On a class of stochastic partial differential equations related to turbulent transport

Abstract

We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C 1,2([0,T]×ℝ d ,(S)*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field, i.e. ρ(t,x) is an L 2-random variable for all time and space parameters (t,x)∈[0,T]×ℝ d .