# Stochastic Least-Action Principle for the Incompressible Navier-Stokes Equation - Mathematical Physics

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Abstract: We formulate a stochastic least-action principle for solutions of theincompressible Navier-Stokes equation, which formally reduces to Hamilton-sprinciple for the incompressible Euler solutions in the case of zero viscosity.We use this principle to give a new derivation of a stochastic Kelvin Theoremfor the Navier-Stokes equation, recently established by Constantin and Iyer,which shows that this stochastic conservation law arises fromparticle-relabelling symmetry of the action. We discuss issues ofirreversibility, energy dissipation, and the inviscid limit of Navier-Stokessolutions in the framework of the stochastic variational principle. Inparticular, we discuss the connection of the stochastic Kelvin Theorem with ourprevious -martingale hypothesis- for fluid circulations in turbulent solutionsof the incompressible Euler equations.

Author: ** Gregory L. Eyink**

Source: https://arxiv.org/