### The 2023 Project on Toy Models of Fluid Equations for Energy Cascades in Fluid Flow

This is a brief introduction to the 2023 Polymath Jr project that will be run by Vincent Martinez.

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As complex and seemingly chaotic a turbulent fluid may look, it is a remarkable fact that such flows possess statistical structure in the form of power laws. The presence of this universality is possible due to existence of both a source and sink of energy separated by a wide-range of length scales and facilitated by a mechanism that transfers energy across this range of length scales. This mechanism is nonlinear. On the other hand, the equations that model a fluid equation are infinite-dimensional. In the projects below, we propose a few scenarios and settings to help understand the mechanism of energy transfer across scale in a systematic way.

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1) Toy models for energy cascade

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The PDE that models an incompressible fluid with constant density is the incompressible Navier-Stokes equations. From these equations, one can formally develop an identity that displays the source of energy and the dissipation of energy that are in balance. One can further decompose this balance into the energy confined to each constituent length scale, across all length scales. It is well-known that although the energy across individual scales are all coupled to each other, the majority of the transfer of energy between scales is localized. By reducing the study of the NSE to the evolution of its energy and subsequently truncating interactions between scales to be local, one arrives at a general class of toy models referred to as the “Shell Models of Turbulence,” which are coupled systems of ordinary differential equations. In this set of projects, we describe a methodology for deriving shell models that will lead to a systematic development of shell models and will subsequently study the models that result from this approach.

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2) Finite-dimensional Steady States

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Different from the toy models, but related in the sense that it involves a certain reduction of the original equation of fluid motion is the problem find non-trivial steady states of a particular form, namely, having compact support in frequency. Interestingly, the problem of search for such steady states is related to finding non-trivial integer solutions of Diophantine equations. In this set of projects, we will identify these Diophantine equations and hope to identify as many situations as possible that lead only to trivial solutions, thereby narrowing the scope of possibilities. Of course there is no computable algorithm for finding solutions of Diophantine equations, but since these arise specifically from a particular PDE, such Diophantine equations may have interesting properties that general Diophantine equations do not possess.

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3) Generalized Inviscid Limits

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A third separate, but related direction is to study certain the long-time behavior of different dissipative perturbations of known shell models with the ultimate goal of understanding what types of behavior could arise in the zero-perturbation limit (generalization of “inviscid limit”) from considering different perturbations. The problem of studying the inviscid limit derives from the intimate connection of this singular limit to turbulent fluids. In particular, since the experiments of O. Reynolds in the 18th century in which a certain parameter, now referred to as the Reynolds number, was identified that effectively characterizes the transition of a fluid flow from a laminar regime to a turbulent one, it has been known that turbulence occurs when the Reynolds number is sufficiently large. This naturally leads one to ask what happens when Reynolds number is taken to be infinitely large? In terms of the equations of motion, this limit is formally refers to the limit of the Navier-Stokes equations to the Euler equations and is referred to as the inviscid limit, that is, the limit of zero viscosity. In this set of projects, we probe various generalizations of the inviscid limit by introducing other forms of dissipative mechanisms that replace the original viscous mechanism in order to develop an understanding this phenomenon in the toy model context of shell models.