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

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.