Seminars and Colloquia by Series

Hardy spaces for Fourier integral operators

Series
PDE Seminar
Time
Tuesday, September 27, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Jan Rozendaal – IMPAN –

It is well known that the wave operators cos(t (−∆)) and sin(t (−∆)) are not bounded on Lp(Rn), for n≥2 and 1≤p≤∞, unless p=2 or t=0. In fact, for 1 < p < ∞ these operators are bounded from W2s(p),p  to Lp(Rn) for s(p) := (n−1)/2 | 1/p − 1/2 |, and this exponent cannot be improved. This phenomenon  is symptomatic of the behavior of Fourier integral operators, a class of oscillatory operators which includes wave propagators, on Lp(Rn).

In this talk, I will introduce a class of Hardy spaces HFIOp (Rn), for p ∈ [1,∞],on which Fourier integral operators of order zero are bounded. These spaces also satisfy Sobolev embeddings which allow one to recover the optimal boundedness results for Fourier integral operators on Lp(Rn).

However, beyond merely recovering existing results, the invariance of these spaces under Fourier integral operators allows for iterative constructions that are not possible when working directly on Lp(Rn). In particular, we shall indicate how one can use this invariance to obtain the optimal fixed-time Lp regularity for wave equations with rough coefficients. We shall also mention the connection of these spaces to the phenomenon of local smoothing.

This talk is based on joint work with Andrew Hassell and Pierre Portal (Aus- tralian National University), and Zhijie Fan, Naijia Liu and Liang Song (Sun Yat- Sen University).

Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations

Series
PDE Seminar
Time
Tuesday, September 20, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lukas Wessels – Georgia Tech and Technische Universität Berlin –

In this talk, we consider a finite-horizon optimal control problem of stochastic reaction-diffusion equations. First we apply the spike variation method which relies on introducing the first and second order adjoint state. We give a novel characterization of the second order adjoint state as the solution to a backward SPDE. Using this representation, we prove the maximum principle for controlled SPDEs. 

As another application of our characterization of the second order adjoint state, we derive additional necessary optimality conditions in terms of the value function. These results generalize a classical relationship between the adjoint states and the derivatives of the value function to the case of viscosity differentials.

The last part of the talk is devoted to sufficient optimality conditions. We show how the necessary conditions lead us directly to a non-smooth version of the classical verification theorem in the framework of viscosity solutions.

This talk is based on joint work with Wilhelm Stannat:  W. Stannat, L. Wessels, Peng's maximum principle for stochastic partial differential equations, SIAM J. Control Optim., 59 (2021), pp. 3552–3573 and W. Stannat, L. Wessels, Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations, https://arxiv.org/abs/2112.09639, 2022.

Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

Series
PDE Seminar
Time
Tuesday, August 30, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Dallas Albritton – Princeton University –

In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.

The mathematical theory of wave turbulence

Series
PDE Seminar
Time
Thursday, March 31, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Zaher Hani – University of Michigan

Please Note: Meeting also available online: https://gatech.zoom.us/j/92742811112

Wave turbulence is the theory of nonequilibrium statistical mechanics for wave systems. Initially formulated in pioneering works of Peierls, Hasselman, and Zakharov early in the past century, wave turbulence is widely used across several areas of physics to describe the statistical behavior of various interacting wave systems. We shall be interested in the mathematical foundation of this theory, which for the longest time had not been established.

The central objects in this theory are: the "wave kinetic equation" (WKE), which stands as the wave analog of Boltzmann’s kinetic equation describing interacting particle systems, and the "propagation of chaos” hypothesis, which is a fundamental postulate in the field that lacks mathematical justification. Mathematically, the aim is to provide a rigorous justification and derivation of those two central objects; This is Hilbert’s Sixth Problem for waves. The problem attracted considerable interest in the mathematical community over the past decade or so. This culminated in recent joint works with Yu Deng (University of Southern California), which provided the first rigorous derivation of the wave kinetic equation, and justified the propagation of chaos hypothesis in the same setting.

Meeting also available online: https://gatech.zoom.us/j/92742811112

On Liouville systems, Moser Trudinger inequality and Keller-Segel equations of chemotaxis

Series
PDE Seminar
Time
Tuesday, March 24, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gershon Wolansky – Israel Institute of Technology –
The Liouville equation is a semi-linear elliptic equation of exponential non-linearity. Its non-local version is a steady state of the Keller-Segel equation representing the distribution of living cells, such as slime molds. I will represent an extension of this equation to multi-agent systems and discuss some associated critical phenomena, and recent results with Debabrata Karmakar on the parabolic Keller segel system and its asymptotics in both critical and non-critical cases.

Thesis Defense: The Maxwell-Pauli Equations

Series
PDE Seminar
Time
Tuesday, March 10, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Thomas Kieffer – Georgia Tech –

Energetic stability of matter in quantum mechanics, which refers to the question of whether the ground state energy of a

many-body quantum mechanical system is finite, has long been a deep question of mathematical physics. For a system of many
non-relativistic electrons interacting with many nuclei in the absence of electromagnetic fields this question traces back
to the seminal works of Tosio Kato in 1951 and Freeman Dyson and Andrew Lenard in 1967/1968. In particular, Dyson and Lenard
showed the ground state energy of the many-body Schrödinger Hamiltonian is bounded below by a constant times the total particle
number, regardless of the size of the nuclear charges. This situation changes dramatically when electromagnetic fields and spin
interactions are present in the problem. Even for a single electron with spin interacting with a single nucleus of charge
$Z > 0$ in an external magnetic field, Jurg Fröhlich, Elliot Lieb, and Michael Loss in 1986 showed that there is no ground state
energy if $Z > Z_c$ and the ground state energy exists if $Z < Z_c$.
 
Another notion of stability in quantum mechanics is that of dynamic stability. Dynamic stability refers to the question of global
well-posedness for a system of partial differential equations that models the dynamics of many electrons coupled to their
self-generated electromagnetic field and interacting with many nuclei. The central motivating question of our PhD thesis is
whether energetic stability has any influence on the global well-posedness of the corresponding dynamical equations. In this regard,
we study the quantum mechanical many-body problem of $N$ non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and $K$ static nuclei. We model the dynamics of the electrons and their self-generated 
electromagnetic field using the so-called many-body Maxwell-Pauli equations. The main result presented is the construction
time global, finite-energy, weak solutions to the many-body Maxwell-Pauli equations under the assumption that the fine structure
constant $\alpha$ and the nuclear charges are sufficiently small to ensure energetic stability of this system. If time permits, we
will discuss several open problems that remain.

Existence and uniqueness to a fully non-linear version of the Loewner-Nirenberg problem

Series
PDE Seminar
Time
Tuesday, February 25, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yanyan Li – Rutgers University –

We consider the problem of finding on a given bounded and smooth
Euclidean domain \Omega of dimension n ≥ 3 a complete conformally flat metric whose Schouten
curvature A satisfies some equation of the form  f(\lambda(-A)) =1. This generalizes a problem
considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of
locally Lipschitz solutions. We also show that the Lipschitz regularity is in general optimal.

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