Variational Principle and Stochastic Lagrangian Formulation of Viscous Hydrodynamic Equations

TL;DR

This paper extends Lagrangian formulations to viscous hydrodynamic models, deriving them from stochastic variational principles and establishing well-posedness results for these fluid equations.

Contribution

It generalizes Constantin-Iyer's Lagrangian approach to a broader class of viscous models using stochastic Hamilton-Pontryagin principles.

Findings

Derived a stochastic variational formulation for viscous fluids.

Proved local well-posedness of generalized viscous fluid models.

Discussed a generalized Kelvin circulation theorem for viscous flows.

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.