The von Kármán-Howarth equations are derived for three-dimensional (3D) Hall magnetohydrodynamics (MHD) in the case of an homogeneous and isotropic turbulence. From these equations, we derive exact scaling laws for the third-order correlation tensors. We show how these relations are compatible with previous heuristic and numerical results. These multi-scale laws provide a relevant tool to inves...

This section is devoted to the discussion of two fundamental (partial) differential equations, that arise in the context of Markov diffusion processes. After giving a brief introduction of continuous-time continuous state Markov processes, we introduce the forward and backward equation, and provide a heuristic derivation of these equations for diffusion processes. We also discuss some examples ...

Self-similar Euler singularities may be useful for understanding some aspects of Navier-Stokes turbulence. Here, a causal explanation for intermittency is given, based on the control of the sudden growth of the gradients by the Euler equations. This explanation uses certain Euler solutions as intermediate asymptotics in Navier-Stokes turbulence [1] – controlling the dynamics over a limited spat...

The study of anharmonic lattices can be considerably simplified by imposing a spatial periodicity condition on solutions. This reduces the infinite dimensional lattice equations to a finite dimensional system of ordinary differential equations for which we have at our disposal Birkhoff normal forms, invariant theory, singular reduction and the Kolmogorov Arnol’d Moser theorem. As an example we ...

The results of the direct numerical simulation of isotropic turbulence of surface gravity waves in the framework of Hamiltonian equations are presented. For the first time, the simultaneous formation of both direct and inverse cascades has been observed in the framework of the primordial dynamical equations. At the same time, a strong long wave background has been developed. It has been shown t...

The main object of the paper is to present a criterion for the minimal semigroup associated with the Kolmogorov differential equations to be stochastic in `1. Our criterion uses a weighted `1space. As an abstract preparation we present a perturbation theorem for substochastic semigroups which generalizes known results to the case of ordered Banach spaces which need not be AL-spaces We also cons...

This Resource Letter provides a guide to the literature on fully developed turbulence in fluids. It is restricted to mechanically driven turbulence in an incompressible fluid described by the Navier-Stokes equations of hydrodynamics, and places greatest emphasis on fundamental physical questions. Journal articles and books are cited for the following topics: The Navier-Stokes equations, qualita...