A Proof of Onsager’s Conjecture for the Incompressible Euler Equations

A Proof of Onsager’s Conjecture for the Incompressible Euler Equations

by Philip Isett

Formal & Physical Sciences Mathematics PBKCalculus and mathematical analysisPBKJDifferential calculus and equations
🎙 Philip Isett 👥 79K 📅 January 14, 2026 ⏱ 72 min 👁 2K 📄 research talk 🧭 2026-08-02
Available in: English (current) Français

Keywords

Onsager's conjectureEuler equationsweak solutionsenergy dissipationHölder regularity

Summary

Philip Isett presents a proof of Onsager’s conjecture for the incompressible Euler equations. The conjecture, proposed by Lars Onsager in 1949, states that weak solutions with Hölder regularity greater than 1/3 conserve energy, while solutions with regularity less than or equal to 1/3 may dissipate energy. Isett begins by deriving the Euler equations from physical principles and introduces weak solutions. He explains the motivation from turbulence theory, particularly Kolmogorov’s theory and the concept of energy cascade. He then surveys previous results: energy conservation above 1/3 was proven by Constantin et al., while constructions of dissipative solutions were given by Scheffer and Shnirelman, and later by De Lellis and Székelyhidi using convex integration. Isett’s own thesis improved the regularity to 1/5, and the talk focuses on the ideas leading to the full proof of the conjecture, achieving (1/3 - epsilon)-Hölder regularity. He discusses the key challenges, such as the transport and oscillation error terms, and the need to abandon perturbative methods. The proof uses a convex integration scheme with intermittent building blocks. The talk concludes with a discussion of the result’s implications and open questions.

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Critical Evaluation

The talk is a masterful presentation of a landmark result in mathematical fluid dynamics. Isett’s exposition is clear and well-structured, guiding the audience from the physical origins of the Euler equations to the technical intricacies of the proof. The mathematical content is rigorous, and the speaker is careful to distinguish between proven results and conjectures. The proof of Onsager’s conjecture is a major achievement, and this talk provides an excellent overview of the ideas involved. The speaker’s use of invariant index notation is consistent and aids in clarity. The talk is aimed at a specialist audience, but the speaker does an admirable job of making the material accessible. The sources cited are appropriate and include the key papers in the field. The talk does not include any promotional content. The title accurately reflects the content. The main strength of the talk is its depth and the speaker’s ability to convey the essence of a complex proof. The only minor weakness is that the talk is a survey and does not provide all the technical details of the proof, but that is expected given the time constraints. Overall, this is an outstanding presentation of a groundbreaking result.

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Title / Content Match

The title accurately reflects the content: the talk presents a proof of Onsager's conjecture for the incompressible Euler equations.

Quality & Reliability

9/10

Talk by a leading expert (Caltech) presenting a proof of a major conjecture, with clear mathematical reasoning and references to prior work. The content is technical and appears rigorous, though the presentation is a survey rather than a full proof.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

  • Carmin.tv — Video platform hosting the talk and other scientific videos.

Concurring Sources

  • Onsager's conjecture - Wikipedia — Background on the conjecture and its status.

Contribution & Novelties

This talk presents the first full proof of Onsager’s conjecture, a major open problem in mathematical fluid dynamics. The proof introduces novel techniques in convex integration, including the use of intermittent building blocks and a careful balance of error terms. The result has significant implications for the understanding of turbulent dissipation and weak solutions of the Euler equations.

Pour aller plus loin :

  • Onsager’s conjecture - Wikipedia — Provides background and context.
  • Convex integration - Wikipedia — Key technique used in the proof.
  • Incompressible Euler equations - Wikipedia — Basic equations and properties.

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Radar Profile

The radar profile shows very high scores in all dimensions, indicating a talk that is rich in information, technically deep, and highly reliable. The balance between quantity and quality is excellent, with a slight emphasis on technical level.

Reliability 9/10