Inertial Manifolds And Its Applications To Dissipative Partial Differential Equations

With the development of the theory of infinite dimensional dynamical systems,many dissipative dynamical systems that generated by mathematical and physical equations present some finite dimensional properties.Thus,a series of extensive researches on the finite dimensional reduction of infinite dimensional dynamical systems have been initiated.The classical inertial manifold theory shows that if a partial differential equation(PDE)has an N-dimensional inertial manifold,the long-time behavior of the solutions of this PDE can be reduced to a system of ordinary differential equations(ODEs)of order N.This essentially simplifies the understanding of the dynamics of the original PDE.At present,the study of inertial manifolds is still one of the most important and challenging problems in infinite dimensional dynamical systems.In this paper,we study the inertial manifolds and its applications to dissipative partial differential equations.Firstly,we investigate the critical modified-Leray-? model in T3 and prove the exis-tence of an N-dimensional inertial manifold for this problem.The interesting thing about this problem is that it is a "double critical" problem in terms of well-posedness and inertial manifolds.On the other hand,since the turbulence exists in the problem,the existence of the inertial manifolds of this problem may have positive inspiration significance for the study of inertial manifolds for the two-dimensional Navier-Stokes equations.Secondly,we give a comprehensive study of inertial manifolds for semilinear parabol-ic equations and their smoothness using the spatial averaging method suggested by J.Mallet-Paret and G.Sell.We propose/design a universal approach/framework which can deal with scalar and vector equations in a unified way and covers the most part of known results obtained via this method as well as gives a number of new ones.In addition,many previous results only obtain Lipschitz continuous inertial manifolds.In this paper,we have improved to C1+?-smoothness.Among our applications are reaction-diffusion equations,various types of generalized Cahn-Hilliard equations,including fractional and 6th order Cahn-Hilliard equations and several classes of modified Navier-Stokes equations including the Leray-? regularization,hyperviscous regularization and their combinations.The exis-tence of inertial manifolds of fractional order Cahn Hilliard equation and the combination of Leray-? regularization and hyperviscous regularization have not been obtained before the present paper.Finally,it is noted that the existing examples of inertial manifolds are only considered to be relatively "good" equations(at least without singularity),so the universality of inertial manifolds for non-autonomous models with singular terms needs to be verified.In Chapter 5,we prove the existence of an N-dimensional inertial manifold for a class of singularly non-autonomous parabolic equations(?) where A(t)? 0 for any t??,and ?(?)Rd is a bounded domain with smooth boundary.Since the operator A(t)may degenerate to zero at some time t,then the inverse of A(t)will be non-existent at these degenerated time.To verify the existence of inertial manifolds,we propose a special admissible class of A(t)and a compatibility condition between A(t)and the nonlinearity F,and extend the strong cone condition to the asymptotic strong cone condition.