SCIENTIFIC PROGRAMS AND ACTIVITIES

PAST SEMINARS

Zhuomin Liu (University of Pittsburgh)

The Liouville Theorem on Conformal Mappings
The celebrated Liouville Theorem from 1850 states that in dimension n?3, the only conformal maps are M\"obius transforms. Liouville's proof, as well as many subsequent proofs, required the mappings to be diffeomorphisms of class C3. However, since C1 regularity is sufficient to define conformal maps, one may inquire whether the Liouville theorem remains true under that, or even weaker conditions, e.g. Sobolev functions. The reduction from C3 regularity turned out to be very difficult. It this talk we will discuss the development of the Liouville Theorem under weaker and weaker regularity assumptions, including results of Gerhing, Reshnetyak, Bojarski and Iwaniec, Iwaniec and Martin on W1,nloc conformal mappings. Furthermore, Iwanice and Martin proved that in even dimensions n?4, W1,n/2loc conformal mappings are M\"{o}bious transforms and they conjectured that it should also be true in odd dimensions. We also discuss a proof of the Liouville Theorem f?W1,1loc in dimension n?3 under one additional assumption that the norm of the first order derivative $

Big frequency cascades in the cubic nonlinear Schrödinger flow on the 2-torus
Smooth solutions of the cubic nonlinear Schrödinger equation on the 2-torus which display a big frequency cascade were constructed in joint work with M. Keel, G. Staffilani, H. Takaoka and T. Tao. These solutions start off oscillating on long spatial scales. Over time, through nonlinear resonant interactions, these solutions begins to oscillate on smaller and smaller spatial scales exhibiting an arbitrary increment in high regularity Sobolev spaces. A strategy has recently been put forward by Z. Hani which aims to construct solutions with an infinite cascade. Recent work by M. Guardia and V. Kaloshin has quantified the speed of the transient cascade. This talk will describe these developments.

Arick Shao, University of Torotno
A Generalized Representation Formula for Covariant Tensor Wave Equations on Curved Spacetimes

In the Minkowksi space $\R^{1+3}$, the solution to an inhomogeneous wave equation is given explicitly by Kirchhoff's Formula. In this talk, we aim to extend Kirchhoff's Formula into a local first-order representation formula for covariant tensorial wave equations on arbitrary curved $(1+3)$-dimensional Lorentzian spacetimes. This formula is a generalization of an analogous Kirchhoff-Sobolev parametrix derived by Klainerman and Rodnianski, both in terms of the types of equations that can be treated as well as the assumptions required. Furthermore, the formula can be directly generalized to certain abstract vector bundles on the spacetime.

Nobu Kishimoto, Kyoto University
Well-posedness and finite-time blowup for the Zakharov system on two-dimensional torus

We consider the Zakharov system on two-dimensional torus. First, we show the local well-posedness of the Cauchy problem in the energy space by a standard iteration argument using the $X^{s,b}$ norms. Our result does not depend on the period of torus. Conservation laws and a sharp Gagliardo-Nirenberg inequality imply an a priori bound of solutions, which enables us to extend the local-in-time solution to a global one if its $L^2$ norm is less than that of the ground state solution of the cubic NLS on $R^2$. We then show that the $L^2$ norm of the ground state is actually the threshold for global solvability, namely, that there exists a finite-time blow-up solution to the Zakharov system on 2d torus with the $L^2$ norm greater than but arbitrarily close to that of the ground state. This is joint work with Masaya Maeda (Tohoku University, Japan).

Jordan Bell, University of Toronto
The KAM theorem

The Kolmogorov-Arnold-Moser (KAM) theorem tells us that if the orbits of a Hamiltonian system are confined to low dimensional tori, then many of the orbits will remain confined to low dimensional tori if we perturb the Hamiltonian. Many presentations of the KAM theorem do not give a precise formulation of the theorem. I will give a precise statement of the KAM theorem and then explain the method of it's proof.

Dorian Goldman, New York University
The Gamma-limit of the Ohta-Kawasaki energy. Derivation of the renormalized energy.

We study the asymptotic behavior of the screened sharp interface Ohta-Kawasaki energy in dimension 2 using the framework of convergence. In that model, two phases appear, and they interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating ``droplets" of that phase in a sea of the other phase. We consider perturbations to the critical volume fraction where droplets first appear, show the number of droplets increases monotonically with respect to the perturbation factor, and describe their arrangement in all regimes, whether their number is bounded or unbounded. When their number is unbounded, the most interesting case we compute the limit of the `zeroth' order energy and yield averaged information for almost minimizers, namely that the density of droplets should be uniform. We then go to the next order, and derive a next order -limit energy, which is exactly the ``Coulombian renormalized energy W" introduced in the work of Sandier/Serfaty, and obtained there as a limiting interaction energy for vortices in Ginzburg-Landau. The derivation is based on their abstract scheme, that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing at all to the Euler-Lagrange equation, we establish here for all configurations which have ``almost minimal energy," the asymptotic roundness and radius of the droplets as done by Muratov, and the fact that they asymptotically shrink to points whose arrangement should minimize the renormalized energy W, in some averaged sense. This leads to expecting to see hexagonal lattices of droplets.

