The APDE seminar on Monday, 3/18, will be given by Zhuolin Li (SLMath/MSRI) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Degenerate variational problems under the constant rank condition

Abstract: Differential expressions involving non-elliptic operators emerge in various PDEs and variational principles that arise from materials science, fluids, differential geometry, etc. Despite their inherent degeneracy, such operators, under the constant rank condition, retain certain good properties of elliptic operators. In this talk, we will first give a short introduction to the study of vectorial problems in the calculus of variations, and then discuss quasi-convex variational problems involving constant rank operators. For clarity, exterior derivatives will be taken as a particular example for illustration. We will consider the existence, which can also be interpreted as a Sobolev-type regularity, as well as the corresponding partial regularity via an excess decay estimate strategy. This talk is based on an ongoing work with Bogdan Raiță.

The APDE seminar on Monday, 2/12, will be given by Steve Shkoller (UC Davis) online via Zoom from 4:10pm to 5:00pm PST (in particular, there will be no in-person talk). To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: The geometry of maximal development and shock formation for the Euler equations

Abstract: We establish the maximal hyperbolic development of Cauchy data for the multi-dimensional compressible Euler equations throughout the shock formation process. For an open set of compressive and generic $H^7$ initial data, we construct unique $H^7$ solutions to the Euler equations in the maximal spacetime region such that at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-$2$ surface of “first singularities” called the pre-shock set; second, a downstream hypersurface emanating from the pre-shock set, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock set, which the Euler solution cannot reach. This talk is based on joint work with Vlad Vicol at NYU.

The APDE seminar on Monday, 1/29, will be given by Joonhyun La (Korea Institute for Advanced Study) in-person in Evans 740, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Null shell solutions – stability and instability

Abstract: In this talk, we study initial value problem for the Einstein equation with null matter fields, motivated by null shell solutions of Einstein equation. In particular, we show that null shell solutions can be constructed as limits of spacetimes with null matter fields. We also study the stability of these solutions in Sobolev space: we prove that solutions with one family of null matter field are stable, while the interaction of two families of null matter fields can give rise to an instability.

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 (sjoh@math.berkeley.edu).

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 (sjoh@math.berkeley.edu).

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 (sjoh@math.berkeley.edu).

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 (sjoh@math.berkeley.edu).

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 (sjoh@math.berkeley.edu).

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 (sjoh@math.berkeley.edu).

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 (sjoh@math.berkeley.edu).

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).