Stochastic Partial Differential Equations in Turbulence Related Problems

Abstract

The theory of stochastic differential equations has become popular because of its importance in applications. An attempt has been made to bridge the distance between divergent ways of thinking that are reflected in theory and practice. This chapter presents the relevant results in probabilistic analysis, especially Gaussian measures in function spaces and the theory of stochastic partial differential equations (PDEs) of Itô type. The problems in turbulence provide the major source of stochastic PDEs. The chapter discusses the analysis of linearized Navier-Stokes equations with a random forcing. It further discusses stochastic equations for waves in random media. It presents certain model equations in turbulent transport theory. Some Markovian model equations in turbulence are treated from the viewpoint of stochastic differential equations. The problems of wave propagation in turbulent media are governed by hyperbolic systems with random coefficients.