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Probability & incompressible Navier-Stokes equations: an overview of some recent developments. (English) Zbl 1189.76424

Summary: This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by Y. Le Jan and A. S. Sznitman [Probab. Theory Relat. Fields 109, No. 3, 343–366 (1997; Zbl 0888.60072)] and extended by R. N. Bhattacharya, L. Chen, S. Dobson, R. B. Guenther, C. Orum, M. Ossiander, E. Thomann and E. C. Waymire [Trans. Am. Math. Soc. 355, No. 12, 5003–5040 (2003; Zbl 1031.35115)]. In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration.

MSC:

76M35 Stochastic analysis applied to problems in fluid mechanics
60H30 Applications of stochastic analysis (to PDEs, etc.)
35Q30 Navier-Stokes equations
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
76D05 Navier-Stokes equations for incompressible viscous fluids
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
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