On the Cauchy Problem - 1st Edition - ISBN: 9780125016605, 9781483269061

On the Cauchy Problem

1st Edition

Editors: William F. Ames
Authors: Sigeru Mizohata
eBook ISBN: 9781483269061
Imprint: Academic Press
Published Date: 26th August 1986
Page Count: 186
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Notes and Reports in Mathematics in Science and Engineering, Volume 3: On the Cauchy Problem focuses on the processes, methodologies, and mathematical approaches to Cauchy problems.

The publication first elaborates on evolution equations, Lax-Mizohata theorem, and Cauchy problems in Gevrey class. Discussions focus on fundamental proposition, proof of theorem 4, Gevrey property in t of solutions, basic facts on pseudo-differential, and proof of theorem 3. The book then takes a look at micro-local analysis in Gevrey class, including proof and consequences of theorem 1. The manuscript examines Schrödinger type equations, as well as general view-points on evolution equations. Numerical representations and analyses are provided in the explanation of these type of equations.

The book is a valuable reference for mathematicians and researchers interested in the Cauchy problem.

Table of Contents

Lecture I Evolution Equations

Lecture II H∞-wellposedness


Lecture III Lax-Mizohata Theorem

§ 1

§ 2

§ 3 Proof of Theorem

§ 4

§ 5 Further Considerations


§ 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.7

Lecture 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



§ A.1 Proof of Lemma 4

§ A.2 Proof of Lemmas 1,2 and 6

§ A.3 Proof Lemma 3

Lecture 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)


§ 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.1

Lecture 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∞


§ A.1

§ A.2 Proof of Lemma 2

Lecture V I I Schrödinger Type Equations

§ 1 Introduction (General View-Points on Evolution Equations)

§ 2 Necessity of (Co)

§ 3 Sufficiency for L2-Wellposedness


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© Academic Press 1985
Academic Press
eBook ISBN:

About the Editor

William F. Ames

About the Author

Sigeru Mizohata

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