Maximum entropy production as a necessary admissibility condition for the fluid Navier-Stokes and Euler equations.

In a particle physics dynamics, we assume a uniform distribution as the physical measure and a measure-theoretic definition of entropy on the velocity configuration space. The dynamics is governed by an assumption of a Lagrangian formulation, with the velocity time derivatives as the momenta conjugate to the velocity configurations. From these definitions and assumptions, we show mathematically that a maximum entropy production principle selects the physical measure as a solution of the fluid Navier-Stokes and Euler equations from among alternate measures on the configuration space. The physical solution is shown to be a solution of the Navier-Stokes or Euler equations in the Lagrangian frame, but its transformation to an Eulerian frame is not established, and depends on the regularity of the solution, a property which is known to be difficult to establish.