pdes fluid-dynamics turbulence convex-integration
Weak solutions of the incompressible Euler equations with Hölder regularity above exponent \(1/3\) conserve kinetic energy, while for every exponent below \(1/3\) there exist weak solutions that dissipate energy. (Resolved 2018: both halves proved.)
Onsager announced the conjecture in 1949 on the basis of Kolmogorov's 1941 turbulence phenomenology: below a critical roughness, the energy cascade should be able to dissipate energy without viscosity. The conservation half (exponent above \(1/3\)) was proved by Constantin–E–Titi (1994). The dissipation half resisted for two decades until De Lellis–Székelyhidi imported convex integration into fluid mechanics; successive constructions (De Lellis–Székelyhidi, Buckmaster, Buckmaster–De Lellis–Székelyhidi–Vicol) pushed the dissipative exponent up to \(1/3\), and Isett (2018) closed the conjecture. The same machinery then attacked Navier–Stokes uniqueness itself (Buckmaster–Vicol 2019, Albritton–Brué–Colombo 2022).
Constantin–E–Titi (1994) proved energy conservation for Euler solutions smoother than Hölder \(1/3\). Isett (2018) constructed dissipative Hölder-continuous solutions for every exponent below \(1/3\) via convex integration, completing the conjecture; the De Lellis–Székelyhidi program and Buckmaster–De Lellis–Székelyhidi–Vicol supplied the intermediate exponents.
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