Research On The Long-time Behavior Of Solutions To Some Equations Of Fluid Mechanics

Navier-Stokes equations are a simplified system,which models the viscoelas-tic incompressible fluid,and reflects the mechanics law as a typical nonlinear system.They are applied in many fields extensively,and many fluid systems are all considered as a coupled equations of Navier-Stokes equations with oth-er equations.The research on well-posedness of 3D Navier-Stokes equations has been one of hot topics,the corresponding theory on attractors plays an impor-tant role in studying turbulence and has great academic value,which presents important references for many industries,such as weather forecast,material,ship and airplane design and so on.In this dissertation,we study the existence of attractors and upper estimate on fractal dimension to several classes of dynami-cal systems,including 2D Navier-Stokes-Voight equations with distributed delay,3D Kelvin-Voight-Brinkman-Forchheimer equations with continuous delay and 3D Navier-Stokes equations with rapidly increasing damping,and we also derive some interesting results,which are stated as follows:(1)The existence of global attractors of 2D Navier-Stokes-Voight equations with distributed delay on the Lipschitz domain is studied.Under some assump-tions on initial datum and the distributed delay ?-h0 G(s,u(t+s))ds,we construc-t the background function such that the model is reduced into a homogeneous system,use the Faedo-Galerkin method,compact theorem,the Sobolev inequal-ity and the Hardy inequality,and obtain the well-posedness of system.Using the decomposition method,we show that the semigroup {S(t)}is asymptotically compact,and the global attractor exists.(2)The existence of global attractors and upper estimate on fractal dimen-sion for 2D Navier-Stokes-Voight equations with distributed delay on the smooth domain Q are studied.After the system is reduced into a homogeneous model,the use of the Faedo-Galerkin approximation method,compactness theorem and the Gagliardo-Nirenberg inequality leads to the well-posedness of system.By the decomposition method of {s(t)} and the basic theory of attractors,we obtain the global attractor A in Xv.Through solving the first variation equation of system,we show that {S(t)} is uniformly differentiable on,A.We extend the generator of evolution system in XV and use the Lieb-Thirring inequality to estimate the fractal dimension of A.(3)The existence of pullback-D attractors of 3D Kelvin-Voight-Brinkman-For-chheimer equations with continuous delay is discussed.Under some assump-tions on delay datum,we use the standard Faedo-Galerkin method,compactness theorem and the Gronwall inequality to obtain the well-posedness of solution.Using the energy method and the decomposition method,we derive the existence of pullback-D attractors.(4)The upper semicontinuity of attractors for 3D Navier-Stokes equations with rapidly increasing damping ?|u|?-1u and non-autonomous perturbation is studied.With some assumptions on the external force,we show that the pullback attractors A?(t)and the global attractor A of system with ?=0 have the property of upper semicontinuity.