Blanc SIMI 1 - Blanc - SIMI 1 - Mathématiques et interactions

Stochastic systems in mathematics and mathematical physics – STOSYMAP

Submission summary

The aim of this project is to unite efforts of three French teams working on mathematical aspects of turbulence in various physical media. Past successes to tackle turbulence mathematically have been scarce and analytic comprehension has been notoriously difficult. Going further requires new results in Hamiltonian PDE's, probability theory, stochastic PDE, a deep qualitative understanding at a physical level, and possibly insights from numerical simulations. In the last few years, this type of knowledge was used independently by members of this project to obtain complementary original results in turbulence problems. The present joint effort should enable a marked progress in this important field.

We will consider mathematical issues related to solutions of the 3D Navier–Stokes equations (the classical turbulence setup) with high Reynolds numbers, and various statistical characteristics of these solutions. We will also consider simpler related models: the two-dimensional stochastic Navier–Stokes equations (2D turbulence, relevant for meteorology and some fields of physics), the stochastic nonlinear Schrödinger equation in dimensions 1, 2, 3 (optical turbulence), the Gross–Pitaevskii equation with stochastic perturbations (turbulence in Bose–Einstein condensation), the stochastic Burgers equation (a popular toy model for the classical turbulence), the Korteweg–de Vries equation with small dissipation and random force (another physical model for turbulence in various media).

The project is formed of analytical researches of qualitative properties of solutions for the equations above and for other similar problems. They are supported by numerical studies of the corresponding models. More specifically, our team plans to consider the following questions:
1. Ergodicity for 2D Navier–Stokes equation in a bounded domain with stochastic perturbations localized in the physical or Fourier space
2. The inviscid limit of stationary measures in special cases, such as damped/driven linear or completely integrable equations
3. Qualitative study of dispersive equations with various types of stochastic interventions, such as a random dispersion or a random amplitude of a potential
4. Ergodic behavior of the 2D Euler and Navier–Stokes flows and large-scale structures
5. 2D Navier–Stokes cascades in curved geometry

In parallel to the above-mentioned questions, we will investigate the more challenging (and unpredictable) problem of qualitative behavior of solutions for the 3D Navier–Stokes system in bounded and unbounded domains. Only few mathematical results related to the phenomenon of turbulence are known in this context, and there is no good understanding of the problem on the physical level of rigor. Our program will include the investigation of following problems for the 3D Navier–Stokes system and other related and/or simplified equations:
1. Uniform bounds for the local energy for particular classes of solutions
2. Rigorous results on approximation of physically relevant flows by models with a good understanding of the behavior of solutions
3. Investigation of space-time stationary solutions for the Navier–Stokes system with the Ekman damping and other related PDEs
4. Ergodic properties of stochastic models of turbulent transport of inertial particles
Further directions for research in the context of the Navier–Stokes system would concern the quantitative study of the direct and inverse cascades and application of the methods of non-equilibrium statistical mechanics to the ergodic theory of nonlinear PDEs

The first aim (and main cost) of this four-year project is to extend the existing research effort, by hiring post-docs to work with the involved researchers. The project will also develop long-term relationships between the partners laboratories, each of them internationally recognized in their own field. Finally, the project will fund workshops and meetings to foster international collaborations and discussions.

Project coordination

Armen SHIRIKYAN (UNIVERSITE DE CERGY-PONTOISE) – Armen.Shirikyan@u-cergy.fr

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

ENS RENNES
CNRS - ENS de Lyon CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION REGIONALE RHONE-AUVERGNE
Université de Cergy-Pontoise UNIVERSITE DE CERGY-PONTOISE
ENS Cachan - Bretagne ECOLE NORMALE SUPERIEURE DE CACHAN

Help of the ANR 226,609 euros
Beginning and duration of the scientific project: December 2011 - 48 Months

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