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.
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.
We show the existence of strong solutions for the nonhomogeneous Navier-Stokes equations in three-dimensional domains with boundary uniformly of class C 3 . Under suitable assumptions, uniqueness is also proved.
Communicated by W. SprößigThis paper is devoted to the study of the Cauchy problem for a semilinear heat equation with nonlinear term presenting a nonlinear source centered in a closed region of the spatial domain . We assume that R n is either a smooth bounded domain or the whole space R n , n 2. The initial data u 0 is assumed to belong to the Lebesgue space L r . /.
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