# uriel FRISCH - Laboratoire Lagrange, Observatoire Cote d'Azur - Keywords: Turbulence, Simulations, Computability

Contribution title: Bridging the mathematician's and the physicist's current vision of turbulence

On October 2-27, 2017 at the Simons Center for Geometry and Physics (Stony Brook, Long Island, USA) took place a Workshop and Program "Geometrical and statistical fluid dynamics" www.oca.eu/etc7/scgp.html,  organized by Uriel Frisch (UCA), Konstantin Khanin (U.-Toronto) and Rahul Pandit (IISc-Bangalore). The present project, which should interest scientists from a number of UCA laboratories, is based on the conclusions of the Workshop and Program. All presentations were video-recorded and are available from the above URL. Below, key references are given by enclosing the last name of the speaker in square brackets.

An important evolution took place during the Workshop, concerning turbulence in the limit of infinite Reynolds number. At the beginning, the focus of the theorem-proving mathematicians was on constructing weak (distributional) energy-dissipating solutions to the Euler equations for nonviscous, incompressible, three-dimensional flow, whose velocity increments over a distance $r$ vary roughly as $r^{1/3}$, more precisely, having a Hoelder exponent arbitrarily close to the value 1/3 predicted by Onsager [Buckmaster]. These are essentially equivalent to the self-similar solutions predicted by Kolmogorov in 1941 (often called K41). As to the theoretical and numerical fluid dynamicists, their focus was on constructing non-self-similar (fractal/multifractal) solutions with anomalous scaling, in which the moments of order $p$ of the (longitudinal) velocity increments over a distance $r$ varies as $r^{\zeta_p}$, where the scaling exponent $\zeta_p$ differs from its K41 value $p/3$ and is some convex, possibly universal, function [Glimm]. Thanks to the numerous discussion sessions (not video-recorded), a conjecture supported by both communities emerged during the final discussion. The construction of K41 weak solutions is done by an iterative process involving a sequence of suitably chosen increasingly slender jets, called mikados. It should be possible to modify this construction in such a way as to obtain intermittent turbulent solutions with arbitrary (convex) scaling functions $\zeta_p$.

During the Program days after the Workshop, there were preliminary discussions on how to do this mathematical construction of mikados and how to implement it numerically [Matsumoto]. It is conceivable that, under small perturbations, such arbitrary "multifractal" solutions will run away to ones with universal exponents that do not depend on the initial and boundary conditions.

Furthermore, the mikado-based construction has the property that, after any finite number of iterations, the flow is spatially smooth, so that one can determine its fluid particle trajectories
(characteristics); in other words one can study its Lagrangian structure and then study the limiting behavior when the iterative process is continued indefinitely. This will help understanding the
process of spontaneous stochasticity, closely related to the nonuniqueness of characteristics in a flow that lacks smoothness [Bec,Eyink,Mailybaev]. It may also shed light on an issue of
algorithmic complexity reported during the Program: It was proven that it is impossible to design a Navier-Stokes solver with arbitrarily specified solution quality (guaranteed bound on the error), in other words the equations are "unsimulable" [Smith].

Preliminary discussions were held on the important open issue of algorithmic decidability of the "blow-up", the spontaneous loss of smoothness after a finite time of initially smooth data for the
Euler and Navier-Stokes equations.

The investigations listed above will require a coordinated effort of a number of mathematicians, numericists, physicists and perhaps logicians and could bring us significantly closer to the solution of the turbulence problem.