We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any Riemannian compact manifold. As a consequence we infer low regularity local well-posedness results in any dimension, as well as global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of cubic defocusing nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H s (M ), s < 1/2, where M is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H s (M ), where s ≥ 1/4 in the case of a boundary less manifold and s ≥ 8/21 in the case of a manifold with boundary.

A refined trilinear Strichartz estimate for solutions to the Schrödinger equation on the flat rational torus T 3 is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic Nonlinear Schrödinger Equation in H s (T 3 ) for all s ≥ 1. This is the first energy-critical global well-posedness result in the setting of compact manifolds.

Abstract. -We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere S 2 and more generally on Zoll surfaces, we prove that, for s > 1/4, NLS is uniformly well-posed in H s , which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces.Résumé. -Onétudie l'équation de Schrödinger non linéaire (NLS) sur une surface compacte. Sur la sphère S 2 et plus généralement sur toute surface de Zoll, on démontre que pour s > 1/4, NLS est uniformément bien posée dans H s , ce qui est optimal sur la sphère. Le principal ingrédient de notre démonstration est une estimation bilinéaire pour les projecteurs spectraux du laplacien sur une surface compacte.

Abstract. The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs measures), and second, we show that the flow that we are able to construct enjoys very nice dynamical properties, including a new notion of probabilistic continuity.

Alzheimer’s disease (AD) is the most common cause of progressive dementia in the elderly. It is characterized by a progressive and irreversible loss of cognitive abilities and formation of senile plaques, composed mainly of amyloid β (Aβ), and neurofibrillary tangles (NFTs), composed of tau protein, in the hippocampus and cortex of afflicted humans. In brains of AD patients the metabolism of Aβ is dysregulated, which leads to the accumulation and aggregation of Aβ. Metabolism of Aβ and tau proteins is crucially influenced by autophagy. Autophagy is a lysosome-dependent, homeostatic process, in which organelles and proteins are degraded and recycled into energy. Thus, dysfunction of autophagy is suggested to lead to the accretion of noxious proteins in the AD brain. In the present review, we describe the process of autophagy and its importance in AD. Additionally, we discuss mechanisms and genes linking autophagy and AD, i.e., the mTOR pathway, neuroinflammation, endocannabinoid system, ATG7, BCL2, BECN1, CDK5, CLU, CTSD, FOXO1, GFAP, ITPR1, MAPT, PSEN1, SNCA, UBQLN1, and UCHL1. We also present pharmacological agents acting via modulation of autophagy that may show promise in AD therapy. This review updates our knowledge on autophagy mechanisms proposing novel therapeutic targets for the treatment of AD.

Abstract. -We estimates the L p norm (2 ≤ p ≤ +∞) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. All our estimates are shown to be optimal for the sphere. Moreover, we sketch their extension to higher dimension Résumé. -On prouve une estimation de la norme L p (2 ≤ p ≤ +∞) de la restrictionà une courbe des fonctions propres de l'opérateur de Laplace Beltrami sur une surface riemannienne. Si la courbe est une géodésique de la sphère, on montre que nos estimations sont optimales. En revanche, si la courbe possède une courbure géodésique non nulle, on améliore le résultat. Toutes nos estimées sont optimales sur la sphère. Nous en esquissons par ailleurs des généralisations aux dimensions supérieures. IntroductionLet (M, g) be a compact smooth Riemannian manifold (without boundary) of dimension d. Let us denote by ∆ the Laplace operator associated to the metric g. Let Σ : [a, b] k → M be a smooth embedded sub-manifold of dimension k. The metric g endows M and Σ with canonical measures and consequently we can define the Lebesgue spaces L p (M ) and L p (Σ), 1 ≤ p ≤ +∞, of functions on M and Σ respectively. Let (ϕ λ ), λ ≥ 0, be the eigenfunctions of ∆ such that −∆ϕ λ = λ 2 ϕ λ . This paper fits in the line of researches dealing with possible concentrations of the eigenfunctions of the Laplace operator on a manifold. There are many ways 2000 Mathematics Subject Classification. -35P20, 35J15, 53C21.

Abstract. -We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in ∩ s<1/2 H s (Θ). We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions.