EXTINCTION OF WEAK SOLUTIONS OF DOUBLY NONLINEAR NAVIER-STOKES EQUATIONS

In this article we discus the doubly nonlinear incompressible Navier-Stokes equations partial derivative b(u)/partial derivative t + div(b(u) circle times u) = -d pi + div(a(del(sym)u) + f, div(u) = 0, 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 (del(sym)u) := (1 + vertical bar del(sym)u vertical bar(p-2)del(sym)u and b (u) := vertical bar u vertical bar(m-2)u with 1 < p < m < infinity.