Analysis and PDE Seminar Homepage, Autumn-Winter-Spring 2019/2020

Location: 384H.

Time: Tuesday, 4pm.

• Tuesday, January 21

  Blair Davey (City College New York)

  ``How to obtain parabolic theorems from their elliptic counterparts''

  Abstract: Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well-known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement. Producing a parabolic statement from an elliptic statement is not as straightforward. In this talk, we demonstrate a method for producing parabolic theorems from their elliptic analogues. Specifically, we show that an $L^2$ Carleman estimate for the heat operator may be obtained by taking a high-dimensional limit of $L^2$ Carleman estimates for the Laplacian. Other applications of this technique will be discussed.

• Tuesday, February 4

  Alexander Volberg (Michigan State)

  ``Unexpected combinatorial property of all positive measures on the unit square and unit cube''

  Abstract: Paraproducts are building blocks of many singular integral operators and the main instrument in proving “Leibniz rule” for fractional derivatives (Kato-Ponce). Also multi-parameter paraproducts appear naturally in questions of embedding of spaces of analytic functions in polydisc into Lebesgue spaces with respect to a measure in the polydisc. The latter problem (without loss of information) can be often reduced to boundedness of weighted dyadic multi-parameter paraproducts. We find the necessary and sufficient condition for this boundedness in n-parameter case, when n is 1, 2, or 3. The answer is quite unexpected-it is a certain combinatorial property of all measures in dimension 2 and 3-and seemingly goes against the well known difference between box and Chang-Fefferman condition that was given by Carleson quilts example of 1974.

• Tuesday, February 11

  Aingeru Fernandez (Bilbao)

  ``Dynamic versions of Hardy's uncertainty principle''

  Abstract: Hardy's uncertainty principle states that a function and its Fourier transform cannot have Gaussan decay simultaneously unless the function is identically zero if the rates of decay are too large. This result can be restated in terms of solutions to the Schrödinger equation with Gaussian decay at two different times, saying that such a solution cannot be different from the identically zero solution. In this talk, we will review this and related results, and study versions of this problem in different settings. We will discuss recent results in collaboration with A. Grecu and L. Ignat (IMAR, Bucharest) where we prove a dynamic version of Hardy's Uncertainty principle in metric trees.

• Tuesday, February 25

  Joey Zou (Stanford)

  ``The Travel Time Tomography Inverse Problem for Transversely Isotropic Elastic Media''

  Abstract: I will discuss the travel time tomography problem for the elastic wave equation, where the aim is to recover elastic coefficients in the interior of an elastic medium given the travel times of the corresponding elastic waves. I will consider in particular the transversely isotropic case, which provides a reasonable seismological model for the interior of the Earth or other planets. By applying techniques from boundary rigidity problems, our problem is reduced to the microlocal analysis of certain operators obtained from a pseudo-linearization argument. These operators are not quite elliptic, but they strongly resemble parabolic operators, for which a symbol calculus first constructed by Boutet de Monvel can be applied. I will describe how to use this calculus to solve the problem given certain global assumptions, and if time permits I will discuss current work to modify this calculus in order to solve the problem more locally.

• Wednesday, February 26: Bay Area Microlocal Analysis Seminar at Stanford

  • 3-4pm, Room 383N

    Eugenia Malinnikova (Stanford)

    ``Landis conjecture in dimension two''

    Abstract: In 1960's Landis conjectured that a nontrivial solution to a time-independent Schrödinger equation with bounded potential cannot decay faster than exponentially. In 1992 the conjecture was disproved by Meshkov, who constructed a counter example and found the optimal rate of decay for complex valued solutions. The conjecture is still open for the case of real valued potentials. We will outline a solution in dimension two. The talk is based on a joint work in progress with A.Logunov, N.Nadirashvili and F.Nazarov.

  • 4:30-5:30pm, Room 380C

    András Vasy (Stanford)

    ``Fredholm theory for the Laplacian near zero energy on asymptotically conic spaces''

    Abstract: In this talk I will discuss and compare two approaches via Fredholm theory to resolvent estimates for the Laplacian of asymptotically conic spaces (such as appropriate metric perturbations of Euclidean space), including in the zero spectral parameter limit.