Thursday, 24 November 2011 at 12:10PM
Fields Institute,
Room 210

Robert McCann, University of Toronto
Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach

With Jochen Denzler (UT Knoxville) and Herbert Koch (Bonn), we quantify the speed of convergence and higher asymptotics of fast diffusion dynamics on Euclidean space to the Barenblatt (self similar) profile. The degeneracy in the parabolicity of the equation is cured by re-expressing the dynamics on a manifold with a cylindrical end, called the cigar. The nonlinear evolution semigroup becomes differentiable with respect to Hoelder initial data on the cigar. The linearization of the dynamics is given by Laplace-Beltrami operator plus a drift term (which can be suppressed by the introduction of appropriate weights into the function space norm), plus a finite-depth potential well with a universal profile. In the limiting case of the (linear) heat equation, the depth diverges, the number of eigenstates increases without bound, and the continuous spectrum recedes to infinity. We provide a detailed study of the linear and nonlinear problems in Hoelder spaces on the cigar, including a sharp boundedness estimate for the semigroup, and use this as a tool to obtain sharp convergence results toward the Barenblatt solution. In finer convergence results (after modding out symmetries of the problem), a subtle interplay between convergence rates and tail behavior is revealed. The difficulties involved in choosing the right functional analytic spaces in which to carry out the analysis can be interpreted as genuine features of the equation rather than mere annoying technicalities.

Marina Chugunova (University of Toronto)
On uniqueness of the waiting-time type solution of the thin film equation
By formal asymptotic methods we analyze persistence and loss of zeros in generalized weak solutions of the thin film equation. We use a new dissipated functional recently constructed by Laugesen to prove an auxiliary energy equality that leads to the proof of the uniqueness of the waiting-type thin film solution under some additional assumptions about the initial data and regularity. Joint work with John King and Roman Taranets.

Brendan Pass (University of Alberta)
Optimal transportation with infinitely many marginals
I will discuss work in progress on an optimal transportation problem with infinitely many prescribed marginals. After formulating the problem and stating the main result, I will show that this result can be interpreted as a type of rearrangement inequality for stochastic processes. I will then discuss some connections with parabolic PDE, derivative pricing in mathematical finance and phase space bounds in quantum physics.

Christian Seis (University of Toronto)
On the coarsening rates in demixing binary viscous liquids
We consider the demixing process of a binary mixture of two liquids after a temperature quench. In viscous liquids, demixing is mediated by diffusion and convection. The typical particle size grows as a function of time t, a phenomenon called coarsening. Simple scaling arguments based on the assumption of statistical self-similarity of the domain morphology suggest the coarsening rates: from ? t^{1/3} for diffusion-mediated to ? t for flow-mediated coarsening. In joint works with Y. Brenier, F. Otto, and D. Slepcev, we derive the crossover of both scaling regimes in the sense of lower bounds on the energy, which scales, heuristically at least, as an inverse length. The mathematical model is a Cahn-Hilliard equation with an additional convection term. The velocity of the convecting fluid is determined by a Stokes equation. Our analysis follows closely a method proposed by R. V. Kohn and F. Otto, which is based on the gradient flow structure of the evolution.

Thursday
October 27, 2011 at 12:10 PM
Fields Institute,
Room 210

We consider the Willmore energy on the space of complete minimal surfaces in H3 and study the possible loss of compactness in the space of such surfaces with energy bounded above. This question has been extensively studied for various energy functionnals for closed manifolds. The first such study was that of Sacks and Uhlenbeck for harmonic maps. The key tools to study the loss of compactness in that case are epsilon-regularity and removability of singularities; the loss of compactness can then occur due to bubbling at a finite number of points where energy concentrates. We find the analogous results in our setting of complete surfaces. These are the first results in this direction for surfaces with a free boundary. joint with R. Mazzeo.

More information on the speaker can be found at: http://www.claymath.org/fas/research_fellows/Alexakis/

Amir Moradifam (University of Toronto) (slides)
Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions

We consider the problem of recovering an isotropic conductivity outside some perfectly conducting or insulating inclusions from the interior measurement of the magnitude of one current density field $|J|$. We prove that the conductivity outside the inclusions, and the shape and position of the perfectly conducting and insulating inclusions are uniquely determined (except in an exceptional case) by the magnitude of the current generated by imposing a given boundary voltage. We have found an extension of the notion of admissibility to the case of possible presence of perfectly conducting and insulating inclusions. This makes it possible to extend the results on uniqueness of the minimizers of the least gradient problem $F(u)=\int_{\Omega} a|\nabla u|$ with $u|_{\partial \Omega}=f$ to cases where u has flatregions (is constant on open sets). This is a joint work with Adrian Nachman and Alexandru Tamasam.

Thursday
September 22, 2011
12:10 p.m.
Fields Insitute,
Room 210

Christian Seis (University of Toronto)
Rayleigh--B\'enard convection. A new bound on the Nusselt number
This talk may be viewed as a follow-up to the UofT analysis seminar from Sept. 16, but also aspires to be self-contained!
We consider Rayleigh--B\'enard convection as modeled by the Boussinesq equations in the infinite-Prandtl-number limit. We are interested in the scaling of the average upward heat transport, the Nusselt number $Nu$, in terms of the non-dimensionalized temperature forcing, the Rayleigh number $Ra$. Experiments, asymptotics and heuristics suggest that $Nu \sim Ra^{1/3}$.
This work is mostly inspired by two earlier rigorous work on upper bounds of $Nu$ in terms of $Ra$: 1.) The work of Constantin and Doering establishing $Nu \sim Ra^{1/3}\ln^{2/3}Ra$ with help of a (logarithmically failing) maximal regularity estimate in $L^{\infty}$ on the level of the Stokes equation. 2.) The work of Doering, Reznikoff and Otto establishing $Nu \sim Ra^{1/3}\ln^{1/3}Ra$ with help of the background field method. We present a new bound on the Nusselt number: Etimates behind the background field method can be combined with the maximal regularity in $L^{\infty}$ to yield $Nu \sim Ra^{1/3}\ln^{1/3}\ln Ra$ --- an estimate that is only a double logarithm away from the supposed optimal scaling.

This is joint work with Felix Otto.