminar3:35pm - ZoomA capillarity model for soap films Darren King, UT Austin We study a variational model for soap films based on capillarity theory and its relation to minimal surfaces. Here, soap films are modelled, not as surfaces, but as regions of small volume satisfying a homotopic spanning condition.Join Zoom Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting ID: 917 6595 9125Passcode: VinH16 |