Isett–Oh optimal energy regularity conjecture for Hölder Euler solutions

Let $v$ be a weak solution of the incompressible Euler equations on $\mathbb R\times\mathbb T^n$ with spatial-temporal Hölder regularity $v \in C^\theta(\mathbb R\times\mathbb T^n)$, where $\theta<\frac{1}{3}$. Write $e_v(t)$ for its kinetic energy profile. The known estimate for $v\in L^\infty((0,T);C^\theta(\mathbb T^3))$ is

$$|e_v(t)-e_v(s)|\leq C|t-s|^{\frac{2\theta}{1-\theta}}.$$

Isett–Oh conjecture. For every $\theta<\frac{1}{3}$, there exists a weak Euler solution $v\in C^\theta(\mathbb R\times\mathbb T^n)$ such that

$$e_v\notin W^{\frac{2\theta}{1-\theta}+\varepsilon,p}(I)$$

for every $\varepsilon>0$, $p\geq1$, and every open interval $I\subset\mathbb R$. Moreover, the set of all such solutions is residual in the space of all $C^\theta(\mathbb R\times\mathbb T^n)$ weak solutions, equipped with the topology induced by the $C^\theta$ norm.

This conjecture asserts optimality of the Hölder exponent $\frac{2\theta}{1-\theta}$ for the energy profile below Onsager's threshold. The corresponding upper regularity estimate is known, as is energy conservation for $\theta>\frac13$; the sharp typicality and failure of every higher fractional Sobolev regularity remain the conjectural content.