Laboratoire AGM Département de Mathématiques Université de Cergy-Pontoise, France Since october 2014, i am PhD student under the supervision of Research interests:Qualitative study of Hamiltonian PDE Invariant Measures Papers:[1] Mouhamadou Sy. Invariant measure and long time behavior of regular solutions of the Benjamin-Ono equation, 2016, submitted. (pdf)[2] Mouhamadou Sy. Invariant measure and large time dynamics of the cubic Klein-Gordon equation in 3D, To appear.Selected talks:July 2 2016: AIMS 11th conference, Orlando (USA). "Invariant measure concentrated on C^\infy(\Bbb T)" for the Benjamin-Ono equation".(slides)April 26 2016: PDE seminar, Brown university (Providence USA). "Invariant measure concentrated on C^\infty for the Benjamin-Ono equation".October 01 2015: Journée de restitution, DIM RDMath IDF, Jussieu. "Mesures invariantes pour des EDPs hamiltoniennes"Invariant measures for Hamiltonian PDE and some results of my thesis:In qualitative study of evolution PDE, invariant measures allow to describe long time behavior of solutions by ergodic theorems (in particular Poincaré recurrence theorem). They give also a probabilistic alternative of the Cauchy theory (Probabilistic well posedness). There are, at least, two approaches to construct such measures: 1-GIBBS MEASURES FOR PDEIn finite dimension, there is a general result on existence of invariant measure for evolution under divergence free vector-fields. The so-called Liouville theorem says that such evolution preserves the Lebesgue measure defined on the underlying phase space. The Hamiltonian vector-fields are concerned by this theorem. It is clear that any integrable function of the Hamiltonian function is an invariant density w.r.t. this measure. The particular case of the function exp(-x) corresponds to the GIBBS MEASURE for the ODE. Gibbs measures for PDE are the extension of the theory described above to infinite dimensional Hamiltonian system. 2- THE FLUCTUATION-DISSIPATION-LIMIT (FDL) APPROACHThis approach proceeds by introducing a "competition" to the Hamiltonian PDE. Namely, a damping term is confronted to the effects of a forcing one. "PDE=a.Damping +f(a).Forcing" a is a viscosity parameter. A balance holds under a good scaling w.r.t. the viscosity parameter. An inviscid limit is the desired measure. SOME RESULTS OF MY THESIS1- Construction of invariant measure concentrated on infinitely smooth solutions (C^\infty)For certain PDE having an infinite sequence of regularity increasing conservation laws (for example KdV, cubic 1D NLS, Benjamin-Ono), an invariant Gaussian type measure is constructed in any Sobolev space (Zidkhov for KdV and NLS 1D, Deng-Tzvetkov-Visciglia for Benjamin-Ono). That, in particular, describes long time behavior of "Sobolev regularity solutions" of these equations. However all these measures neglect the space C^\infty, i.e the infinitely smooth solutions. In [1], we construct an invariant measure for BO concentrated on C^\infty. Our strategy is based on the FDL approach and applies to KdV and cubic 1D NLS, we prove some qualitative properties for this measure. 2-Invariant measure for a general Klein-Gordon equation in 3D, extension of the FDL approach to PDE having one conservation law The FDL approach is known to require two "good" conservation laws, it was needed in previous works to work compactness arguments. We consider in [2] the 3D Klein-Gordon equation (including the masless and even the imaginary mass cases in certain situations), only one conservation law is known for this equation. We give a strategy based on the FDL approach to construct an invariant measure concentrated on $H^2xH^1$ when the equation is posed both on the torus T^3 or on a bounded domain with smooth boundary in R^3. We show qualitative properties of this measure. |