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Wed Sep 12

PDE Seminar

3:30pm - Vincent Hall 570
Estimates of the entropy numbers of the Sobolev embedding near the limit case
Yuri Netrusov, Bristol

In 1967 M.Birman and M.Solomyak proved the following remarkable result. Let l,d be positive integers in [1,\infty], and T be the embedding of W^l_p((0,1)^d) to L^q ((0,1)^d). Suppose that 0 < 1/p-1/q < l/d.
Then there are positive constants c_1, c_2 such that for all positive interges k the estimates c_1 k^{-l/d} \leq e_k(T) \leq c_2 k^{-l/d} hold. Here e_k(T) is the k-entropy number of T. Some generalizations and extensions of this result will be given. For instance under some restrictions on the parameters behavior of c_1, c_2
(as function of l/d-1/p+1/q) were investigated.

Wed Sep 19

PDE Seminar

3:30pm - Vincent Hall 570
On sparse bounds for some biparameter operators
Jose Conde Alonso, Department of Mathematics, Brown University

The last few years have seen a great deal of work around the concept of sparse domination. This is a technique that allows one to study quantitative inequalities for a wide range of important operators in harmonic analysis in a unified and precise way. We briefly review what sparse domination is and we will explain the very different situation that one en- counters when the operators under study are biparametric, that is, when we consider spaces whose dilations need not be isotropic. Based on joint work (in progress) with Alex Barron, Yumeng Ou and Guillermo Rey.

Wed Oct 10

PDE Seminar

3:35pm - Vincent Hall 570
Soliton resolution for the critical nonlinear heat equation
Hiroshi Matano, Meiji University

In this talk, I will discuss the asymptotic behavior of radially symmetric solutions of the nonlinear heat equation on ${\bf R}^N$ $(N > 2)$ with the Sobolev critical power nonlinearity.

In the case of time-global solutions, we show that the solution is asymptotically decomposed into a finite sum of rescaled ground states that hardly interact with each other because the ratio of the rescaling parameters of different solitons tends to infinity. The total energy of the solution then converges to an integer multiple of the energy of the ground state. We call this behavior ``soliton resolution" of the solution.

In the case where the solution blows up in finite time, and if the blow-up is of type II in a certain sense, we show that a similar soliton resolution occurs near the blow-up point.

Finally we will show the existence of a multi-soliton time-global solution for the case $N > 6$. This is joint work with Frank Merle.

Wed Oct 17

PDE Seminar

3:30pm - Vincent Hall 570
Continuum Limits of Semi-Supervised Learning on Graphs
Matthew Thorpe, University of Cambridge

Given a data set $\{x_i\}_{i=1}^n$ with labels $\{y_i\}_{i=1}^N$ on the first $N$ data points the goal of semi-supervised is to infer labels on the remaining $\{x_i\}_{i=N+1}^n$ data points. In this talk we use a random geometric graph model with connection radius $r(n)$. The framework is to consider objective functions which reward the regularity of the estimator function and impose or reward the agreement with the training data, more specifically we will consider discrete p-Laplacian and fractional Laplacian regularization.

The talk concerns the asymptotic behaviour in the limit where the number of unlabelled points increases while the number of training points remains fixed. The results are to uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. I will give almost optimal ranges on the scaling of $r(n)$ for asymptotic consistency to hold. Furthermore, I will setup the Bayesian interpretation of this problem.

This is joint work with Matt Dunlop (Caltech), Dejan Slepcev (CMU) and
Andrew Stuart (Caltech).

Mathew Thorpe is a Research Fellow at the Cantab Capital Institute for the Mathematics of Information, Department of Applied Mathematics and Theoretical Physics, University of Cambridge.


Wed Oct 31

PDE Seminar

3:30pm - Vincent Hall 570
On the global regularity for Einstein-Klein-Gordon coupled system
Alexandru Ionescu, Princeton University

I will discuss the Einstein-Klein-Gordon coupled system of General
Relativity, and the problem of global stability of the Minkowski space-time.
This is joint work with Benoit Pausader.

