Scaling laws and exact results in turbulence

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.