On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations

Ulrik Skre Fjordholm; Siddhartha Mishra; Franziska Weber

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arXiv · 2110.04674

On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations

Ulrik Skre Fjordholm·Siddhartha Mishra·Franziska Weber

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Abstract

We study statistical solutions of the incompressible Navier-Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures, which are probability measures accounting for spatial correlations, and moment equations is equivalent to statistical solutions in the Foias-Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov's self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Karman-Howarth-Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures.

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Ulrik Skre Fjordholm, Siddhartha Mishra, Franziska Weber. 2021-10-10. On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations. https://arxiv.org/abs/2110.04674

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