Wed Nov 28

PDE Seminar

3:30pm - Vincent Hall 570
An Epiperimetric Approach to Isolated Singularities
Max Engelstein, MIT

The presence of singular points (i.e. points around which the object in question does not look flat at any scale) is inevitable in most minimization problems. One fundamental question is whether minimizers have a unique tangent object at singular points i.e., is the minimizer increasingly well approximated by some other minimizing object as we “zoom in” at a singular point. This question has been investigated with varying degrees of success in the settings of minimal surfaces, harmonic maps and obstacle problems amongst others.

In this talk, we will give the first uniqueness of blowups result for minimizers of the Alt-Caffarelli functional. In particular, we prove that the tangent object is unique at isolated singular points in the free boundary. Our main tool is a new approach to proving (log-)epiperimetric inequalities at isolated singularities. This epiperimetric inequality differs from previous ones in that it holds without any additional assumptions on the symmetries of the tangent object.

If we have time, we will also discuss how this method allows us to recover some uniqueness of blow-ups results in the minimal surfaces setting, particularly those of Allard-Almgren (’81) and Leon Simon (’83). This is joint work with Luca Spolaor (MIT) and Bozhidar Velichkov (U. Grenoble Alpes).

Wed Dec 05

PDE Seminar

3:35pm - Vincent Hall 570
Barotropic instability of shear flows
Zhiwu Lin, Georgia Institute of Technology

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop an approach by using the Hamiltonian structures of the linearized equation and an instability index formula to find the sharp stability conditions. We studied the flow with Sinus profile in details and found the sharp stability boundary in the whole parameter space, which corrected previous results in the fluid literature. The addition of the Coriolis force brings some fundamental changes to the stability of shear flows. Moreover, we also study the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping near the shear flows. This is joint work with Hao Zhu and Jincheng Yang.

Wed Dec 12

PDE Seminar

3:35pm - Vincent Hall 570
Scattering for the 3D Gross-Pitaevskii equation
Zihua Guo, Monash University

We study the Cauchy problem for the 3D Gross-Pitaevskii equation. Global well-posedness in the natural energy space was proved by Gerard.
We prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.
The crucial ingredients are new generalized Strichartz estimates and some new observed "NULL" structures of the Gross-Pitaevskii equation after some normal form type transform.
This is a joint work with Zaher Hani and Kenji Nakanishi.

Mon Feb 11

PDE Seminar

3:35pm - Vincent Hall 313
Computer-Assisted Proofs in Partial Differential Equations
Javier Gomez-Serrano, Princeton University

In the last 50 years computing power has experienced an enormous development: every two years the number of transistors has doubled since the 1970s. However, even nowadays when we can perform very fast computations it is not clear a priori if one can obtain rigorous results based on the output of computer calculations. In this talk I will explain the basics of interval analysis and how it can be used to prove theorems in different contexts of PDE, ranging from fluid dynamics to spectral geometry.

Tue Feb 12

PDE Seminar

10:00am - Vincent Hall 570
Global existence in incompressible fluid equations
Javier Gomez-Serrano, Princeton University

There has been high scientific interest to understand the behavior of the surface quasi-geostrophic (SQG) equation because it is a possible model to explain the formation of fronts of hot and cold air and because it also exhibits analogies with the 3D incompressible Euler equations. It is not known at this moment if this equation can produce singularities or if solutions exist globally. In this talk, I will discuss some recent works on the existence of global solutions for the SQG and modified SQG equations.

Fri Feb 22

PDE Seminar

3:35pm - Vincent Hall 311
Special PDE Seminar - How to obtain parabolic theorems from their elliptic counterparts
Blair Davey, City College of New York

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 analogs. 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.

