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arXiv · 2606.12825

Optimal minimax formula for bounds on ensemble averages of statistically stationary three-dimensional Navier-Stokes flows

Anne Bronzi·Cecília F. Mondaini·Ricardo M. S. Rosa
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Abstract

We establish an optimal upper bound formula for ensemble averages of flow quantities associated with the three-dimensional incompressible Navier-Stokes equations. The formula takes the form of a minimax problem, extending the framework developed by Tobasco, Goluskin, and Doering (2018) for finite-dimensional systems and by Rosa and Temam (2022) for the two-dimensional Navier-Stokes equations. The lack of global well-posedness for the 3D case presents a significant challenge, which we overcome by working within the space of Foias-Prodi stationary statistical solutions. The minimax formula is derived by exploiting suitable compactness and continuity properties of specific subspaces of probability measures under the weak topology of the phase space. A distinguishing feature of our result is the characterization of the maximizing measures: unlike the previous cases, the optimal bounds in 3D are achieved on extreme points that are specific convex combinations of at most two Dirac delta measures, instead of exactly one, a structure that naturally appears from the constraints given by the mean energy dissipation inequalities in the characterization of the Foias-Prodi stationary statistical solutions.

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Anne Bronzi, Cecília F. Mondaini, Ricardo M. S. Rosa. 2026-06-11. Optimal minimax formula for bounds on ensemble averages of statistically stationary three-dimensional Navier-Stokes flows. https://arxiv.org/abs/2606.12825

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