Global existence of strong solutions to micropolar equations in cylindrical domains

Abstract

The micropolar equations are a useful generalization of the classical Navier-Stokes model for fluids with microstructure. We prove the existence of global and strong solutions to these equations in cylindrical domains in R 3 . We do not impose any restrictions on the magnitude of the initial and external data, but we require that they cannot change in the x 3 -direction too fast.

On global regular solutions to magnetohydrodynamics in axi-symmetric domains

“…The method we use does not lead to exponential decay of the external data (for similar ideas see e.g. [15][16][17][18]). …”

Long-time behavior of micropolar fluid equations in cylindrical domains

“…With the above notation the following Theorem was proved in [Now12a]. It is fundamental for further considerations.…”

“…In this article we present some remedy which is based on [CD98]. It takes into account that global and strong solutions exist if some smallness of L 2 -norms on the rate of change of the external and the initial data is assumed (see [Now12a]). This leads to a restriction of the uniform attractor to a proper phase space.…”

“…In other words the data need not to be small but small must be their rate of change. With the above notation the following Theorem was proved in [Now12a]. It is fundamental for further considerations.…”