Stochastic Lagrangian formulations for damped Navier-Stokes equations and Boussinesq system, with applications

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Abstract

We obtain stochastic Lagrangian formulations of solutions to some partial differential equations in fluid mechanics with diffusion, specifically damped Navier-Stokes equations, as well as the viscous and thermally diffusive Boussinesq system. As a byproduct of our discussion, we deduce stochastic Lagrangian formulations for other models, namely viscous and forced Burgers' equation, micropolar and magneto-micropolar fluid systems with zero vortex viscosity while positive and possibly distinct kinematic and angular viscosities, Bénard problem, as well as Leray-α magnetohydrodynamics model. Kelvin's circulation theorem is extended for the damped Navier- Stokes equations and the viscous and thermally diffusive Boussinesq system. The Cauchy formula for vorticity is extended from the damped Euler equations to the damped Navier-Stokes equations. The global well-posedness of the three-dimensional Euler equations with damping is proven for small initial data in critical Besov space. Finally, the global well-posedness of the four-dimensional Navier-Stokes equations with partial damping in only third and fourth components of the velocity field is also proven.

Original languageEnglish
Pages (from-to)447-471
Number of pages25
JournalCommunications on Stochastic Analysis
Volume12
Issue number4
DOIs
StatePublished - 2018

Keywords

  • Brownian motion
  • Characteristics
  • Feynman-Kac formula
  • Kelvin's circulation theorem
  • Lagrangian formulation

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