Forthcoming events in this series
Global Nonlinear Stability of Minkowski Space for the Massless Einstein-Vlasov System
Abstract
A rigidity phenomenon for the Hardy-Littlewood maximal function
Abstract
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.
On geometry of stationary solutions of Euler equations
Abstract
Higher gradient integrability for σ -harmonic maps in dimension two
Abstract
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.)
Can we compute everything?
Abstract
Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).
Fluids at a high Reynolds number
Abstract
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.
On quantitative compactness estimates for hyperbolic conservation laws and Hamilton-Jacobi equations
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.
Approximate current-vortex sheets near the onset of instability
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.
Construction of a macroscopic model of phase-transformation for the modeling of superelastic Shape Memory Alloys
Abstract
Optimal shape and location of actuators or sensors in PDE models
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).
Stability in exponential time of Minkowski Space-time with a translation space-like Killing field
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.
Linear inviscid damping for monotone shear flows.
Abstract
exhibit damping close to linear shear flows.
The mechanism behind this "inviscid
In this talk I give a proof of linear stability,
Twinning in Strained Ferroelastics: Microstructure and Statistics
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.
The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^\infty$ coefficients.
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.
HYPOCOERCIVITY AND GEOMETRIC CONDITIONS IN KINETIC THEORY.
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.
Regularity for double phase variational integrals
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.
Higher regularity of the free boundary in the elliptic thin obstacle problem
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.
Interface motion in ill-posed diffusion equations
Abstract
ill-posed PDE, thus any limit dynamics might feature measure-valued solutions, phases interfaces, and hysteretic interface motion.
interfaces with non-trivial dynamics and study the rigorous passage to the limit for a piecewise affine nonlinearity.
Towards an effective theory for nematic elastomers in a membrane limit
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.