Ergodic Properties of the Levy Stochastic Shell Modells

To model homogeneous incompressible fluids one usually uses the Navier-Stokes equations. In most practical situations the numerical investigation of the three dimensional Navier-Stokes equations at high Reynolds` number is ubiquitous. However, it is well-known that it is difficult to compute analytically or via direct numerical simulations these kinds of fluids. To overcome this problem, many efforts have been made to construct models which can exhibit the physical properties of the three dimensional Navier-Stokes equations at high Reynolds` number. These are the so called models of turbulence and they are designed in such a way that they can capture the statistical properties of Navier-Stokes equations at a lower computability cost. Two examples are the so called sabra shell and GOY models which are simpler than the Navier-Stokes equations and are very promising for the investigation of turbulence in hydrodynamics. For the mathematical study towards the understanding of turbulence in hydrodynamics mathematicians use very often stochastic partial differential equations. These stochastic equations are usually obtained by adding a noise term in the dynamical equations of the fluid models. In this proposed project we intend to study the stochastic shell models of turbulence. For stochastic shell models with a Gaussian noise, the problem has been extensively studied. Therefore, we will mainly assume that the noise term is represented by random perturbation with discontinuous paths. Our objective is to study the long-time behavior of the stochastic shell models. For this purpose we need to establish some results related to the existence and uniqueness of the solution of our models. After that we will investigate the existence of invariant measure. We also want to study the uniqueness of an ergodic invariant measure and the rate of convergence towards this invariant measure. This is a very challenging task and it has never been done before for the stochastic shell models driven by Lévy noise. The problems we want to address are out of reach of the current state of the art. Therefore, to tackle these problems we have to elaborate new and sophisticated tools. It follows that the proposed project will potentially have a great impact on the development of the theory of stochastic partial differential equations. We also hope that our project will shed some light on the turbulence in hydrodynamics.

To model homogeneous incompressible fluids one usually uses the Navier-Stokes equations. In most practical situations the numerical investigation of the three dimensional Navier- Stokes equations at high Reynolds number is ubiquitous. However, it is well-known that it is difficult to compute analytically or via direct numerical simulations these kinds fluids. To overcome this problem, many efforts have been made to construct models which can exhibit the physical properties of the three dimensional Navier-Stokes equations at high Reynolds number. These are the so called models of turbulence and they are designed in such a way that they can capture the statistical properties of Navier-Stokes equations at a lower computability cost. Two examples are the so called Sabra shell and GOY models whichare simpler than the Navier-Stokes equations and are very promising for the investigation of turbulence in hydrodynamics. For the mathematical study towards the understanding of turbulence in hydrodynamics mathematicians use very often stochastic partial differential equations. These stochastic equations are usually obtained by adding a noise term in the dynamical equations of the fluid models. In this proposed project we studied the stochastic shell models of turbulence. For stochastic shell models with a Gaussian noise, the problem has been extensively studied. Therefore, we mainly assumed that the noise term is represented by random perturbation with discontinuous paths. Our objective was to study the long-time behavior of the stochastic shell models. For this purpose we established some results related to the existence and uniqueness of the solution of our models. After that we investigated the existence of invariant measure. We also studied the uniqueness of an ergodic invariant measure and the rate of convergence towards this invariant measure. Finally, we designed numerical methods to approximate the solution of the stochastic models. This last investigation will enable in the near future to perform simulation to confirm and validate our theoretical results. These are challenging tasks and they have never been done before for the stochastic shell models driven by Lévy noise. The problems we addressed were out of reach of the current state of the art. Therefore, to tackle these problems we elaborated new tools. It follows that the proposed project will potentially have a great impact on the development of the theory of stochastic partial differential equations. We also hope that our project will shed some light on the turbulence in hydrodynamics.