Lax representations for the Navier--Stokes equations
Abstract
We construct a family of Lax representations for the three-dimensional incompressible Navier--Stokes equations in vorticity form. The key idea is to introduce an auxiliary vector field, termed the \emph{vorticity-diffusion potential}, which recasts the viscous Laplacian diffusion term as a Lie-bracket advection term. On solutions of the defining equation for this potential, the viscous vorticity equation takes the form of a pure Lie-transport equation along a modified divergence-free velocity field, while the defining constraint itself is encoded as the curl-free compatibility condition for an auxiliary scalar potential. We further show that the reduction of this three-dimensional system to two spatial dimensions yields a Lax representation for the two-dimensional Navier--Stokes vorticity equation, which we generalize to a family parametrized by an arbitrary smooth function of one variable.
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BibTeX
@misc{morozov2026representations,
title = {{Lax representations for the Navier--Stokes equations}},
author = {Oleg I. Morozov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26668},
eprint = {2610.26668},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26668},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lax representations for the Navier--Stokes equations
AU - Oleg I. Morozov
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26668
UR - https://arxiv.org/abs/2610.26668
AB - We construct a family of Lax representations for the three-dimensional incompressible Navier--Stokes equations in vorticity form. The key idea is to introduce an auxiliary vector field, termed the \emph{vorticity-diffusion potential}, which recasts the viscous Laplacian diffusion term as a Lie-bracket advection term. On solutions of the defining equation for this potential, the viscous vorticity equation takes the form of a pure Lie-transport equation along a modified divergence-free velocity field, while the defining constraint itself is encoded as the curl-free compatibility condition for an auxiliary scalar potential. We further show that the reduction of this three-dimensional system to two spatial dimensions yields a Lax representation for the two-dimensional Navier--Stokes vorticity equation, which we generalize to a family parametrized by an arbitrary smooth function of one variable.
ER -