PDE-Applied Math Archives for Academic Year 2019

Stability for Higher-Order Hamiltonians

When: Thu, September 19, 2019 - 3:30pm
Where: Kirwan Hall 0101
Speaker: Shijun Zheng (Georgia Southern University) -
I will address the orbital stability problem for a class of dispersive Hamiltonian systems arising in nonlinear quantum media. In particular, I will elaborate on the construction of ground states as well as the threshold dynamics for the bi-harmonic NLS with a second-order perturbation. The proof relies on the profile decomposition that might be an efficient tool treating other systems like the magnetic NLS, fractional NLS and Dirac equations.

Separation of time-scales in fluid mechanics

When: Thu, September 26, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Michele Coti Zelati (Imperial College of London) -
Abstract: We present recent results on time-scales separation in fluid
mechanics. The fundamental mechanism to detect in a precise
quantitative manner is commonly referred to as fluid mixing. Its
interaction with advection, diffusion and nonlocal effects produces a
variety of time-scales which explain many experimental and numerical
results related to hydrodynamic stability and turbulence theory.

Stable singularity formation for the critical Keller-Segel equation

When: Thu, October 17, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Charles Collot (New York University) -
Abstract: The Patlak-Keller-Segel models chemotactic aggregation, and is a nonlocal non-
linear reaction diffusion equation. In dimension two, the problem is said to be critical as the conservation law (the mass of the solution), is left invariant by the scaling symmetry of the equation. Solutions do not exist for all times for this equation, and singularities may form. As in other critical settings (nonlinear Schrödinger, waves, wave maps etc...) the blow up phenomenon is not a truly self-similar one, but displays degenerate self-similarity: solutions concentrate at a point in finite time a bubble consisting of a stationary state, at a speed that is to be determined. After formal computations based essentially on matched asymptotics [Herrero-Velazquez, Dejak-Lushnikov-Yu-Ovchinnikov-Sigal, Dyachenko-Lushnikov-Vladimirova], a rigorous proof was provided [Raphael-Schweyer] relying on the precise control near the stationary state and the so-called tail dynamics. We propose a new proof of this result, conciliating the two approaches, which has at its heart a perturbative spectral problem encoding precisely the behaviour of the solution both near the stationary state and in the parabolic zone. This enables us to show the full nonradial stability of the dynamics, and to obtain refined asymptotics for the scaling law. This is joint work with T. Ghoul, N. Masmoudi and V.-T. Nguyen.

Finite Energy Weak Solutions of the Navier-Stokes-Korteweg equations

When: Thu, November 7, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Stefano Spirito (Gran Sasso Science Institute) -
In this talk I will present some results concerning the analysis of the existence of finite energy weak solutions of the Navier-Stokes-Korteweg equations, which model the dynamic of a viscous compressible fluid with degenerate viscosity and capillarity tensor. These kind of model are useful to study the dynamic of fluid near vacuum regions. A general theory of global existence is still missing, however for some particular cases of physical interest, it is possible to prove global existence of weak solutions. In particular, I will present two results regarding the case when the capillarity coefficient is constant and when the capillarity coefficient gives the Bohm potential. Moreover, in the cases under consideration the viscosity coefficient in the stress tensor are degenerating at the vacuum. The talk is based on a series of joint works with Paolo Antonelli (GSSI - Gran Sasso Science Institute, L’Aquila)

Schrodinger solutions on sparse and spread-out sets

When: Thu, November 21, 2019 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Xiumin Du (UMD) -
Abstract: If we want the solution to the Schrodinger equation to converge to its initial data pointwise, what's the minimal regularity condition for the initial data should be? I will present recent progress on this classic question of Carleson. This pointwise convergence problem is closely related to other problems in PDE and geometric measure theory, including spherical average Fourier decay rates of fractal measures, Falconer's distance set conjecture, etc. All these problems essentially ask how to control Schrodinger solutions on sparse and spread-out sets, which can be partially answered by several recent results derived from induction on scales and Bourgain-Demeter's decoupling theorem.

Elliptic integrands in geometric variational problems

When: Tue, November 26, 2019 - 3:30pm
Where: 1310.0
Speaker: Antonio De Rosa (New York University) - https://sites.google.com/view/antonioderosa
Abstract: Elliptic integrands are used to model anisotropic energies in
variational problems. These energies are employed in a variety of
applications, such as crystal structures, capillarity problems and
gravitational fields, to account for preferred inhomogeneous and
directionally dependent configurations. After a brief introduction to
variational problems involving elliptic integrands, I will present an
overview of the techniques I have developed to prove existence,
regularity and uniqueness properties of the critical points of
anisotropic energies. In particular, I will present the anisotropic
extension of Allard's rectifiability theorem and its applications to the
Plateau problem. Furthermore, I will describe the anisotropic
counterpart of Alexandrov's characterization of volume-constrained
critical points. Finally, I will mention some of my ongoing and future
research projects.

Beyond Bogoliubov Dynamics

When: Thu, January 16, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Petrat Soeren (Jacobs University Bremen) -
Abstract: Abstract: In this talk we consider a quantum system of N interacting
bosons in the mean field scaling regime. We construct corrections to the
Bogoliubov dynamics that approximate the true N-body dynamics in norm to
arbitrary precision. The corrections are such that they can be
explicitly computed in an N-independent way from the solutions of the
Bogoliubov and Hartree equations and satisfy a generalized form of
Wick's theorem. We determine the n-point correlation functions of the
excitations around the mean field, as well as the reduced densities of
the N-body system to arbitrary precision, given only the knowledge of
the two-point functions of a quasi-free state and the solution of the
Hartree equation.

The combined mean field and homogenization limit for interacting particle systems

When: Thu, February 6, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Matias Delgadino (Pontifical Catholic University of Rio de Janeiro) -
Abstract: In this talk, we will start by motivating the study of interacting particle systems as a way to model stochastic gradient descent in supervised learning. Then, we will analyse the statistical behavior of a large number of weakly interacting diffusion processes with highly oscillatory periodic interaction potentials. We study the combined limit of taking the number of particles to infinity, also known as the mean field limit, and taking the period of the potential to zero, also known as the homogenisation limit. In particular, we show that these limits do not commute if the system undergoes a phase transition, which will be properly defined in the talk.

Mean-field disordered systems and Hamilton-Jacobi equations

When: Thu, February 13, 2020 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Jean-Christophe Mourrat (Courant Institute - New York University) - https://cims.nyu.edu/~jcm777/
Abstract: Spin glasses are the most basic examples of disordered systems of
statistical mechanics with mean-field interactions. The infinite-volume
limit of their free energies are described by the celebrated Parisi
formula. I will describe a connection between this result and certain
Hamilton-Jacobi equations. The talk will then mostly focus on a simpler
setting arising from the problem of infering a large rank-one matrix, in
which the corresponding Hamilton-Jacobi equation is posed in a
finite-dimensional space.