Wed Mar 06

PDE Seminar

3:35pm - Vincent Hall 570
Quantitative estimates of propagation of chaos for large systems of interacting particles
Zhenfu Wang, University of Pennsylvania

We present a new method to derive quantitative estimates proving the propagation of chaos for large stochastic or deterministic systems of interacting particles. Our approach requires to prove large deviations estimates for non-continuous potentials modified by the limiting law. But it leads to explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit; and it can be applied to very singular kernels that are only in negative Sobolev spaces and include the Biot-Savart law for 2D Navier-Stokes and 2D Euler. Joint work with P.-E. Jabin.

Wed Mar 27

PDE Seminar

3:30pm - Vincent Hall 570
PDE Seminar
Luis Vega, Serrin Lecture
Wed Apr 03

PDE Seminar

3:30pm - Vincent Hall 570
Convexity of Whitham's highest cusped wavear
Bruno Vergara, Madrid

Whitham's model of shallow water waves is a non-local dispersive equation that features traveling wave solutions and also singularities. I will discuss a conjecture of Ehrnström and Wahlén on the profile of solutions of extreme form and show that there exists a highest, cusped and periodic solution convex between consecutive crests of $C^{1/2}$-regularity. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.

Wed Apr 10

PDE Seminar

3:30pm - Vincent Hall 570
Lagrangian chaos and scalar mixing in stochastic fluid mechanics
Samuel Punshon-Smith, Brown University

Lagrangian chaos refers to the chaotic behavior of
Lagrangian trajectories in a fluid. This chaotic behavior often
characterized by the dynamics having a positive Lyapunov exponent,
namely the property that initially close trajectories will separate at
an exponential rate after long time. In this talk we will consider a
variety of stochastically forced fluid models, including the 2
dimensional stochastic Navier-Stokes equations, and show that under
certain non-degeneracy conditions on the noise, the Lagrangian flow
possesses a positive Lyapunov exponent. The proof crucially uses the
theory of random dynamical systems as well as tools from Malliavin
calculus and control theory to satisfy a certain non-degeneracy
criterion originally due to Furstenberg. We will explore several
important consequences of a positive exponent for passive scalars
advected by the fluid. Most importantly, we show that these velocity
fields are almost sure exponential mixers. Specifically, using the
positive Lyapunov exponent, we obtain almost sure exponential decay in
time of passive scalars in any negative Sobolev norm through a
detailed study of ergodicity of the two point Lagrangian motion. This
work is joint with Jacob Bedrossian and Alex Blumenthal.

Wed Apr 17

PDE Seminar

3:35pm - Vincent Hall 570
On some recent results of Collisional Plasma in bounded domains
Chanwoo Kim, University of Wisconsin

In this talk we discuss Vlasov-Poisson-Boltzmann system in bounded domains. Some recent results on regularity and large time behavior of solutions will be presented.

Wed Apr 24

PDE Seminar

3:35pm - Vincent Hall 570
SERRIN LECTURE - Hydrodynamic stability and coherent structure at high Reynolds number
Jacob Bedrossian, Maryland

In this talk I will discuss recent work towards understanding certain stability questions for incompressible Euler or Navier-Stokes at high Reynolds number, the regime in which the viscous effects are weak. We will discuss the mechanisms which stabilize the most basic classes of equilbria: vortices and shear flows in 2D Navier-Stokes/Euler. These stabilization mechanisms, sometimes known as inviscid damping and enhanced dissipation arise from the skew-symmetric transport and the interaction between transport and the weak viscosity. We go on to discuss the roles such dynamics play in 3D, in particular, the manner in which two-dimensional stability can influence coherent structures in 3D. First, we discuss the subcritical transition of 3D shear flows -- the problem of obtaining quantitative estimates on the basin of stability of classical equilibria and the classification of solutions on the borderline of the stability domain. Second, the dynamics of vortex filaments in the 3D Navier-Stokes equations, which begins with understanding the local-in-time well-posedness with certain singular initial data, specifically, vorticity in a scale-invariant Morrey space of measures (in such a class, one does not have, nor does one expect, a general theory of local well-posedness). Connections to kinetic theory problems arising originally in plasma physics regarding nonlinear Landau damping will be addressed if time permits.