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Session S37 - New Developments in Mathematical Fluid Dynamics

Friday, July 16, 12:00 ~ 12:25 UTC-3

Optimal minimax bounds for the Navier-Stokes equations and other infinite dimensional dissipative systems

Ricardo Rosa

Universidade Federal do Rio de Janeiro, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Obtaining sharp estimates for quantities involved in a given model is an integral part of a modeling process. For dynamical systems whose orbits display a complicated, perhaps chaotic, behaviour, the aim is usually to estimate time or ensemble averages of given quantities. This is the case, for instance, in turbulent flows. In this talk, the aim is to present a minimax optimization formula that yields optimal bounds for time and ensemble averages of dissipative infinite-dimensional systems, including the two- and three-dimensional Navier-Stokes equations. The results presented here are extensions to the infinite-dimensional setting of a recent result on the finite-dimensional case given by Tobasco, Goluskin, and Doering in 2017. The optimal result occurs in the form of a minimax optimization problem and does not require knowledge of the solutions, only the law of the system. The minimax optimization problem appears in the form of a maximization over a portion of the phase space of the system and a minimization over a family of auxiliary functions made of cylindrical test functionals defined on the phase space. The function to be optimized is the desired quantity plus the duality product between the law of the system and the derivative of the auxiliary function.

Joint work with Roger Temam (Indiana University).

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