Forthcoming events in this series


Thu, 14 May 2015

12:00 - 13:00
L6

On quantitative compactness estimates for hyperbolic conservation laws and Hamilton-Jacobi equations

Fabio Ancona
(University of Padova)
Abstract
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.

Thu, 07 May 2015

12:00 - 13:00
L5

Approximate current-vortex sheets near the onset of instability

Paolo Secchi
(University of Brescia)
Abstract

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.

Thu, 30 Apr 2015

12:00 - 13:00
L6

Construction of a macroscopic model of phase-transformation for the modeling of superelastic Shape Memory Alloys

Kim Pham
(Paris)
Abstract
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. 
 
Thu, 05 Mar 2015

12:00 - 13:00
L6

Optimal shape and location of actuators or sensors in PDE models

Yannick Privat
(Laboratoire Jacques-Louis Lions)
Abstract
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).
Thu, 26 Feb 2015

12:00 - 13:00
L6

Stability in exponential time of Minkowski Space-time with a translation space-like Killing field

Cecile Huneau
(Ecole Normale Superieure)
Abstract
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.
Thu, 19 Feb 2015

12:00 - 13:00
L6

Linear inviscid damping for monotone shear flows.

Christian Zillinger
(University of Bonn)
Abstract
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.
Thu, 12 Feb 2015

12:00 - 13:00
L6

Twinning in Strained Ferroelastics: Microstructure and Statistics

Xiangdong Ding
(xi'an Jiatong University)
Abstract

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.

Thu, 05 Feb 2015

12:00 - 13:00
L6

The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^\infty$ coefficients.

Andrew Morris
(Oxford University)
Abstract

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.

Thu, 22 Jan 2015

12:00 - 13:00
L6

HYPOCOERCIVITY AND GEOMETRIC CONDITIONS IN KINETIC THEORY.

Harsha Hutridurga
(Cambridge University)
Abstract
We shall discuss the problem of the 'trend to equilibrium' for a 

degenerate kinetic linear Fokker-Planck equation. The linear equation is 

assumed to be degenerate on a subregion of non-zero Lebesgue measure in the 

physical space (i.e., the equation is just a transport equation with a 

Hamiltonian structure in the subregion). We shall give necessary and 

sufficient geometric condition on the region of degeneracy which guarantees 

the exponential decay of the semigroup generated by the degenerate kinetic 

equation towards a global Maxwellian equilibrium in a weighted Hilbert 

space. The approach is strongly influenced by C. Villani's strategy of 

'Hypocoercivity' from Kinetic theory and the 'Bardos-Lebeau-Rauch' 

geometric condition from Control theory. This is a joint work with Frederic 

Herau and Clement Mouhot.
Thu, 15 Jan 2015

12:00 - 13:00
L3

Regularity for double phase variational integrals

Giuseppe Mingione
(Parma)
Abstract
Those mentioned in the title are integral functionals of the Calculus of Variations

characterized by the fact of having an integrand switching between two different

kinds of degeneracies, dictated by a modulating coefficient. They have introduced

by Zhikov in the context of Homogenization and to give new examples of the related

Lavrentiev phenomenon. In this talk I will present some recent results aimed at

drawing a complete regularity theory for minima.
Thu, 04 Dec 2014

12:00 - 13:00
L4

Higher regularity of the free boundary in the elliptic thin obstacle problem

Wenhui Shi
(Bonn University)
Abstract

In this talk, I will describe how to use the partial hodograph-Legendre transformation to show the analyticity of the free boundary in the elliptic thin obstacle problem. In particular, I will discuss the invertibility of this transformation and show that the resulting fully nonlinear PDE has a subelliptic structure. This is based on a joint work with Herbert Koch and Arshak Petrosyan.

Thu, 27 Nov 2014

12:00 - 13:00
L4

Interface motion in ill-posed diffusion equations

Michael Helmers
(Bonn University)
Abstract
We consider a discrete nonlinear diffusion equation with bistable nonlinearity. The formal continuum limit of this problem is an
ill-posed PDE, thus any limit dynamics might feature measure-valued solutions, phases interfaces, and hysteretic interface motion.
Based on numerical simulations, we first discuss the phenomena that occur for different types of initial. Then we focus on the case of
interfaces with non-trivial dynamics and study the rigorous passage to the limit for a piecewise affine nonlinearity.
Thu, 06 Nov 2014

12:00 - 13:00
L4

Towards an effective theory for nematic elastomers in a membrane limit

Paul Plucinsky
(Caltech)
Abstract
 

For nematic elastomers in a membrane limit, one expects in the elastic theory an interplay of material and structural non-linearities. For instance, nematic elastomer material has an associated anisotropy which allows for the formation of microstructure via nematic reorientation under deformation. Furthermore, polymeric membrane type structures (of which nematic elastomer membranes are a type) often wrinkle under applied deformations or tractions to avoid compressive stresses. An interesting question which motivates this study is whether the formation of microstructure can suppress wrinkling in nematic elastomer membranes for certain classes of deformation. This idea has captured the interest of NASA as they seek lightweight and easily deployable space structures, and since the use of lightweight deployable membranes is often limited by wrinkling.

 

In order to understand the interplay of these non-linearities, we derive an elastic theory for nematic elastomers of small thickness. Our starting point is three-dimensional elasticity, and for this we incorporate the widely used model Bladon, Terentjev and Warner for the energy density of a nematic elastomer along with a Frank elastic penalty on nematic reorientation. We derive membrane and bending limits taking the thickness to zero by exploiting the mathematical framework of Gamma-convergence. This follows closely the seminal works of LeDret and Raoult on the membrane theory and Friesecke, James and Mueller on the bending theory.

 

Thu, 23 Oct 2014

12:00 - 13:00
L4

J.C. Maxwell's 1879 Paper on Thermal Transpiration and Its Relevance to Contemporary PDE

Marshall Slemrod
(University of Wisconsin - Madison)
Abstract
In his 1879 PRSL paper on thermal transpiration J.C.MAXWELL addressed the problem of steady flow of a dilute gas over a flat boundary. The experiments of KUNDT and WARBURG had demonstrated that if the boundary is heated with a temperature gradient , say increasing from left to right, the gas will flow from left to right. On the other hand MAXWELL using the continuum mechanics of his (and indeed our) day solved the ( compressible) NAVIER- STOKES- FOURIER equations for balance of mass, momentum, and energy and found a solution: the gas has velocity equal zero. The Japanese fluid mechanist Y. SONE has termed this the incompleteness of fluid mechanics. In this talk I will sketch MAXWELL's program and how it suggests KORTEWEG's 1904 theory of capillarity to be a reasonable “ completion” of fluid mechanics. Then to push matters in the analytical direction I will suggest that these results show that HILBERT's 1900 goal expressed in his 6th problem of passage from the BOLTZMANN equation to the EULER equations as the KNUDSEN number tends to zero in unattainable.