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  • 5 - Quasi-invariance for the Navier–Stokes equations

  • Edited by Charles L. Fefferman, Princeton University, New Jersey, James C. Robinson, University of Warwick, José L. Rodrigo, University of Warwick
  • Book:
    Partial Differential Equations in Fluid Mechanics
    Published online:
    15 August 2019
    Print publication:
    27 September 2018, pp 97-112

  • Summary

    In this contribution we focus on a few results regarding the study of the three-dimensional Navier-Stokes equations with the use of vector potentials. These dependent variables are critical in the sense that they are scale invariant. By surveying recent results utilising criticality of various norms, we emphasise the advantages of working with scale-invariant variables. The Navier-Stokes equations, which are invariant under static scaling transforms, are not invariant under dynamic scaling transforms. Using the vector potential, we introduce scale invariance in a weaker form, that is, invariance under dynamic scaling modulo a martingale (Maruyama-Girsanov density) when the equations are cast into Wiener path-integrals. We discuss the implications of this quasi-invariance for the basic issues of the Navier-Stokes equations.