Past 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
26 February 2015
Cecile Huneau
In the presence of a translation space-like Killing field
the 3 + 1 vacuum Einstein equations reduce to the 2 + 1
Einstein equations with a scalar field. We work in
generalised wave coordinates. In this gauge Einstein
equations can be written as a system of quaslinear
quadratic wave equations. The main difficulty is due to
the weak decay of free solutions to the wave equation in 2
dimensions. To prove long time existence of solutions, we
have to rely on the particular structure of Einstein
equations in wave coordinates. We also have to carefully
choose the behaviour of our metric in the exterior region
to enforce convergence to Minkowski space-time at
time-like infinity.
  • PDE CDT Lunchtime Seminar
19 February 2015
Christian Zillinger
While the 2D Euler equations incorporate
neither dissipation nor entropy increase and
even possess a Hamiltonian structure, they
exhibit damping close to linear shear flows.
The mechanism behind this "inviscid
damping" phenomenon is closely related to
Landau damping in plasma physics.
In this talk I give a proof of linear stability,
scattering and damping for general
monotone shear flows, both in the setting
of an infinite periodic channel and a finite
periodic channel with impermeable walls.
  • PDE CDT Lunchtime Seminar
12 February 2015
Xiangdong Ding

The generation of functional interfaces such as superconducting and ferroelectric twin boundaries requires new ways to nucleate as many interfaces as possible in bulk materials and thin films. Materials with high densities of twin boundaries are often ferroelastics and martensites. Here we show that the nucleation and propagation of twin boundaries depend sensitively on temperature and system size. The geometrical mechanisms for the evolution of the ferroelastic microstructure under strain deformation remain similar in all thermal regimes, whereas their thermodynamic behavior differs dramatically: on heating, from power-law statistics via the Kohlrausch law to a Vogel-Fulcher law.We find that the complexity of the pattern can be well characterized by the number of junctions between twin boundaries. Materials with soft bulk moduli have much higher junction densities than those with hard bulk moduli. Soft materials also show an increase in the junction density with diminishing sample size. The change of the complexity and the number density of twin boundaries represents an important step forward in the development of ‘domain boundary engineering’, where the functionality of the materials is directly linked to the domain pattern.

  • PDE CDT Lunchtime Seminar
5 February 2015
Andrew Morris

We consider the layer potentials associated with operators $L=-\mathrm{div}A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent. A "Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi-Nash-Moser type estimates. In particular, we prove that $L^2$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.

  • PDE CDT Lunchtime Seminar