Past PDE CDT Lunchtime Seminar

15 October 2015
Martin Taylor
Given an initial data set for the vacuum Einstein equations which is suitably close to that of Minkowski space, the monumental work of Christodoulou—Klainerman guarantees the corresponding solution exists globally and asymptotically approaches the Minkowski solution.  The aim of the talk is to put this theorem in context, emphasising the importance of the null condition, before briefly discussing a new result on the corresponding problem in the presence of massless matter described by the Vlasov equation.
  • PDE CDT Lunchtime Seminar
18 June 2015
Stefan Steinerberger

I will discuss a puzzling theorem about smooth, periodic, real-valued functions on the real line. After introducing the classical Hardy-Littlewood maximal function (which just takes averages over intervals centered at a point), we will prove that if a function has the property that the computation of the maximal function is simple (in the sense that it's enough to check two intervals), then the function is already sin(x) (up to symmetries). I do not know what maximal local averages have to do with the trigonometric function. Differentiation does not help either: the statement equivalently says that a delay differential equation with a solution space of size comparable to C^1(0,1) has only the trigonometric function as periodic solutions.

  • PDE CDT Lunchtime Seminar
4 June 2015
Mariapia Palombaro
I will present some recent results concerning the higher gradient integrability of σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability exponent of the gradient field is known thanks to the work of Astala and Leonetti & Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise unconstrained and show that the optimal exponent is attained on the class of two-phase conductivities σ: Ω⊂R27→ {σ1,σ2} ⊂M2×2. The optimal exponent is established, in the strongest possible way of the existence of so-called exact solutions, via the exhibition of optimal microgeometries. (Joint work with V. Nesi and M. Ponsiglione.)
  • PDE CDT Lunchtime Seminar
28 May 2015
Jonathan Ben-Artzi
It is often desirable to solve mathematical problems as a limit of simpler problems. However, are such techniques always guaranteed to work? For instance, the problem of finding roots of polynomials of degree higher than three starting from some initial guess and then iterating was only solved in the 1980s (Newton's method isn't guaranteed to converge): Doyle and McMullen showed that this is only possible if one allows for multiple independent limits to be taken, not just one. They called such structures "towers of algorithms". In this talk I will apply this idea to other problems (such as computational quantum mechanics, inverse problems, spectral analysis), show that towers of algorithms are a necessary tool, and introduce the Solvability Complexity Index. An important consequence is that solutions to some problems can never be obtained as a limit of finite dimensional approximations (and hence can never be solved numerically). If time permits, I will mention connections with analogous notions in logic and theoretical computer science.

Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).             

  • PDE CDT Lunchtime Seminar
21 May 2015
Toan Nguyen

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small. In boundary-layer variables, this yields an exponentially growing sublayer near the boundary and hence instability of the asymptotic expansions, within an arbitrarily small time, in the inviscid limit. On the other hand, we show that the Prandtl asymptotic expansions hold for certain steady flows. Our proof involves delicate construction of approximate solutions (linearized Euler and Prandtl layers) and an introduction of a new positivity estimate for steady Navier-Stokes. This in particular establishes the inviscid limit of steady flows with prescribed boundary data up to order of square root of small viscosity. This is a joint work with Emmanuel Grenier and Yan Guo.

  • PDE CDT Lunchtime Seminar
14 May 2015
Fabio Ancona
Inspired by a question posed by Lax, in recent years it has received  
an increasing attention the study of quantitative compactness  
estimates for the solution operator $S_t$, $t>0$ that associates to  
every given initial data $u_0$ the corresponding solution $S_t u_0$ of  
a conservation law or of a first order Hamilton-Jacobi equation.

Estimates of this type play a central roles in various areas of  
information theory and statistics as well as of ergodic and learning  
theory. In the present setting, this concept could provide a measure  
of the order of ``resolution'' of a numerical method for the  
corresponding equation.

In this talk we shall first review the results obtained in  
collaboration with O. Glass and K.T. Nguyen, concerning the  
compactness estimates for solutions to conservation laws. Next, we  
shall turn to the  analysis of the Hamilton-Jacobi equation pursued in  
collaboration with P. Cannarsa and K.T.~Nguyen.
  • PDE CDT Lunchtime Seminar
7 May 2015
Paolo Secchi

In this talk I present a recent result about the free-boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. We study such amplitude equation and prove its nonlinear well-posedness under a stability condition given in terms of a longitudinal strain of the fluid along the discontinuity. This is a joint work with A.Morando and P.Trebeschi.

  • PDE CDT Lunchtime Seminar
30 April 2015
Kim Pham
Shape Memory Alloys (SMA) e.g. NiTi display a superelastic behavior at high temperature. Initially in a stable austenite phase, SMA can transform into an oriented martensite phase under an applied mechanical loading. After an unloading, the material recovers its initial stress-free state with no residual strain. Such loading cycle leads to an hysteresis loop in the stress-strain diagram that highlights the dissipated energy for having transformed the material. 
In a rate-independent context, we first show how a material stability criterion allows to construct a local one-dimensional phase transformation model. Such models relies on a unique scalar internal variable related to the martensite volume fraction. Evolution problem at the structural scale is then formulated in a variational way by means of two physical principles: a stability criterion based on the local minima of the total energy and an energy balance condition. We show how such framework allows to handle softening behavior and its compatibility with a regularization based on gradient of the internal variable.
We then extend such model to a more general three dimensional case by introducing a tensorial internal variable. We derive the evolution laws from the stability criterion and energy balance condition. Second order conditions are presented. Illustrations of the features of such model are shown on different examples. 
  • PDE CDT Lunchtime Seminar
5 March 2015
We investigate the problem of optimizing the shape and
location of actuators or sensors for evolution systems
driven by a partial differential equation, like for
instance a wave equation, a Schrödinger equation, or a
parabolic system, on an arbitrary domain Omega, in
arbitrary dimension, with boundary conditions if there
is a boundary, which can be of Dirichlet, Neumann,
mixed or Robin. This kind of problem is frequently
encountered in applications where one aims, for
instance, at maximizing the quality of reconstruction
of the solution, using only a partial observation. From
the mathematical point of view, using probabilistic
considerations we model this problem as the problem of
maximizing what we call a randomized observability
constant, over all possible subdomains of Omega having
a prescribed measure. The spectral analysis of this
problem reveals intimate connections with the theory of
quantum chaos. More precisely, if the domain Omega
satisfies some quantum ergodic assumptions then we
provide a solution to this problem.

These works are in collaboration with Emmanuel Trélat
(Univ. Paris 6) and Enrique Zuazua (BCAM Bilbao, Spain).
  • PDE CDT Lunchtime Seminar