Energy conservation at Onsager’s critical Besov regularity

Establish whether any weak solution of the incompressible Euler equations on the periodic domain that satisfies u ∈ L∞(0,T; L2(𝕋d)) and u ∈ Lp(0,T; Bp,∞1/3(𝕋d)) for some p ≥ 3 conserves kinetic energy, i.e., has vanishing Duchon–Robert energy defect [u] = 0 over (0,T).

Background

The paper reviews Onsager’s program relating anomalous dissipation to the regularity threshold of one-third of a derivative. The positive side (energy conservation above 1/3, e.g. Constantin–E–Titi) and negative side (energy dissipation for regularity strictly below 1/3, e.g. Isett and subsequent works) are established.

The marginal case at exactly Bp,∞1/3 remains unresolved. The author formulates the precise question of energy conservation for weak Euler solutions with L∞tL2x energy and LptBp,∞1/3 spatial regularity, emphasizing its significance for physically relevant turbulent flows and noting that, until this case is settled, Onsager’s conjecture remains incomplete.

References

However, an important question about the excluded middle remains open (and therefore, so does "Onsager's conjecture"): Suppose $u\in L\infty(0,T; L2(\mathbb{T}d)) \cap Lp(0,T; B_{p,\infty}{1/3}(\mathbb{T}d))$ for any $p\geq 3$ is a weak solution of Euler. Does it conserve energy?

— Mathematical Theorems on Turbulence  (2601.09619 - Drivas, 14 Jan 2026) in Section “Onsager’s ideal turbulence,” Question (label consques)