Global existence and exponential stability for the micropolar fluid system

Abstract

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.

Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

“…Recently, Ferreira and Villamizar-Roa [5] considered the existence and stability of solutions to the micropolar fluids in exterior domains. Villamizar-Roa and Rodríguez-Bellido [22] studied the micropolar system in a bounded domain using the semigroup approach in L p , showing the global existence of strong solutions for small data and the asymptotic behavior and stability of the solutions.…”

A uniform error estimate in time for spectral Galerkin approximations of the magneto‐micropolar fluid equations

“…Yamaguchi [10] proved the existence theorem of global in time solution for small initial data. Later, Villamizar-Roa and Rodríguez-Bellido [11] by using semigroups theory in L p spaces, proved results concerning 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. Ortega-Torres and Rojas-Medar [12] gave sufficient conditions on the kinematics pressure to obtain regularity and uniqueness of the weak solutions for the micropolar fluid equations.…”

Exponential stability for magneto-micropolar fluids

“…More recently, in [23] the authors studied the existence and exponential stability in L p of stationary Navier-Stokes flows with prescribed flux in infinite cylindrical domains through analytic semigroup theory, perturbation theory and L p -L q estimates for a perturbation of the Stokes operator in L q -spaces. For the micropolar fluid case, in [30] the authors used semigroup theory to obtain results on L p -stability.…”

“…In [12], the authors use methods of Clifford analysis to write the system of asymmetric fluids in the hypercomplex formulation and to represent its solution in terms of Clifford operators. The evolution case was studied in [31] through a semigroup approach (see also [30]). Linearization and successive approximations have been considered in [4,19] to give sufficient conditions on the kinematics pressure in order to obtain regularity and uniqueness of the weak solutions to the micropolar fluid equations.…”