Geodesic
Passive scalar equation in a turbulent incompressible Gaussian velocity field
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 2, pp. 297-312

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The time evolution of a passive scalar in a turbulent homogeneous incompressible Gaussian flow is considered. The turbulent nature of the flow results in non-smooth coefficients of the corresponding evolution equation. A strong solution (in the probabilistic sense) of the equation is constructed by using the Wiener Chaos expansion, and properties of the solution are studied. In particular, a certain $L_p$-regularity of the solution and a representation formula of Feynman–Kac type (or a Lagrangian formula) are among the results obtained. The results can be applied to both viscous and conservative flows. 
@article{RM_2004_59_2_a5,
     author = {S. V. Lototskii and B. L. Rozovskii},
     title = {Passive scalar equation in a turbulent incompressible {Gaussian} velocity field},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {297--312},
     publisher = {mathdoc},
     volume = {59},
     number = {2},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2004_59_2_a5/}
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S. V. Lototskii; B. L. Rozovskii. Passive scalar equation in a turbulent incompressible Gaussian velocity field. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 2, pp. 297-312. http://geodesic.mathdoc.fr/item/RM_2004_59_2_a5/