June 18, 2024

Thematic Program in Partial Differential Equations

Young Mathematicians Conference in PDE and Dynamical Systems

January 31 - February 1, 2004


Plenary speakers:
Niky Kamran
(McGill University)
Wave Equations in General Relativity

Adrian Nachman (University of Toronto)
Introduction to Inverse Problems in Partial Differential Equations

M. Agueh
(University of British Columbia)
Asymptotic Behavior for Doubly Degenerate Parabolic Equations
We use mass transportation inequalities to study the asymptotic behavior for a class of parabolic equations which contains the Heat,
Fokker-Planck, porous-medium, fast diffusion, and parabolic $p$-Laplacian equations, and some generalizations of these equations. We establish an exponential decay in "relative entropy" and in the $p$-Wasserstein distance of solutions -- or self-similar solutions -- of these equations to equilibrium, and we give the explicit rates of convergence. In particular, we generalize to all $p>1$ the HWI inequalities obtained by Otto and Villani (J.Funct.Anal.173 (2) (2002) 361-400) when $p=2$.

N. Ahmad (University of Toronto)
Boundary Matching via Optimal Transportation

R. Akila (University of Guelph)
Hopf Bifurcation of Coupled Oscillator Systems
Two-dimensional arrays of regularly spaced oscillators, with symmetric nearest neighbor coupling, may synchronize in patterns that are periodic in both space and time. This work presents a complete list of invariants, equivariants, normal forms, isotropy subgroups and fixed-point subspaces, for the cases with spatial periodicity N = 2,3,4, both with and without a Z2-internal symmetry. The analysis includes all the generic equivariant Hopf bifurcations in this setting, and determines the onset and stability of the patterns. This study is motivated by flow- induced vibrations of arrays of tubes in heat exchangers with cross-flow, and leads to predictions of the spatio-temporal patterns in such arrays.

W. Bahsoun (Concordia University)
Ergodic Properties of Random Maps and Their Applications
A random map is a discrete dynamical system consisting of a collection of maps which are selected randomly by means of probabilities at each iteration. We prove the existence of absolutely continuous invariant measures for a general class of position dependent random maps under mild conditions. Moreover, we prove that these measures are stable under small stochastic perturbations. We also apply these results to forecasting in financial markets.

A. Biryuk (The Fields Institute & McMaster University)
Flows Without Pressure, Geometry and Turbulence
We start with the Cauchy problem for the multidimensional Burgers type equation with periodic boundary conditions. We introduce the notion of degeneracy for vector fields. Vector field $u$ is degenerate iff one of the following equivalent conditions holds: (*) the Jacoby matrix of $u$ is everywhere nilpotent (*) the pressureless Euler equation with the initial state $u$ is globally solvable in the class of $C1$-continuous functions. Non-degenerate initial states develops large spatial derivatives (turbulence). In 2D there is a nice geometric criterion for degeneracy due to Pogorelov cylinder theorem.

M. Braverman (University of Toronto)
On the Computability of Julia Sets
While the computer is a discrete device, it is often used to solve problems of a continuous nature. The field of Real Computation addresses the issues of computability in the continuous setting. As in the discrete case, we would like to define the notion of a computable subset of R^n. The definition we use has a computer graphics interpretation (in the case n=2), as well as a deeper mathematical meaning.
The Julia sets are particularly well studied sets arising from complex dynamics. In the talk we will discuss efficient computability of real sets, and describe some results about the computability of Julia sets. Our computability results come in contrast to the Julia sets noncomputability results presented by Blum/Cucker/Shub/Smale. This discrepancy follows from the fact that we are using a different computability model.

M. Cojocaru (University of Guelph)
Projected Dynamical Systems: Overview and Recent Developments
A projected dynamical system (PDS) is defined by an ODE with a nonlinear and discontinuous righthand side whose solutions are constrained to evolve only within a closed and convex subset K of a Hilbert space X. A PDS was first introduced in 1992, however references to the ODE defining it go back to the 70's. It is worth mentioning that it was first formulated in the context of applied economic problems. In this talk, we present a brief overview of this kind of systems, the mathematical questions that were studied so far in its case and some recent developments on the topic.

