A Hamiltonian Interacting Particle System for Compressible Flow

Abstract

The decomposition of the energy of a compressible fluid parcel into slow (deterministic) and fast (stochastic) components is interpreted as a stochastic Hamiltonian interacting particle system (HIPS). It is shown that the McKean–Vlasov equation associated to the mean field limit yields the barotropic Navier–Stokes equation with density-dependent viscosity. Capillary forces can also be treated by this approach. Due to the Hamiltonian structure, the mean field system satisfies a Kelvin circulation theorem along stochastic Lagrangian paths.

Probabilistic representation of helicity in viscous fluids

“…The stochastic Hamiltonian approach that is presented in this section has been developed in [9,10,11]. This approach is a Hamiltonian analogy to the mean field Weber formula theory of Constantin and Iyer [4], and it is also related to Holm's variational principle for stochastic fluid mechanics ( [12]).…”

“…This note studies helicity in the viscous case from the point of view of stochastic Hamiltonian interacting particle systems (SHIPS) as in [9,10,11]. These systems can be viewed as a stochastic perturbation (along Hamiltonian vector fields) of ideal fluid mechanics.…”

“…t ] satisfies the Navier-Stokes equation (Theorem 2.4). The IPS approach in [9,10,11] is a Hamiltonian analogue of the Constantin and Iyer [4] representation of solutions to the Navier-Stokes equation via the stochastic Weber formula. In fact, (1.5) is equivalent to [10] via the stochastic Noether theorem (cf.…”

Probabilistic representation of helicity in viscous fluids

“…This note studies helicity in the viscous case from the point of view of stochastic Hamiltonian interacting particle systems (SHIPS) as in [10][11][12]. These systems can be viewed as a stochastic perturbation (along Hamiltonian vector fields) of ideal fluid mechanics.…”

“…t ] satisfies the Navier-Stokes equation (Theorem 2.4). The IPS approach in [10][11][12] is a Hamiltonian analogue of the Constantin and Iyer [16] representation of solutions to the Navier-Stokes equation via the stochastic Weber formula. In fact, Equation (1.5) is equivalent to [11] via the stochastic Noether theorem (cf.…”