Variational Principle and Stochastic Lagrangian Formulation of Viscous Hydrodynamic Equations

TL;DR

This work develops a unifying stochastic variational framework for viscous fluid dynamics by extending the Constantin–Iyer stochastic Lagrangian representation to a broad class of viscous models. Central to the approach is a stochastic Hamilton–Pontryagin principle that yields a coupled stochastic Lagrangian–Eulerian system, a nonlinear Feynman–Kac representation, and generalized circulation theorems arising from particle relabeling symmetry. The authors illustrate the framework through multiple examples (generalized SQG, Boussinesq–MHD, Hall–MHD) and connect it to Brenier’s generalized flows, offering both pathwise and statistical Kelvin-type conservation laws. They also provide a self-contained local well-posedness theory via Picard iteration for a deterministic Boussinesq–MHD system, highlighting the method’s robustness and potential for broader viscous models in fluid dynamics.

Abstract

In this manuscript, we extend Constantin-Iyer's Lagrangian formulation of Navier-Stokes Equation to a wider class of hydrodynamic models. Moreover, we prove that such Lagrangian formulation is naturally derived from a stochastic Hamilton-Pontryagin type variational principle. Generalized version of Kelvin circulation theorem in viscous fluids is also discussed. We also derive self-contained local well-posedness results of fluid models based on Lagrangian-Eulerian formulation using fixed point argument.