F. Colin (Universite de Montreal)
Decomposition Lemmas Applied to Minimization Problems

D. Gaidashev (University of Toronto)
Renormalization of Isoenergetically Degenerate Hamiltonian Flows
KAM proofs of existence of invariant surfaces for isoenergetically degenarate (non-twist) Hamiltonian flows are known to present significant difficulties. We will demonstrate how the renormalization group transformation, a non-perturbative technique, can be used to
study problems which are traditionally the subject of the KAM theory. We will also mention some recent results about universality and self-similarity of critical Hamiltonian flows whose invariant tori are at the break-up.

P. Guyenne (The Fields Institute & Mcmaster University)
Wave Turbulence in One-dimensional Models
A two-parameter nonlinear dispersive wave equation proposed by Majda, McLaughlin and Tabak (1997) is studied analytically and numerically as a model for testing the validity of weak turbulence theory. Kolmogorov-type solutions for the energy spectrum are determined explicitly and compared with numerical results. These show a strong dependence on the sign of the nonlinear term. In one case, there is agreement with the theory. In the other, there is disagreement. Possible explanations for this discrepancy will be given such as the emergence of coherent structures: quasi-solitons and wave collapses.

M. Harada (University of Toronto)
Integrable systems, polytopes, and skew-hermitian matrices
One of the main themes of symplectic geometry in the past few decades has been the link between torus actions and polytopes. I will discuss this phenomenon in cases where the symplectic manifolds are familiar from linear algebra: spaces of skew-hermitian matrices (over either \C or the quaternions \H) conjugate to a given diagonal matrix. When one works over \C, a construction of Gel'fand-Cetlin, Guillemin-Sternberg links, via a torus action, a maximal polytope to these spaces of matrices, giving an integrable system. I will explain how to get an integrable system also for the skew-hermitian matrices over \H.

Z. Hu (Entrust Inc. & Carleton University)
Favard's Theorem for Stepanov Almost Periodic Differential Equations
In the theory of Bohr almost periodic differential equations, Bochner's Theorem plays an important role in discussing the existence of almost periodic solutions, in particular, in establishing Favard's Theorem. In this paper, we, firstly, extend Bochner's Theorem to the case of Stepanov almost periodic functions and then we establish a result about the existence of Stepanov almost periodic solutions for
linear Stepanov almost periodic differential equations on $R^n$, that is, we extend Favard's Theorem to the case of Stepanov almost periodic differential equations.

K. Kang (University of British Columbia)
On Regularity to the Stokes System and Navier-Stokes Equations Near Boundary
We study regularity problem for the Stokes system (SS) and the Navier-Stokes equations (NSE) near boundary. For the steady-state case we obtained local estimates of the SS "without pressure" and as an application we prove the partial regularity up to the boundary for the stationary NSE in five dimension. For the non-stationary SS we constructed an example, which shows, unlike in the interior case,
H\"{o}lder continuity does not imply smoothness in the spatial variable near boundary. For the NSE We proved that weak solutions,
which is locally in $L^{p,q}$ with $3/p+n/q=1, q>n$ near boundary are regular up to the boundary. In three dimension we also observed
that "suitable weak solutions" of the NSE are regular near boundary if the scaled $L^{r,s}$-norm with $3/r+2/s=2, 2<s<\infty$ of the velocity is sufficiently small.

T. Kolokolnikov (University of British Columbia)
Stripe Instabilities in the Two-Dimensional Gray-Scott Model
The Gray-Scott model is a reaction-diffusion system that is known to exhibit complicated spatial patterns. These include: stripes, rings, spots, domain-filling curves and any combination thereof. In this talk, we consider the stripe solutions in two dimensions. Such a solution can exhibit three different types of instability: a splitting instability, whereby a stripe self-replicates into two parallel stripes; a breakup instability, where a stripe breaks up into spots; and a zigzag instability, whereby a stripe develops a wavy perturbation in the transversal direction. We derive explicit thresholds for all three types of instability. Some open problems will be discussed.