• Tuesday, March 3

  Jacob Bedrossian (Maryland)

  ``Almost-sure exponential mixing of scalars by stochastic Navier-Stokes and other stochastic fluid models with applications to passive scalar turbulence''

  Abstract: In 1959, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity kappa should display a |k|-1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimentally and numerically tested extensively in the physics and engineering literature and is a core prediction of passive scalar turbulence. Together with Alex Blumenthal and Sam Punshon-Smith, we have provided the first mathematically rigorous proof of this prediction for a scalar field evolving by advection-diffusion in a fluid governed by the 2D Navier-Stokes equations and 3D hyperviscous Navier-Stokes equations in a periodic box subjected to stochastic forcing at arbitrary Reynolds number. The main mathematical step in this proof is a precise understanding of the mixing of passive scalars by the Lagrangian flow map. In particular, we show that almost-surely and uniformly in diffusivity, the advection diffusion equation transfers information from low to high frequency exponentially fast -- known as (almost-sure) exponential mixing. These results are proved by studying the Lagrangian flow map using infinite dimensional extensions of ideas from random dynamical systems.

• Tuesday, March 10 (cancelled!)

  Jussi Berndt (Stanford/TU Graz)

  ``Sharp Boundary Trace Theory and Schrödinger Operators on Bounded Lipschitz Domains''

  Abstract: We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows forbidden endpoints in the Sobolev scale and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian. This boundary trace theory serves as a platform for developing a spectral theory for Schrödinger operators on bounded Lipschitz domains, along with their associated Weyl-Titchmarsh operators. The talk is based on joint work with Fritz Gesztesy and Marius Mitrea.

• Tuesday, April 7

  Mihaela Ifrim (Wisconsin/Berkeley)

  ``Almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems in two space dimensions''

  Abstract: We prove almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems with small and localized data in two space dimensions. We assume only mild decay on the data at infinity as well as minimal regularity. We systematically investigate all the possible quadratic null form type quasilinear strong coupling nonlinearities, and provide a new, robust approach for the proof. In a second paper we will complete the present results to full global well-posedness.

• Tuesday, April 14

  Jussi Berndt (Stanford/TU Graz)

  ``Sharp Boundary Trace Theory and Schrödinger Operators on Bounded Lipschitz Domains''

  Abstract: We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows forbidden endpoints in the Sobolev scale and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian. This boundary trace theory serves as a platform for developing a spectral theory for Schrödinger operators on bounded Lipschitz domains, along with their associated Weyl-Titchmarsh operators. The talk is based on joint work with Fritz Gesztesy and Marius Mitrea.

• Wednesday, April 15: Bay Area Microlocal Analysis Seminar virtually at Berkeley via zoom

  • 3-4pm

    Maciej Zworski (UC Berkeley)

    ``Analytic hypoellipticity of Keldysh operators''

    Abstract: For operators modeled by $P = x_1 D^2_{x_1} + D^2_{x_2} + a D_{x_1}$ we show that if $u$ is smooth and $P u$ is analytic then u is analytic. This is motivated by the question of analyticity of quasinormal modes of black holes across event horizons and is a consequence of a general microlocal result. Joint work with J Galkowski.

  • 4:15-5:15pm

    Sung-Jin Oh (UC Berkeley)

    ``On the Cauchy problem for the Hall magnetohydrodynamics''

    Abstract: In this talk, I will describe a recent series of work with I.-J. Jeong on the 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. 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 bicharacteristic flow associated with the appropriate principal symbol.

• Tuesday, April 21

  Tristan Buckmaster (Princeton)

  ``Stable shock wave formation for the isentropic compressible Euler equations''

  Abstract: I will talk about recent work with Steve Shkoller, and Vlad Vicol, regarding shock wave formation for the isentropic compressible Euler equations.

• Tuesday, April 28

  Oliver Lindblad (Stanford)

  ``Compact Cauchy horizons in vacuum spacetimes''

  Abstract: Moncrief and Isenberg conjectured in 1983 that any compact Cauchy horizon in a smooth vacuum spacetime is a smooth Killing horizon. We present a proof of this conjecture, under the assumption that the surface gravity of the horizon is a non-zero constant. One main ingredient in the proof is a new unique continuation theorem for wave equations through compact horizons.

