The K41–Onsager conjecture for vanishing-viscosity Navier–Stokes flows

Let $I$ be a finite open interval, let $v_{\nu_j}$ be suitable weak solutions to the incompressible Navier–Stokes equations on $I \times \mathbb{T}^3$, and let $\nu_j>0$ be viscosity parameters tending to zero. The norm $L_t^\infty C_x^{1/3}$ is the space-time norm appearing in the statement, and $\varepsilon$ is a positive constant.

K41–Onsager conjecture for Navier–Stokes. There exists a finite open interval $I$ and a sequence of suitable weak solutions $v_{\nu_j}$ with $\nu_j\to0$ that is uniformly bounded in the $L_t^\infty C_x^{1/3}$ norm and whose mean rate of kinetic-energy dissipation on $I$ satisfies

$$\limsup_{j \to \infty} \frac{1}{|I|} \int_I \left[-\frac{d}{dt} \int_{\mathbb{T}^3} \frac{|v_{\nu_j}|^2}{2}(t,x)\,\operatorname{d}x\right]\operatorname{d}t \geq \varepsilon>0.$$

This formulation expresses persistence of a positive energy-dissipation rate in the inviscid limit despite critical Hölder control. It is presented as a motivation arising from the Kolmogorov–Onsager picture for turbulence; the supplied text does not state a resolution.