Publications

2013

• Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics
  
  • Brenier Yann
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2013, 47 (1-2), pp.55-64. We consider L2 minimizing geodesics along the group of volume preserving maps SDiff(D) of a given 3-dimensional domain D. The corresponding curves describe the motion of an ideal incompressible fluid inside D and are (formally) solutions of the Euler equations. It is known that there is a unique possible pressure gradient for these curves whenever their end points are fixed. In addition, this pressure field has a limited but unconditional (internal) regularity. The present paper completes these results by showing: (1) the uniqueness property can be viewed as an infinite dimensional phenomenon (related to the possibility of relaxing the corresponding minimization problem by convex optimization), which is false for finite dimensional configuration spaces such as O(3) for the motion of rigid bodies; (2) the unconditional partial regularity is necessarily limited. (10.1007/s00526-012-0510-7)
  DOI : 10.1007/s00526-012-0510-7
• Poincaré et la déconstruction du négatif
  
  • Paul Thierry
  , 2013. Nous présentons l'attitude d'Henri Poincaré face aux résultats "négatifs" tels qu'il les a analysés en mécanique céleste et en théorie des quanta.
• Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension
  
  • Côte Raphaël
  • Zaag Hatem
  Communications on Pure and Applied Mathematics, Wiley, 2013, 66 (10), pp.1541-1581. We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blow-up solution with a characteristic point, we refine the blow-up behavior first derived by Merle and Zaag. We also refine the geometry of the blow-up set near a characteristic point, and show that except may be for one exceptional situation, it is never symmetric with the respect to the characteristic point. Then, we show that all blow-up modalities predicted by those authors do occur. More precisely, given any integer $k\ge 2$ and $\zeta_0 \in \m R$, we construct a blow-up solution with a characteristic point $a$, such that the asymptotic behavior of the solution near $(a,T(a))$ shows a decoupled sum of $k$ solitons with alternate signs, whose centers (in the hyperbolic geometry) have $\zeta_0$ as a center of mass, for all times. (10.1002/cpa.21452)
  DOI : 10.1002/cpa.21452
• Optimal Regularizing Effect for Scalar Conservation Laws
  
  • Golse François
  • Perthame Benoît
  Revista Matemática Iberoamericana, European Mathematical Society, 2013, 29 (2013), pp.1477-1504. We investigate the regularity of bounded weak solutions of scalar conservation laws with convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. The proof is based on the kinetic formulation of scalar conservation laws and on an interaction estimate in physical space. (10.4171/rmi/765)
  DOI : 10.4171/rmi/765
• Sur la densite des representations cristallines du groupe de Galois absolu de Q_p
  
  • Chenevier Gaëtan
  Mathematische Annalen, Springer Verlag, 2013, 335, pp.1469-1525. Let X_d be the p-adic analytic space classifying the d-dimensional (semisimple) p-adic Galois representations of the absolute Galois group of Q_p. We show that the crystalline representations are Zarski-dense in many irreducible components of X_d, including the components made of residually irreducible representations. This extends to any dimension d previous results of Colmez and Kisin for d = 2. For this we construct an analogue of the infinite fern of Gouvêa-Mazur in this context, based on a study of analytic families of trianguline (phi,Gamma)-modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline (phi,Gamma)-modules, as well as the density of the crystalline (phi,Gamma)-modules in this family. These results may be viewed as a local analogue of the theory of p-adic families of finite slope automorphic forms, they are new already in dimension 2. The technical heart of the paper is a collection of results about the Fontaine-Herr cohomology of families of trianguline (phi,Gamma)-modules.