Is there a Lagrangian $L$ (equivalently an action functional $S$) which yields the Navier-Stokes equation?

Question

The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws.

However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\overrightarrow{v}]$ of velocity vector fields $\overrightarrow{v}$ which yield the NS / Euler equations as the equation of motion. By the equation of motion, I mean the Euler-Lagrange equation.

Also, is it possible to realize the incompressibility condition $\nabla \cdot \overrightarrow{v}=0$ as a constraint by means of some Lagrange multiplier as well?

Could anyone please provide relevant reference, or the form of such action $S$?

This post imported from StackExchange Physics at 2024-06-18 14:50 (UTC), posted by SE-user Keith

Comments

Related: physics.stackexchange.com/q/14652/2451

This post imported from StackExchange Physics at 2024-06-18 14:50 (UTC), posted by SE-user Qmechanic

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One reference to look at is arxiv.org/abs/0810.0817 See also Chapter 13 of "Variational Principles In Dynamics And Quantum Theory" and references therein. Also see the authors Yourgrau and Mandelstam. It appears that dissipative systems of equations don't seem to be well represented by a variation of an action.

This post imported from StackExchange Physics at 2024-06-18 14:50 (UTC), posted by SE-user John Bussoletti