| We study the long-time behavior of certain dissipative nonlinear partial differential equations as expressed by their global attractor, in particular when these equations are weakly dissipative or, more generally, when their semigroup is not compact. We study the existence and regularity of the global attractor for a forced Korteweg-deVries equation with weak damping and for the 2D Navier-Stokes equation on some unbounded spatial domains. In both cases, we use appropriate energy equations in order to cope with the lack of compactness of the solution operator.;In the second topic of this dissertation, we study the existence and regularity of inertial manifolds for general abstract semilinear equations under the so-called spectral gap condition and prove the normal hyperbolicity of such inertial manifolds, which is associated with their persistence under perturbations. Finally, the third topic of this work is devoted to constructing a family of approximate inertial manifolds converging exponentially fast (i.e., as an exponentially small function of the distance) to an exact inertial manifold in a form more suitable for numerical implementations. |
