Variational Principle and Stochastic Lagrangian Formulation of Viscous Hydrodynamic Equations

2025/11/26 by Mazzucato, Anna, Pan, Anping
#35C99 #35Q35 #58E30 #60H30 #76D05 #76M30 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.21498

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.

Citations

• Euler-Lagrangian approach to stochastic Euler equations in Sobolev Spaces
• Geometry of generalized fluid flows
• Euler-Lagrangian approach to 3D stochastic Euler equations
• p-Euler equations and p-Navier-Stokes equations
• Stochastic variational principles for dissipative equations with advected quantities
• Generalized surface quasi-geostrophic equations with singular velocities
• A Stochastic Representation for Backward Incompressible Navier-Stokes Equations
• Stochastic Least-Action Principle for the Incompressible Navier-Stokes Equation
• Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms
• An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid Flow
• A paradigm for joined Hamiltonian and dissipative systems

Related

• Application of Stochastic Variational Method to Hydrodynamics
• Lagrangians for variational formulations of the Navier-Stokes equation
• Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers
• Stochastic Least-Action Principle for the Incompressible Navier-Stokes Equation
• Variational Principles of Lagrangian Averaged Fluid Dynamics