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.

# Past PDE CDT Lunchtime Seminar

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.