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.