I: SUFFICIENT CONDITIONS FOR LOCAL SCALING LAWS IN 3-D TURBULENCE II: WELL-POSEDNESS FOR NONLINEAR STOCHASTIC KINETIC EQUATIONS

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## Abstract

Incompressible fluids at high Reynolds number quickly transition to turbulence. This implies that precise predictions for many flows in the physical world are extremely difficult - if not outright impossible. The field of statistical fluid mechanics aims to mitigate this difficulty by studying averaged quantities associated with turbulent flows.

In the mathematical physics literature, turbulent flow is often described in the language of stochastic partial differential equations. The appropriate models are stochastic perturbations of well-known deterministic equations of fluid mechanics, so that the apparent randomness of turbulent flow is modeled via the tools of stochastic analysis.

In the first part of this dissertation, this point of view of stochastic fluid mechanics is employed. We focus on the three dimensional case, with the goal of obtaining a conditional theorem for Kolmogorov's celebrated 4/5 law to hold in the presence of boundaries. The dimensionality enforces the use of a weak notion of solution to our model, in particular we work with families of \say{stationary martingale solutions} to the stochastic Navier-Stokes equations parametrized by the inverse Reynolds number. The main result of the first part of this dissertation provides a sufficient condition for a local version of the 4/5 law in the limit of infinite Reynolds number.

In the second part of the dissertation, the focus is shifted to kinetic theory. Kinetic equations have played a prominent role in statistical mechanics since the 19th century; typically, the kinetic viewpoint represents an intermediate step of coarse-graining between the particle level, governed by Newtonian or Hamiltonian mechanics, and the hydrodynamic level, governed by continuum or fluid mechanics.

The solution of a kinetic equation represents the normalized phase-space density of a large number of particles which might be interacting and potentially diffusing. The evolution of the density of an ensemble of particles interacting electrostatically is modeled by the Vlasov-Poisson equation. Thermal noise on the particles is modeled by the inclusion of a kinetic Fokker-Planck term.

To incorporate the effect of macroscopic fluctuating force fields into kinetic modeling, we perturb the Vlasov-Poisson-Fokker-Planck equation by a stochastic kinetic transport term. We modify and exploit a popular scheme of stochastic fluid mechanics relying on the Gy"ongy-Krylov lemma and construct local strong solutions to the stochastic Vlasov-Poisson-Fokker-Planck equation.