The first APDE seminar on Monday, 2/6, will be given by our own Sung-Jin Oh in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Mengxuan Yang ().

Title: Codimension one stability of the catenoid under the hyperbolic vanishing mean curvature flow

Abstract: The catenoid is one of the simplest examples of minimal hypersurfaces next to the hyperplane. In this talk, we will view the catenoid as a stationary solution to the hyperbolic vanishing mean curvature flow, which is the hyperbolic analog of the (elliptic) minimal hypersurface equation, and study its nonlinear stability under no symmetry assumptions. The main result, which is a recent joint work with Jonas Luhrmann and Sohrab Shahshahani, is that with respect to a “codimension one” set of initial data perturbations of the n-dimensional catenoid, the corresponding flow asymptotes to an adequate translation and Lorentz boost of the catenoid for n greater than or equal to 5. Note that the codimension condition is necessary and sharp in view of the fact that the catenoid is an index 1 minimal hypersurface.

In a broader context, our result is a part of stability theory for solitary waves (i.e., stationary solutions) in the presence of modulation (i.e., symmetries, such as translation and Lorentz boosts, create nearby stationary solutions), which was mostly studied for semilinear PDEs in the past. Among the key challenges of the present problem compared to the more classical context are: (1) the quasilinearity of the equation, (2) slow (polynomial) decay of the catenoid at infinity, and (3) lack of symmetry assumptions. To address these challenges, we introduce several new ideas, such as a geometric construction of modulated profiles, smoothing of modulation parameters, and a robust framework for proving decay for the radiation part, which are hoped to be useful in the larger context of stability theory for solitary waves for quasilinear wave equations in the presence of modulation. 

Our last talk in the APDE seminar will be given by Toan Nguyen (Penn State U) on Monday, Dec. 5th both in person at Evans 740 and online (via Zoom) from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: The roller coaster through Landau damping

Abstract: Of great interest is to address the final state conjecture for the dynamics of charged particles near spatially homogeneous equilibria in a plasma, where particles are transported by the self-consistent electric field generated by the meanfield Coulomb’s interaction. The long-range interaction generates waves that oscillate in time and disperse in space through the dispersion of a Schrodinger type equation, known as plasma oscillations or Langmuir waves. The classical notion of Landau damping refers to the damping of oscillations when particles travel at a resonant speed with the waves. The talk is to address this classical picture for the Vlasov-Poisson system with relativistic or bounded velocities. Based on a joint work with E. Grenier and I. Rodnianski.  

The APDE seminar on Monday, 11/28, will be given by Lili He (Johns Hopkins) both in person at Evans 740 and online (via Zoom) from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

Abstract: I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.

The APDE seminar on Monday, 11/21, will be given by Alexis Drouot (U Washington) online (via Zoom) from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Dirac operators and topological insulators.

Abstract: I will discuss a 2×2 semiclassical Dirac equation that
emerges from the effective analysis of topological insulators, and
specifically focus on the evolution of coherent states initially
localized on the crossing set of the eigenvalues of the symbol.
Standard propagation of singularities results do not apply; instead,
we discover a surprising phenomenon. The dynamics breaks down in two
parts, one that immediately collapses, and one that propagates along a
seemingly novel quantum trajectory. This observation is consistent
with the bulk-edge correspondence, a principle that coarsely describes
features of transport in topological insulators. We illustrate our
result with various numerical simulations.

The APDE seminar on Monday, 11/14, will be given by Leonardo Abbrescia (Vanderbilt) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: A localized picture of the maximal development for shock forming solutions of the 3D compressible Euler equations

Abstract: It is well known that solutions to the inviscid Burgers’ equation form shock singularities in finite time, even when launched from smooth data. A far less documented fact, at least in the popular works on 1D hyperbolic conservation laws, is that shock singularities are intimately tied to a lack-of-uniqueness for the classical Burgers’ equation.

We prove that, locally, solutions to the Compressible Euler equations do not suffer from the same lack-of-uniqueness, even though they can be written as a coupled system of Burgers’ in isentropic plane-symmetry. Roughly, the saving grace is that Euler flow involves two speeds of propagation, and one of them “prevents” the mechanism driving the lack-of-uniqueness. Analytically, this is done by explicitly constructing a portion of the boundary of classical hyperbolic development for shock forming data. This boundary is a connected co-dimension 1 submanifold of Cartesian space, and we will discuss the delicate geo-analytic degeneracies and difficulties involved in its construction. This is joint work with Jared Speck.

The APDE seminar on Monday, 11/7, will be given by Sameer Iyer (UC Davis) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Reversal in the stationary Prandtl equations

Abstract: We discuss a recent result in which we investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by spatio-temporal regions in which $u>0$ and $u<0$. The classical point of view of regarding the Prandtl equations as an evolution completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. Joint work with Nader Masmoudi.

The APDE seminar on Monday, 10/31, will be given by Frédéric Klopp (Sorbonne U & UC Berkeley) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: A complex WKB analysis for finite difference Schrödinger operators and adiabatic perturbations of periodic Schrödinger operators

Abstract: I will explain the basic ideas and constructions of the methods underlining the similarities between the two classes of equations.

The APDE seminar on Monday, 10/17, will be given by Sanchit Chaturvedi (Stanford) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Vanishing viscosity and shock formation in Burgers equation

Abstract: I will talk about the shock formation problem for 1D Burgers equation in the presence of small viscosity. Although the vanishing viscosity problem till moments before the first shock and in presence of a fully developed shocks is very classical, little is known about the moment of shock formation. We develop a matched asymptotic expansion to describe the solution to the viscous Burgers equation (with small viscosity) to arbitrary order up to the first singularity time. The main feature of the work is the inner expansion that accommodates the viscous effects close to the shock location and match it to the usual outer expansion (in viscosity). We do not use the Cole-Hopf transform and hence we believe that this approach works for more general scalar 1D conservation laws. Time permitting, I will talk about generalizing to vanishing viscosity limit from compressible Navier–Stokes to compressible Euler equations. This is joint work with Cole Graham (Brown university).

The APDE seminar on Monday, 10/3, will be given by our own Mengxuan Yang in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: Wave trace and resonances of the Aharonov–Bohm Hamiltonian

Abstract: I want to discuss propagation of singularities of the
magnetic Hamiltonian with singular vector potentials, which is related
to the so-called Aharonov–Bohm effect. In addition, I shall discuss a
Duistermaat–Guillemin type trace formula, as well as some
applications to scattering resonances in this setting.

The APDE seminar on Monday, 9/26, will be given by Mihai Tohaneanu (U Kentucky) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ().

Title: The weak null condition on Kerr backgrounds

Abstract: Understanding global existence for systems satisfying the weak null condition plays a crucial role in the proof of stability of Minkowski in harmonic coordinates. In this talk I will present a proof of global existence for a semilinear system of equations on Kerr spacetimes satisfying the weak null condition. This is joint work with Hans Lindblad.