Scaling laws and exact results in turbulence

Abstract: In this note, we address the validity of certain exact results from turbulence theory in the deterministic setting. The main tools, inspired by the work of Duchon-Robert (Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13(249), 2000) and Eyink (Local 4/5-law and energy dissipation anomaly in turbulence, Nonlinearity, 16(137), 2003), are a number of energy balance identities for weak solutions of the incompressible Euler and Navier-Stokes equations. As a consequence, we show that certain weak solutions of the Euler and Navier-Stokes equations satisfy deterministic versions of Kolmogorov's 4/5, 4/3, 4/15 laws. We apply these computations to improve a recent result of Hofmanova et al. (Kolmogorov 4/5 law for the forced 3D Navier-Stokes equations, arXiv:2304.14470), which shows that a construction of solutions of forced Navier-Stokes due to Bru`e et al. (Onsager critical solutions of the forced Navier-Stokes equations, arXiv:2212.08413) and exhibiting a form of anomalous dissipation satisfies asymptotic versions of Kolmogorov's laws. In addition, we show that the globally dissipative 3D Euler flows recently constructed by Giri, Kwon, and the author (The $L^3$-based strong Onsager theorem, arXiv:2305.18509) satisfy the local versions of Kolmogorov's laws.