By using the spectral Galerkin method, we prove a result on global existence in time of strong solutions for the motion of magneto-micropolar fluid without assuming that the external forces decay with time. We also derive uniform in time estimates of the solution that are useful for obtaining error bounds for the approximate solutions

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Key words asymmetric fluid, strong solution, interactive approach, convergence rates MSC (2000) 35Q35, 65M15, 76D03In this work we present a new proof of the existence and uniqueness of strong solutions for the equations of a viscous asymmetric fluids. We use an interactive approach and we prove that the approximate solutions constructed by using this method converge to a strong solution of these equations. We also give convergence-rate bounds for this method.

-We give sufficient conditions on the kinematics pressure in order to obtain regularity and uniqueness of the weak solutions to the micropolar fluid equations.

We study an optimal boundary control problem for the two-dimensional stationary micropolar fluids system with variable density. We control the system by considering boundary controls, for the velocity vector and angular velocity of rotation of particles, on parts of the boundary of the flow domain. On the remaining part of the boundary, we consider mixed boundary conditions for the vector velocity (Dirichlet and Navier conditions) and Dirichlet boundary conditions for the angular velocity. We analyze the existence of a weak solution obtaining the fluid density as a scalar function of the stream function. We prove the existence of an optimal solution and, by using the Lagrange multipliers theorem, we state first-order optimality conditions. We also derive, through a penalty method, some optimality conditions satisfied by the optimal controls.

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SUMMARYWe study the existence and uniqueness of strong solutions for the equations of non-homogeneous asymmetric uids. We use an iterative approach and we prove that the approximate solutions constructed by this method converge to the strong solution of these equations. We also give bounds for the rate of convergence.