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

Extinction of weak solutions of doubly nonlinear Navier-Stokes equations

Jochen Merker

Abstract:
In this article we discus the 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 $1 < p < m < \infty$I.

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

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Jochen Merker
Universitäat Rostock - Institut für Mathematik
Ulmenstr. 69 (Haus 3)
18057 Rostock, Germany
email: jochen.merker@uni-rostock.de

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