Applications of a Cole–Hopf transform to the 3D Navier–Stokes equations

ABSTRACT

The Navier–Stokes equations written in the vector potential can be recast as non-linear Schrödinger equations at imaginary times, i.e. heat equations with a potential term, using the Cole–Hopf transform. On this basis, we study two kinds of Navier–Stokes flows by means of direct numerical simulations. In an experiment on vortex reconnection, it is found that the potential term takes large negative values in regions where intensive reconnection takes place, whereas the signature of the non-linear term is more broadly spread. For decaying turbulence starting from a random initial condition, such a correspondence is also observed in the early stage when the flow is dominated by vorticity layers. At later times, when the flow features several tubular vortices, this correspondence becomes weaker. Finally, a similar set of transformations is presented for the magneto–hydrodynamic equations, which reduces them to a set of heat equations with suitable potential terms, thereby obtaining new criteria for the regularity of their solutions.

KEYWORDS:

• Navier–Stokes equations
• Cole–Hopf transform
• Feynman–Kac formula
• Duhamel principle

View correction statement:

Corrigendum

Acknowledgments

This work has been supported by an EPSRC grant EP/N022548/1, including a research associate position for Riccardo Vanon.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Engineering and Physical Sciences Research Council [grant number EP/N022548/1].