Variational and Topological Methods: Theory, Applications, 
Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 223-234.

Title: Extinction of weak solutions of doubly nonlinear Navier-Stokes equations

Author: Jochen Merker (Univ. Rostock,  Germany)
 
Abstract: 
  In this article doubly nonlinear incompressible Navier-Stokes equations
  $$
  \frac{\partial b(u)}{\partial t} + \hbox{div}(b(u) \otimes u)
  = - d\pi + \hbox{div}(a(\nabla^{\rm sym} u)) + f \,, \quad
  \hbox{div}(u) = 0 
  $$
  are discussed, where u models the velocity vector field
  of a homogeneous incompressible non-Newtonian fluid
  whose momentum $b(u)$ depends nonlinearly on u.
  Particularly, under certain regularity assumptions it is shown
  that u becomes extinct in finite time
  for sufficiently small initial values u(0),
  if $a(\nabla^{\rm sym} u) := (1 + |\nabla^{\rm sym} u|^{p-2})
  \nabla^{\rm sym} u$ and $b(u) := |u|^{m-2} u$ with $1 < p < m < \infty$.

Published February 10, 2014.
Math Subject Classifications: 58F15, 58F17, 53C35.
Key Words: Navier-Stokes equations; doubly nonlinear evolution equations;
           extinction.