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. This distribution is labeled as the physical solution in the remainder of the article. The dynamics are 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 from among alternate solutions of the Navier–Stokes and Euler equations, but its transformation to an Eulerian frame is not established here, a topic that will be considered separately.