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October 17, 2025

Notes on Navier–Stokes Regularity and Energy Conservation

Key Points

The research establishes regularity criteria involving the strain matrix's eigenvalue in Besov space, enhancing our understanding of flow behavior.
It highlights two energy conservation criteria for weak solutions, emphasizing the control over the integrability of the strain tensor.
This marks the first result of energy conservation criteria linked to strain tensor integrability in three-dimensional flow dynamics.
These findings potentially pave the way for new approaches in fluid mechanics and mathematical fluid dynamics.

Abstract

ABSTRACT This paper first proves the regularity criteria of the weak solution for the 3D Navier–Stokes equations involving the positive part of the intermediate eigenvalue of the strain matrix in Besov space. And then, it shows two energy conservation criterion of very weak solution via control on integrability of strain tensor. To our best knowledge, for three‐dimensional flow, it is the first result of energy conservation criterion by control on the integrability of strain tensor in the framework.

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Notes on Navier–Stokes Regularity and Energy Conservation

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