Remarks concerning modified Navier-Stokes equations

Abstract

We discuss some historical background concerning a modified version of the Navier-Stokes equations for the motion of an incompressible fluid. The classical (Newtonian) linear relation between the Cauchy stress tensor and the rate of strain tensor yields the Navier-Stokes equations. Certain nonlinear relations are also consistent with basic physical principals and result in equations with "stronger" dissipation. We describe a class of models that has its genesis in Kolmogorov's similarity hypothesis for 3-dimensional isotropic turbulence and was formulated by Smagorinsky in the meteorological context of rapidly rotating fluids and more generally by Ladyzhenskaya. These models also describe the motion of fluids with shear dependent viscosities and have received considerable attention. We present a dyadic model for such modified Navier-Stokes equations. This model is an example of a hierarchical shell model. Following the treatment of a (non-physically motivated) linear hyper-dissipative model given by Katz-Pavlović, we prove for the dyadic model a bound for the Hausdorff dimension of the singular set at the first time of blow up. The result interpolates between the results of solvability for sufficiently strong dissipation of Ladyzhenskaya, (later strengthened by Nečas et al) and the bound for the dimension of the singular set for the Navier-Stokes equations proved by Caffarelli, Kohn and Nirenberg. We discuss the implications of this dyadic model for the modified Navier-Stokes equation themselves.