The Existences Of Global Strong Solution And Of Attractor For Navier-Stokes Equations In 3D Thin Domains With Navier-friction Boundary Conditions

In this master thesis, we consider the solutions of Navier-Stokes equations on thin 3D domains of the formΩ_ε=Ω×(0,εg(x₁,x₂)) = (0,l₁)×(0, l₂)×(0,εg(x₁,x₂)), where the top boundary is non-flat. We consider the following boundary conditions:" periodic boundary conditions on the lateral boundary and general Navier-friction boundary conditions on the top and the bottom boundary". Under the assumption that‖u₀‖H¹â‰¤Cε^-1/2,‖M_εu₀‖L²â‰¤C andsimilar assumptions on the forcing term, we show global existence of strongsolutions, here M_εu₀=(Mu₀¹,Mu₀²,0),Mu₀ⁱ=1/εg∫₀^εgu₀ⁱ(x₁,x₂,x₃)dx₃,i=1,2.Under the assumption that f is translation compact in L^∞(0,∞；L(Ω_ε)³),byusing the method in [24], we show the existence of (local) attractor of strong solutions for 3D Navier-Stokes equations.