Time Periodic Solutions of a Burgers'' Type Equation

在一維的開區間上，我們考慮一個帶有黏滯項與時間週期的外力的Burgers''型方程式，且方程式滿足狄利克雷邊界條件。我們將利用伽遼金方法與Schaefer不動點定理，證明時間週期解的存在性； 並證明在某種程度夠小的時間週期外力下，此時間週期解是唯一的，且滿足$H^1$漸進穩定。We consider the Burgers'' type equation with a viscosity term and a time periodic force defined on a one dimensional open interval domain with Dirichlet boundary condition. We prove the existence of the time periodic solution with some regular time periodic force by using Galerkin method and Schaefer''s fixed point theorem, and moreover, we show that this time periodic solution is unique and globally asymptotically stable in $H^1$ sense under additional condition with small force in some sense.Table of Contents 口試委員會審定書 i 誌謝 ii 中文摘要 iv 英文摘要 v 1 Introduction 1 2 Preliminary 3 2.1 The Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Fundamental Functional Space . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 Space With Time Evolution . . . . . . . . . . . . . . . . . . . . 5 2.3 Mathematics Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Approximate Solution and Prior Estimate 12 3.1 Galrekin Method and L2 Estimate . . . . . . . . . . . . . . . . . . . . . 12 3.2 H1 and High Order Estimate . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Existence, Uniqueness and Stability of Time Periodic Solution 25 4.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 H1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 References 3