Mathematical Sciences: Partial Differential Equations and Harmonic Analysis for the Sublaplacians

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Abstract Lu The principal investigator intends to continue to develop the theory of structural stability for infinite dimensional dynamical systems originating in science and engineering, which are generated by, for example, parabolic equations and to study the related problems such as Floquet theory for parabolic equations, to develop the theory of the persistence of normally hyperbolic invariant manifolds, the existence of stable and unstable manifolds of the normally hyperbolic invariant manifolds, and the theory of invariant foliations for infinite dimensional dynamical systems generated by partial differential equations. However, the infinite dimensional dynamical systems generated by, for example, parabolic equations are not reversible and the phase spaces are not locally compact. This characteristic creates difficulties not encountered in the study of finite dimensional dynamical systems and new methods need to be developed to understand the nature of these systems. A theory of structural stability for scalar reaction-diffusion equations, the Cahn-Hilliard Equation and Phase-Field System has recently been obtained. Techniques found by the PI in his previous works will be employed in the current studies. It is expected that this work will contribute to a better understanding of the dynamics of the models of physical systems described by partial differential equations. Mathematical models for the state of an evolving physical system are the subject of investigation of dynamical systems. The main goal of the study of dynamical systems is to understand the long term behavior of states in the systems. In applications, the mathematical models (say differential equations) approximately describe physical reality. To understand the qualitative properties of a physical system, one needs to investigate not only the mathematical model but also the perturbations of the model. One also needs to study how the qualitative properties of the perturbed models are related to the qualitative properties of the original model. This is especially important to the numerical computations for the models. Because of round off error and numerical schemes, the model studied by the numerical computations actually is a perturbation of the original model.The theory for dynamical systems has been considered by many mathematicians and scientists starting with Poincare, Liapunov, and Birkhoff. One of the fundamental problems in the theory of dynamical systems is the structural stability of dynamical systems. For a structurally stable system, the qualitative properties are preserved under small perturbations of the system. To understand the dynamics of a system, one needs to investigate the existence of invariant sets, in particular, such as equilibrium points, periodic orbits, invariant tori, and attractors, to study their structures and to know what happens in their vicinity (do the nearby solutions approach the invariant set, or stay nearby, or leave the neighborhood). A fundamental problem is to study the persistence of invariant manifolds and to study the qualitative properties of the flow nearby invariant manifolds. The theory of invariant manifolds and invariant foliations has become a fundamental tool for the study of dynamical systems.