Analogue of the Cole-Hopf transform for the incompressible Navier–Stokes equations and its application

ABSTRACT

We consider the Navier–Stokes equations written in the stream function in two dimensions and vector potentials in three dimensions, which are critical dependent variables. On this basis, we introduce an analogue of the Cole-Hopf transform, which exactly reduces the Navier–Stokes equations to the heat equations with a potential term (i.e. the nonlinear Schrödinger equation at imaginary times). The following results are obtained. (i) A regularity criterion immediately obtains as the boundedness of condition for the potential term when the equations are recast in a path-integral form by the Feynman-Kac formula. (ii) This in turn gives an additional characterisation of possible singularities for the Navier–Stokes equations. (iii) Some numerical results for the two-dimensional Navier–Stokes equations are presented to demonstrate how the potential term captures near-singular structures. Finally, we extend this formulation to higher dimensions, where the regularity issues are markedly open.

KEYWORDS:

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

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. For an inviscid fluid, the well-known Madelung transform connects the nonlinear Schrödinger equation (at real times) with the Euler equations with the quantum pressure term. See [6] and references cited therein.

2. Another approach is to write the vorticity and ‘tensor-potential’ as skew-symmetric matrices and consider matrix logarithms. It appears that this does not work, because of non-commutative nature of matrix multiplications.

Additional information

Funding

This work was supported by Engineering and Physical Sciences Research Council [grant number EP/N022548/1].