Weak Solutions and Energy Conservation in Fluid Dynamics

Weak solutions represent generalised interpretations of the fundamental equations governing fluid motion, such as the Euler and Navier–Stokes systems, in which classical differentiability may fail. These solutions permit discontinuities or oscillations intrinsic to turbulent flows, enabling the mathematical description of phenomena beyond smooth regimes. Central to the modern theory is the dichotomy between conservation and dissipation of kinetic energy: while classical solutions strictly conserve total energy, weak solutions may exhibit anomalous dissipation, a feature intimately linked to the cascade of energy across scales in turbulent regimes. Pioneering work on the Onsager conjecture established a threshold of 1/3-Hölder regularity below which energy conservation can break down, inspiring a broad programme combining harmonic analysis, geometric measure theory and convex integration. Convex integration has since demonstrated the existence of infinitely many non-unique weak solutions, some preserving energy and others dissipating it, thereby illuminating the subtle interplay between regularity, admissibility criteria and physical realism. Progress in this field carries global significance, informing turbulence modelling in engineering design, predicting mixing in geophysical flows and constraining numerical schemes in climate science. Recent advances continue to bridge rigorous analysis with computational and experimental studies, offering concrete guidance on when solutions remain physically admissible and how anomalous dissipation emerges in complex flow configurations.

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Weak Solutions and Energy Conservation in Fluid Dynamics publication trend

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Technical terms

Weak solution: A function satisfying a differential equation in an integrated (distributional) sense, allowing limited regularity.

Convex integration: A constructive method that builds highly oscillatory weak solutions by piecing together simple local patterns.

Onsager conjecture: A principle predicting that energy conservation holds for solutions above a 1/3-Hölder regularity threshold and may fail below it.

Hölder continuity: A measure of function regularity characterised by bounded differences scaling as a fixed power of separation distance.

Besov space: A function space capturing joint information on smoothness, integrability and oscillatory behaviour across scales.

References

1. Anomalous Dissipation and Lack of Selection in the Obukhov–Corrsin Theory of Scalar Turbulence. Annals of PDE (2023).
2. Energy conservation for weak solutions of incompressible fluid equations: The Hölder case and connections with Onsager's conjecture. Journal of Differential Equations (2023).
3. Three results on the energy conservation for the 3D Euler equations. Nonlinear Differential Equations and Applications NoDEA (2024).