ABSTRACT
This work examines some decay properties of solutions to the Cauchy problem for the magneto-micropolar fluid equations on (where
and
real). We show, for each
, that
as
for Leray global solutions with an arbitrary initial data in
. When the vortex viscosity is present, we obtain a (faster) decay for the micro-rotational field:
. Some related results are also included.
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Acknowledgements
We would like to express our gratitude to Prof. P. Zingano for his interest in this work and various helpful suggestions. We also thank the Brazilian agencies CAPES and CNPq for their financial support to this work.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1. For example, we actually have with
, and so on. All interpolation inequalities we use have constants
, and so we simply take
throughout, except where indicated otherwise.
2. Because a monotonic function has to satisfy
as
(see e.g. [Citation20, p.236]).
3. However, after we show (Equation3a(3a)
(3a) ) for
it can be done just repeating the same steps to obtain (Equation12
(12)
(12) ).
4. Observe that, if , for
, then
, for each
and