On the Cauchy Problem

Lecture I Evolution EquationsLecture II H∞-wellposednessAppendixLecture III Lax-Mizohata Theorem § 1 § 2 § 3 Proof of Theorem § 4 § 5 Further ConsiderationsAppendix § A.1 Preliminaries § A.2 Proof of (12) § A.3 Partition of Unity § A.4 Estimates of αn(D)b(x,D)χn ±(D) § A.5 Proof of (9), (10), (11) § A.6 § A.7Lecture IV Cauchy Problems in Gevrey Class § 1 Introduction and Results § 2 Fundamental Proposition § 3 Proof of Theorem 4 § 4 Gevrey Property in t of Solutions § 5 CommentsAppendix § A.1 Proof of Lemma 4 § A.2 Proof of Lemmas 1,2 and 6 § A.3 Proof Lemma 3Lecture V Micro-local Analysis in Gevrey Class (I) § 1 Introduction § 2 Definition of {αn(D),βn(X)} § 3 Criterion of WFs(u) by Sn § 4 Some Comments on WF(u) § 5 Some comments on WFA(u)Appendix § A.1 Partition of Unity § A.2 Proof of Theorem 1 § A.3 Proof of (18) § A.4 Pseudo-Local Property in y(s) § A.5 Proof of Theorem A.1Lecture VI Micro-local Analysis in Gevrey Class (II) § 1 Preliminaries § 2 Proof of Theorem 1 § 3 Some Consequence of Theorem 1 § 4 Propagation of Singularities in the sense of C∞Appendix § A.1 § A.2 Proof of Lemma 2Lecture V I I Schrödinger Type Equations § 1 Introduction (General View-Points on Evolution Equations) § 2 Necessity of (Co) § 3 Sufficiency for L2-Wellposedness