We analyze the well-posedness of the initial value problem for the generalized micropolar fluid system in a space of tempered distributions and also prove the existence of the stationary solutions. The asymptotic stability of solutions is showed in this space, and as a consequence, a criterium for vanishing small perturbations of initial data (stationary solution) at large time is obtained. A fast decay of the solutions is obtained when we assume more regularity on the initial data.

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We analyse the well-posedness of the initial value problem for a convection problem. Mild solutions are obtained in the weak-L p (R n ) spaces and the existence of self-similar solutions is shown, while the only small self-similar solution in the Lebesgue space L p (R n ) is the null solution. The asymptotic stability of solutions is analysed and, as a consequence, a criterion of selfsimilarity persistence at large times is obtained.

We consider the micropolar fluid system in a bounded domain of R 3 and prove the existence and the uniqueness of a global strong solution with initial data being a perturbation of the stationary solution, whose existence is also obtained. We prove that these solutions converge uniformly to the stationary solutions with exponential decay rate. The technique of our analysis is the semigroups approach in L p −spaces.

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