Dyadic models for fluid equations: a survey

Abstract

Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years, including the groundbreaking work of Ladyzhenskaya on the Navier-Stokes equations. However crucial questions such as the existence, uniqueness and regularity of the three dimensional Navier-Stokes equations remain open. Partly because of this mathematical challenge and partly motivated by the phenomena of turbulence, insights into the full PDEs have been sought via the study of simpler approximating systems that retain some of the original nonlinear features. One such simpler system is an infinite dimensional coupled set of nonlinear ordinary differential equations referred to a dyadic model. In this survey we provide a brief overview of dyadic models and describe recent results. In particular, we discuss results for certain dyadic models in the context of existence, uniqueness and regularity of solutions.

arXiv comment31 pages

References in corpus (18)

• Dissipative continuous Euler flows
• An optimal shell model of turbulence
• Shell Models of Magnetohydrodynamic Turbulence
• Analytic Study of Shell Models of Turbulence
• Sharp nonuniqueness for the Navier-Stokes equations
• Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system
• The vanishing viscosity limit for a dyadic model
• Smooth solutions for the dyadic model
• Hidden scale invariance of intermittent turbulence in a shell model
• Simple Systems with Anomalous Dissipation and Energy Cascade
• Uniqueness for an inviscid stochastic dyadic model on a tree
• Anomalous dissipation in a stochastic inviscid dyadic model
• Invariant Gibbs measures of the energy for shell models of turbulence; the inviscid and viscous cases
• Shell model intermittency is the hidden self-similarity
• Solvable intermittent shell model of turbulence
• A Blow-Up Result for Dyadic Models of the Euler Equations
• Stochastic inviscid shell models: well-posedness and anomalous dissipation
• Dyadic models for ideal MHD

Cited by in corpus (1)

• RG approach to the inviscid limit for shell models of turbulence