Mathematical analysis of stability ofviscous incompressible flows in several unbounded domains

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The motion of a viscous incompressible fluids can be understood via analysis of the Navier-Stokes equation. The aim of this research is to study stability of solutions to this equation in the Mowing two important unbounded domains: aperture domain and exterior domain of a rotating body. First, the aperture domain is a compact perturbation of two separated half-spaces, which are connected by an aperture. It is remarkable that, even for the Stokes equation, many solutions may exist subject to usual boundary conditions. In this research, when a flux through the aperture is prescribed, we prove the existence of a unique global solution and deduce its large time behavior provided that the data are small Secondly, concerning the exterior problem, the case where the body is at rest or translating has been discussed in many papers, while the rotating case has been less studied because, in a reference frame, a drift term with unbounded coefficient appears and causes a lot of difficulties. The present research clarifies some structure of the reduced equation in the reference frame and, thereby, the most crucial step is overcome so that the desired theorem is finally obtained. In short, we derive both time decay of the semigroup generated by the linear part and spatial decay of a small steady flow and, by use of them, we prove the asymptotic stability of the steady flow. Moreover, we derive some definite decay rates of the disturbance in various Lebesgue spaces. The decay of the semigroup is described as L_p-L_q estimate, while the decay of the steady flow is understood in terms of Lorentz spaces, especially, weak-L_q spaces.