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Concerning to the non-stationary Navier-Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier-Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339-367, 2005), in L p spaces for p ≥ 3. In this article, we first extend their result to the case 3 2 < p by modifying the method in Bae and Jin (J Math Fluid Mech 10: [423][424][425][426][427][428][429][430][431][432][433] 2008) that was used to obtain weighted estimates for the Navier-Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier-Stokes flow with a nonzero velocity at infinity.Mathematics Subject Classification (2010). Primary 35Q30; Secondary 76D05.

We present a new coupled kinetic-fluid model for the interactions between Cucker-Smale (C-S) flocking particles and incompressible fluid on the periodic spatial domain T d. Our coupled system consists of the kinetic C-S equation and the incompressible Navier-Stokes equations, and these two systems are coupled through the drag force. For the proposed model, we provide a global existence of weak solutions and a priori time-asymptotic exponential flocking estimates for any smooth flow, when the kinematic viscosity of the fluid is sufficiently large. The velocity of individual C-S particles and fluid velocity tend to the averaged time-dependent particle velocities exponentially fast.

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We show that the time decay rate of L 2 norm of weak solution for the Stokes equations and for the Navier-Stokes equations on the half spaces are t − nWe also show that the decay rate is determined by the linear part of the weak solution. We use the heat kernel and Ukai's solution formula for the Stokes equations. It has been known up to now that the decay rate on the half space was t − n 2 ( 1 r − 1 2 ) , which was obtained by Borchers and Miyakawa [1] and Ukai [9].