Seminars and Colloquia by Series

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 WolanskyIsrael 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 KiefferGeorgia 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 LiRutgers 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.

Convection in a coupled free-flow porous media flow system

Series
PDE Seminar
Time
Tuesday, January 21, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Xiaomin WangSouthern University of Science and Technology

We show that the Principle of Exchange of Stability holds for convection in a layer of fluids overlaying a porous media with proper interface boundary conditions and suitable assumption on the parameters. The physically relevant small Darcy number regime as well as the dependence of the convection on various parameters will be discussed. A theory on the dependence of the depth ratio of the onset of deep convection will be put forth together with supporting numerical evidence. A decoupled uniquely solvable, unconditionally stable numerical scheme for solving the system will be presented as well.

Benjamin-Ono soliton dynamics in a slowly varying potential

Series
PDE Seminar
Time
Thursday, January 16, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zhiyuan ZhangBrown University

We consider the Benjamin Ono equation, modeling one-dimensional long interval waves in a stratified fluid, with a slowly-varying potential perturbation. Starting with near soliton initial data, we prove that the solution remains close to a soliton wave form, with parameters of position and scale evolving according to effective ODEs depending on the potential. The result is valid on a time-scale that is dynamically relevant, and highlights the effect of the perturbation. It is proved using a Lyapunov functional built from energy and mass, Taylor expansions, spectral estimates, and estimates for the Hilbert transform.

Continued gravitational collapse for Newtonian stars

Series
PDE Seminar
Time
Thursday, January 9, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mahir HadzicUniversity College London

The basic model of an isolated self-gravitating gaseous star is given by the gravitational Euler-Poisson system. For any value of the adiabatic index strictly between 1 and 4/3 we construct an infinite-dimensional family of collapsing solutions to the Euler-Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface in the Lagrangian coordinates. The leading order singular behaviour is driven by collapsing dust solutions. This is a joint work with Yan Guo (Brown) and Juhi Jang (USC).

Invariant Gibbs measures and global strong solutions for 2D nonlinear Schrödinger equations

Series
PDE Seminar
Time
Tuesday, November 19, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrea R. NahmodUniversity of Massachusetts Amherst

In this talk I'll first give an background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic nonlinear Schrodinger equation in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. I will then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence /with uniqueness/) for the periodic nonlinear Schrödinger equation (NLS) in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call /random averaging operators /which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is work with Yu Deng (USC) and Haitian Yue (USC).

Quantitative estimates of propagation of chaos for stochastic systems

Series
PDE Seminar
Time
Tuesday, November 5, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pierre-Emmanuel JabinUniversity of Maryland

We study the mean field limit of large stochastic systems of interacting particles. To treat more general and singular kernels, we propose a modulated free energy combination of the method that we had previously developed and of the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the most singular terms involving the divergence of the flow. Our modulated free energy allows to treat singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller-Segel system in the subcritical regimes, is obtained. This is a joint work with D. Bresch and Z. Wang.

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