Onsager’s conjecture states that the conservation of energy may fail for
three-dimensional incompressible Euler flows with Hölder regularity below
. This
conjecture was recently solved by the author, yet the endpoint case remains an
interesting open question with further connections to turbulence theory. In this work,
we construct energy nonconserving solutions to the three-dimensional incompressible
Euler equations with space-time Hölder regularity converging to the critical
exponent at small spatial scales and containing the entire range of exponents
.
Our construction improves the author’s previous result towards the endpoint case.
To obtain this improvement, we introduce a new method for optimizing the regularity
that can be achieved by a convex integration scheme. A crucial point is to avoid loss
of powers in frequency in the estimates of the iteration. This goal is achieved using
localization techniques of Isett and Oh (Arch. Ration. Mech. Anal. 221:2 (2016),
725–804) to modify the convex integration scheme.
We also prove results on general solutions at the critical regularity
that may not conserve energy. These include a theorem on intermittency
stating roughly that energy dissipating solutions cannot have absolute
structure functions satisfying the Kolmogorov–Obukhov scaling for any
if
their singular supports have space-time Lebesgue measure zero.
