Solution Theory for Evolutionary Equations

Christian Seifert,Marcus Waurick,Sascha Trostorff

doi:10.1007/978-3-030-89397-2_6

Abstract

AbstractIn this chapter, we shall discuss and present the first major result of the manuscript: Picard’s theorem on the solution theory for evolutionary equations which is the main result of Picard (A structural observation for linear material laws in classical mathematical physics. In Mathematical Methods in the Applied Sciences, vol 32, 2009, pp 1768–1803). In order to stress the applicability of this theorem, we shall deal with applications first and provide a proof of the actual result afterwards. With an initial interest in applications in mind, we start off with the introduction of some operators related to vector calculus.

Highlights

Proposition 6.1.1 The relations div, div0, grad, grad0, curl and curl0 are all densely defined, closed linear operators
It follows from integration by parts that gradc ⊆ grad, divc ⊆ div and curlc ⊆ curl
This, in turn, implies that gradc, divc and curlc are closable by Lemma 2.2.7 with respective closures grad0, div0 and curl0 by Lemma 2.2.4

Summary

Chapter 6

We shall discuss and present the first major result of the manuscript: Picard’s theorem on the solution theory for evolutionary equations which is the main result of [82]. With an initial interest in applications in mind, we start off with the introduction of some operators related to vector calculus

First Order Sobolev Spaces

Well-Posedness of Evolutionary Equations and Applications

Proof of Picard’s Theorem

Solution Theory for Evolutionary Equations readily implies

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