Remarks concerning modified NavierStokes equations
Susan Friedlander  Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 606077045, United States (email) Abstract: We discuss some historical background concerning a modified version of the NavierStokes 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 NavierStokes 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 3dimensional 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 NavierStokes equations. This model is an example of a hierarchical shell model. Following the treatment of a (nonphysically motivated) linear hyperdissipative model given by KatzPavlović, 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 NavierStokes equations proved by Caffarelli, Kohn and Nirenberg. We discuss the implications of this dyadic model for the modified NavierStokes equation themselves.
Keywords: Modiﬁed NavierStokes equations, dyadic model.
Received: January 2002; Revised: August 2002; Available Online: October 2003. 
2014 IF (1 year).972