• Tuesday, May 5

  Toan T. Nguyen (Penn State/Princeton)

  ``Landau damping and plasma echoes''

  Abstract: The talk presents an elementary proof of the nonlinear Landau damping for analytic and Gevrey data, that was first obtained by Mouhot and Villani and subsequently extended by Bedrossian, Masmoudi, and Mouhot, coupled with a presentation of echo solutions to the classical Vlasov-Poisson system which in particular exhibit an infinite cascade of echoes of smaller and smaller amplitude. The constructed echo solutions do not belong to the analytic or Gevrey classes studied by Mouhot and Villani, but do, nonetheless, exhibit damping phenomena for large times. This is a joint work with Emmanuel Grenier (ENS Lyon) and Igor Rodnianski (Princeton).

• Tuesday, May 12

  Benjamin Foster (U Penn)

  ``The Inhomogeneous Wave Equation with L^p Data''

  Abstract: The Fourier transform associates a polynomial to each linear differential operator with constant coefficients, and a formal calculation shows that elements in the kernel of such a differential operator have their Fourier transforms supported on the vanishing set of that polynomial. For example, the Helmholtz, wave, and linear Schrödinger operators correspond to polynomials whose vanishing sets are the sphere, cone, and paraboloid respectively; thus, we have a duality between such operators and certain geometric subsets of Euclidean space. When studying partial differential equations with inhomogeneous data, taking the Fourier transform gives a formal solution whose Fourier transform is instead potentially singular on that geometric region. Assuming the inhomogeneous data lies in a suitable L^p space with a vanishing condition on its Fourier transform, Michael Goldberg showed that this Fourier-analytic solution operator is bounded for the Helmholtz equation by exploiting the compactness and curvature properties of the associated geometric region, the sphere. In this talk, I'll discuss how to extend the result to wave operators, for which the associated geometric object is a noncompact cone, and how the geometry of the cone affects the possible estimates for the operator.

• Tuesday, May 19

  Dean Baskin (Texas A and M)

  ``Propagation of singularities for the Dirac-Coulomb system''

  Abstract: The Dirac equation describes the relativistic evolution of electrons and positrons. We consider the (time-dependent!) Dirac equation in three dimensions coupled to a potential with Coulomb-type singularities. We prove a propagation of singularities result for this equation and show that singularities are diffracted by the singularities of the potential and compute the symbol of the diffracted wave, which is typically non-zero. If time permits, I will describe how similar techniques can be used to characterize the asymptotic behavior of solutions of this system. This talk is based on joint work with Jared Wunsch as well as work with Bob Booth and Jesse Gell-Redman.

• Tuesday, May 26

  Andrew Hassell (Australian National University)

  ``Wave equations with $C^{1,1}$ coefficients''

  Abstract: We consider solutions to the wave equation in $\mathbb{R}^{n+1}$ with coefficients that are $C^{1,1}$ functions of space and time. We work with Hardy spaces of functions adapted to Fourier integral operators recently introduced by the speaker with Portal and Rozendaal. These are modifications of $L^p$ spaces, on which we showed classical FIOs of order zero act as bounded operators (in contrast to $L^p$ spaces, where there is a loss of derivatives for $p \neq 2$). I will review this result and then discuss work in progress showing that, at least in 2 spatial dimensions, the same is true for solution operators to wave equations with $C^{1,1}$ coefficients. These can be thought of as "rough Fourier integral operators". In 3 dimensions, we need epsilon more differentiability in the coefficients to get the same result. I shall also indulge in some reckless speculation about what might be true for even rougher Fourier integral operators.

• Schedule for 2018-2019.
• Schedule for 2017-2018.
• Schedule for 2016-2017.
• Schedule for 2015-2016.
• Schedule for 2014-2015.
• Schedule for Spring 2014.
• Schedule for Winter 2014.
• Schedule for Autumn 2013.
• Schedule for Spring 2013.
• Schedule for Winter 2013.
• Schedule for Autumn 2012.
• Schedule for Winter/Spring 2012.
• Schedule for Autumn/Winter 2010-2011.
• Schedule for Winter-Spring 2010.
• Schedule for Autumn 2009.
• Schedule for Spring 2009.
• Schedule for Winter 2009.
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• Schedule for Autumn 2007.