[Paper Review] Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers

Ana Bela Cruzeiro|arXiv (Cornell University)|Jul 4, 2018

Nonlinear Waves and Solitons13 references3 citations

TL;DR

This paper derives the deterministic and stochastic Navier-Stokes equations via a stochastic variational principle where pressure acts as a Lagrange multiplier enforcing incompressibility. By modeling fluid particle trajectories as stochastic diffusions with drift $ v $, the Navier-Stokes equation emerges as the critical point of an action functional involving kinetic energy and a constraint term, with constants of motion identified through a stochastic Noether-type theorem.

ABSTRACT

We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.

Motivation & Objective

• To establish a stochastic variational principle for the deterministic Navier-Stokes equation on the d-dimensional torus.
• To interpret the viscous term $ \nu\Delta v $ as arising from underlying stochastic diffusion processes rather than ad hoc dissipation.
• To derive a corresponding stochastic Navier-Stokes equation as a critical point of a random action functional.
• To identify conserved quantities in the stochastic setting using a generalized Noether theorem with space-averaged symmetries.
• To unify geometric fluid dynamics with stochastic processes by extending Arnold's and Ebin-Marsden's approaches to dissipative systems.

Proposed method

• Formulates the Navier-Stokes equation as the critical point of an action functional $ S(g,p) = S^1(g,p) + S^2(g,p) $, where $ g_t(x) $ is a stochastic diffusion process with drift $ v $ and diffusion coefficient $ \sqrt{2\nu} $.
• Introduces pressure $ p $ as a Lagrange multiplier enforcing the incompressibility constraint $ \det \nabla g_t(x) = 1 $ via the term $ \int p(t,g_t(x))(\det \nabla g_t(x) - 1) \, dtdx $.
• Applies variations to the action functional with respect to both the path $ g_t $ and the pressure field $ p $, leading to the Euler-Lagrange equations that yield the Navier-Stokes PDE.
• Uses the generalized derivative $ D_t $ to define the drift of the stochastic process, with $ D_t g_t(x) = v(t,g_t(x)) $, and applies Itô's formula to compute $ D_t v(t,g_t(x)) $.
• Derives a stochastic Navier-Stokes equation by considering a random action functional $ \tilde{S}(\xi,p) $ that includes stochastic integrals with respect to a Wiener process $ W_t $, leading to a Stratonovich-type SDE.
• Applies a stochastic Noether theorem by integrating symmetry variations over space, showing that conserved quantities are martingales under the action of the stochastic differential operator $ \mathcal{L}_t = \partial_t + (v\cdot\nabla) + \nu\Delta $.

Experimental results

Research questions

• RQ1 Can the deterministic Navier-Stokes equation be derived from a stochastic variational principle with pressure as a Lagrange multiplier?
• RQ2 How does the viscous term $ \nu\Delta v $ emerge from a stochastic diffusion process rather than being imposed phenomenologically?
• RQ3 What is the form of the stochastic Navier-Stokes equation derived from a random action functional involving Itô and Stratonovich integrals?
• RQ4 How can conserved quantities be identified in the stochastic setting, and how do they differ from classical Noether invariants?
• RQ5 What is the role of space-averaged symmetries in deriving conserved quantities for stochastic fluid flows?

Key findings

• The Navier-Stokes equation is derived as the critical point of a stochastic action functional where pressure acts as a Lagrange multiplier enforcing incompressibility of the stochastic flow $ g_t(x) $.
• The viscous term $ \nu\Delta v $ arises naturally from the Itô correction in the stochastic diffusion process $ dg_t(x) = \sqrt{2\nu} dW_t + v(t,g_t(x)) dt $, without assuming dissipation a priori.
• A stochastic Navier-Stokes equation is derived as $ dv + (v\cdot\nabla)v \, dt = \sqrt{2\nu} \nabla v \circ dW_t \, dt - \nabla p \, dt $, with divergence-free $ v $, in Stratonovich form.
• Conserved quantities are identified via a stochastic Noether theorem: if the Lagrangian is invariant under a symmetry $ \eta $, then $ \int \mathcal{L}_t (v\eta - G) \, dx = 0 $, where $ \mathcal{L}_t $ is the stochastic differential operator.
• The result generalizes classical Noether’s theorem by integrating over space, showing that the conserved quantity is a martingale under the stochastic evolution.
• The method extends to general Riemannian manifolds via the construction in [2], suggesting broader geometric applicability beyond the flat torus.