The Long-Time Behavior Of Solutions For A Nonclassical Navier-Stokes Equations

In this paper, using the recent theoretical results about the existence of global attractors and combining with some estimates of energy functional, we investigate the long-time behaviors of the solution for the following non-classical Navier-Stokes equations and where Ω is a bounded domain in Rn(n≥3) with smooth boundary (?)Ω. Equation (1) is differ-ent from the classical Navier-Stokes equations essentially. As with a term-△ut, the regularity of system (1) is not as high as Navier-Stokes equations, so we cannot prove the existence of attractor of the traditional method to solve the non classical Navier-Stokes equations, the equa-tion (2) is in equation (1) on the basis of the generalized derivative, because of the limitations of the generalized derivative, so we can’t and then as a test function and equation (2) as the inner product, therefore, part of the results derived in this paper is more difficult.This paper prepare to divide into five chapters:(Ⅰ) Introduction.In the first chapter, we introduce the background of dynamical systems and the nonclassical Navier-Stokes equations.(Ⅱ) Preliminary knowledgeIn the second chapter, some necessary preliminary concepts we will use is given.(Ⅲ) The existence and uniqueness of solutions of the non-classical Navier-Stokes equa-tions.In the third chapter, we mainly use the Galerkin method and the energy estimate method,the non classical Navier-Stokes equations and the existence and uniqueness of solutions depend continuously on the initial value was studied.(Ⅳ) The existence of attractors of the non-classical Navier-Stokes equations.In the forth chapter, we mainly discuss the long-time behaviors of the strong solution of the system (1), Firstly, By texting (C) condition, we get that the solution of the system (1) are w-limit compact in the space H01(Ω) and H01(Ω) n H2(Ω). Sequentially, we obtain the existence of global attractor of system (1) in the space H01(Ω) and H01(Ω)∩H2(Ω). Secondly, By texting Lipschitz continuity and squeezing property, we obtain the existence of the exponential attractors of system (1) in the space H01(Ω).(V) The existence of the global attractors of the non-classical Navier-Stokes equations with generalized derivative.In the fifth chapter, Firstly, we discuss the existence of the global attractors of the system (2) in the space H01(Ω), By texting (C) condition, using the Poincare inequality and energy inequality, the conclusion is obtained.Secondly, By texting Lipschitz continuity and squeezing property, we obtain the existence of the exponential attractors of system (2) in the space H01(Ω).