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Abstract:

I will give a brief overview of the development of the probabilistic method for the classical incompressible Navier-Stokes equations introduced by Le Jan and Sznitman in 1997 and further improved by Bhattacharya, Waymire, Thomann, Ossiander, Dascaliuc and others. The solution constructed by this method coincides with the mild solutions. The probabilistic construction gives rise to interesting problems in branching processes which can be applied to the existence and uniqueness of mild solutions. For example, the stochastic explosion infers the non-uniqueness of self-similar solutions of the "cheap" Navier-Stokes equation introduced by Montgomery-Smith. Another example is that the conservation of frequencies infers the global existence of mild solutions given any initial data in L^2 whose Fourier transform is supported in the half-space. Joint work with Radu Dascaliuc, Enrique Thomann, and Edward Waymire.

Stochastic Cascade, Symmetry And Comparison Method For The Navier-Stokes Equations