A. Kuznetsov (University of Toronto)
Solvable Markov Models with Stochastic Volatility and Jumps
There are only few known examples of Markov processes for which the transitional probability density can be computed in analytically closed form. Our method gives a general framework for building a family of analytically tractable models, which include all the previously known processes, and enables us to construct new solvable martingale processes having such important properties as state-dependent volatility and/or stochastic volatility and/or jumps. The probability kernel is expressed as expansion in orthogonal polynomials. The method can be summarized as the change of measure, change of phase space for the process and the spectrum deformation for the generator of the Markov semigroup. I will also discuss the lattice approximations for these models and some applications to Mathematical Finance.

S. Lawi (University of Toronto)
Generating Functions for Stochastic Integrals
Generating functions for stochastic integrals have been known in analytically closed form for just a handful of stochastic processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which generating functions for stochastic integrals and transition probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. We also extend this scheme to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey-Wilson classification tree.

B. Moore (McGill University)
Numerical Methods that Preserve a Multi-Symplectic Conservation Law
A useful way to understand symplectic integration of Hamiltonian ODEs is through the system of equations, known as the modified equations, which are solved by the numerical solution. Here, the ideas of symplectic integration are extended to Hamiltonian PDEs, such that the symplectic structure in both space and time is exactly preserved. This paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these methods. In particular, the modified equations are used to derive
modified conservation laws of energy and momentum that are preserved to higher order along the numerical solution. For linear PDEs it is possible to show that the modified equations converge to the numerical scheme, and one is able to derive dispersion relations that completely describe the numerical solution behavior.

V. Panferov (University of Victoria)
Comparison Principles and Pointwise Upper Bounds for Solutions of Boltzmann-type Equations
I will describe a technique for obtaining pointwise upper bounds for solutions of the Boltzmann equation for elastic and inelastic particles. The results are relevant for description of tails of velocity distributions in rarefied gases and granular media.

A. Savu (University of Toronto)
Hydrodynamic Scaling Limit of the Fourth Order Ginzburg-Landau Model
The fourth order Ginzburg-Landau model is a microscopic model for the diffusion of particles on a surface relaxing to equilibrium. In my talk I will discusss how the evolution of the surface, on the macroscopic scale, given by a fourth order nonlinear evolution equation, emerges as a scaling limit of the particle dynamics. Since the model is of non-gradient type a major step in the computation of the limit is finding the right decomposition of the Hilbert space of "closed functions".

D. Slepcev (University of Toronto)
Gradient-Flow Structure and Stability of Selfsimilar Solutions of Nonlinear Parabolic PDE's
Nonlinear diffusion equations (porous-medium, fast-diffusion equation) and thin-film equations (with certain types of nonlinearities) can be recast as a gradient flows on an infinite-dimensional manifold.
The gradient-flow structure of these equations suggests a framework in which to study linear stability of selfsimilar solutions. Furthermore, in some cases it provides a way to show the asymptotic stability of the selfsimilar solutions, with optimal rates of convergence.
Of particular interest to us will be the stability of blow-up profiles of long-wave unstable thin-film equations. Future directions and open problems will also be discussed.

P. Tupper (McGill University)
Ergodicity and Numerical Simulation
An outstanding problem in numerical analysis is to justify long-time simulations of molecular systems. Typically, these systems are chaotic and numerically computed trajectories are accurate for only short periods of time. Nevertheless, qualitative features of computed trajectories are often quite accurate. I will discuss some recent work on this problem. The main innovation is the introduction of a weak ergodicity that suffices for the purposes of numerical simulation but is easier to work with than the standard mathematical definition.

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