A probabilistic representation of solutions of the incompressible Navier-Stokes equations in \(\mathbb R^3\). (English) Zbl 1077.35107
Summary: A new probabilistic representation is presented for solutions of the incompressible Navier-Stokes equations in \(\mathbb R^3\) with given forcing and initial velocity. This representation expresses solutions as scaled conditional expectations of functionals of a Markov process indexed by the nodes of a binary tree. It gives existence and uniqueness of weak solutions for all time under relatively simple conditions on the forcing and initial data. These conditions involve comparison of the forcing and initial data with majorizing kernels.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35C15 | Integral representations of solutions to PDEs |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 60J85 | Applications of branching processes |
| 60G50 | Sums of independent random variables; random walks |
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