Past PDE CDT Lunchtime Seminar

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.

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.)

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

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.