Approximation of the pressure term in the Stochastic Navier Stokes Equation

Stochastic Partial Differential Equations have become one of the most popular tools in understanding and investigating mathematically the hydrodynamic turbulence. To model turbulent fluids, mathematicians often use the stochastic Navier-Stokes equation obtained from adding a noise term in the dynamical equations of the fluids. Since the pioneering work of Bensoussan and Temam on stochastic Navier-Stokes equations with Wiener noise and related problems has been the object of intense analytical investigations which have generated several important results. However, the numerical study of these equations is really at its infancy and most of the results so far are about numerical approximation of the velocity field only. In this project we address the approximation of the pressure term in the stochastic Navier-Stokes equations. For this purpose we mainly use the projection method. The Projection method, initiated by Chorin and Temam, consists in splitting the original equation in two sub-equations which have lower complexity and are easier to solve numerically. At a first step we will first study numerically the pressure term for the linear stochastic Stokes equations with divergence free noise. For the second step of our investigation we want to extend our result obtained in the first step to extend to the fully stochastic Navier-Stokes equations by adding the nonlinearity. With the help of our results obtained in our first goal, our objective is to verify the convergence rate for the pressure term. However, due to the incompressibility constraint, the non-Lipschitz nonlinearity and the stochastic forcing we expect that we will encounter some problems. Finally, if time permits, one can investigate other methods, and verify if there is a possibility of improving the order of rate of convergence. Also, the question of non--divergence free noise will be addressed.

Stochastic Partial Differential Equations have become one of the most popular tools in under- standing and investigating the hydrodynamic turbulence mathematically. To model turbulent fluids, mathematicians often use the stochastic Navier-Stokes equation obtained from adding a noise term in the dynamical equations of the fluids. Since the pioneering work of Bensoussan and Temam on stochastic Navier-Stokes equations with Wiener noise and related problems has been the object of intense analytical investigations which have generated several significant results. However, the numerical study of these equations is really at its infancy, and most of the results so far are about numerical approximation of the velocity field only. On one side, to tackle the problem numerically one can search for similar systems having the same feature as the stochastic Navier Stokes, this is done in the first part of the project. In the second part of this project, we address the approximation of the pressure term in the stochastic Navier-Stokes equations. For this purpose, we mainly use the projection method. The Projection method, initiated by Chorin and Temam, consists in splitting the original equation into two sub-equations which have lower complexity and are easier to solve numerically. At a first step, we will first study the pressure term for the linear stochastic Stokes equations with divergence-free noise numerically. For the second step of our investigation, we want to extend our result obtained in the first step to extend to the fully stochastic Navier- Stokes equations by adding the nonlinearity. With the help of our results obtained in our first goal, our objective is to verify the convergence rate for the pressure term. However, due to the incompressibility constraint, the non-Lipschitz nonlinearity and the stochastic forcing we expect that we will encounter some problems. Finally, if time permits, one can investigate other methods, and verify if there is a possibility of improving the order of rate of convergence. Also, the question of nondivergence free noise will be addressed. Therefore, to tackle these problems we have to elaborate new and sophisticated tools. It follows that the proposed project will potentially have a high 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.