Application of Onsager's Variational Principle to the Navier-Stokes Equations

Hamid A. Said

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arXiv · 2509.24033

Application of Onsager's Variational Principle to the Navier-Stokes Equations

Hamid A. Said

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Abstract

In this note we propose a basic $L^2$-based approach for studying the global and local energy equalities of the incompressible 3D Navier-Stokes equations in the standard energy class on $\mathbb{T}^3 \times (0,T]$. Motivated by L. Onsager's principle of least dissipation of energy (1931), we give a new sufficient condition for the energy equalities in terms of the limit of a sequence of minimizers. In particular, we show that the equalities are attained if this limit is non-vanishing. We observe that the indeterminacy in the vanishing case is reminiscent of turbulence-driven energy transfer dominating other transport processes.

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Hamid A. Said. 2025-09-28. Application of Onsager's Variational Principle to the Navier-Stokes Equations. https://arxiv.org/abs/2509.24033

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