Existence of self-similar homogeneous statistical solutions to Navier–Stokes

Establish the existence of self-similar homogeneous statistical solutions for the three-dimensional incompressible Navier–Stokes equations, i.e., a family of time-parameterized, spatially homogeneous probability measures that satisfy the equations in an averaged sense and capture Kolmogorov’s decay behavior for fully developed turbulence.

Background

In the Kolmogorov framework, self-similar homogeneous statistical solutions encode the decay of turbulence via energy spectra and dissipation rates. Foias and collaborators proposed such solutions to resolve paradoxes associated with stationary statistical solutions and finite energy dissipation.

The authors emphasize that, despite their foundational role in turbulence theory, the existence of these self-similar homogeneous statistical solutions remains unresolved, motivating the maximum entropy approach developed in the paper.