Summary.
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]×ℝd,(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]×ℝd.
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Received: 27 March 1995 / In revised form: 15 May 1997
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Deck, T., Potthoff, J. On a class of stochastic partial differential equations related to turbulent transport. Probab Theory Relat Fields 111, 101–122 (1998). https://doi.org/10.1007/s004400050163
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DOI: https://doi.org/10.1007/s004400050163
- Mathematics Subject Classification (1991): 60G20
- 60H15
- 60H99