An Introduction to Maximal Regularity for Parabolic Evolution Equations

Denk, Robert

An Introduction to Maximal Regularity for Parabolic Evolution Equations

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Date

2021

Publication type

Contribution to a conference collection

Publication status

Published

Published in

Nonlinear Partial Differential Equations for Future Applications / Koike, Shigeaki; Kozono, Hideo; Ogawa, Takayoshi et al. (ed.). - Singapore : Springer, 2021. - pp. 1-70. - ISSN 2194-1009. - eISSN 2194-1017. - ISBN 978-981-3348-21-9

Abstract

In this note, we give an introduction to the concept of maximal L^p-regularity as a method to solve nonlinear partial differential equations. We first define maximal regularity for autonomous and non-autonomous problems and describe the connection to Fourier multipliers and R-boundedness. The abstract results are applied to a large class of parabolic systems in the whole space and to general parabolic boundary value problems. For this, both the construction of solution operators for boundary value problems and a characterization of trace spaces of Sobolev spaces are discussed. For the nonlinear equation, we obtain local in time well-posedness in appropriately chosen Sobolev spaces. This manuscript is based on known results and consists of an extended version of lecture notes on this topic.

Subject (DDC)

510 Mathematics

Keywords

Maximal regularity, Fourier multipliers, Parabolic boundary value problems, Quasilinear evolution equations

Conference

International workshop on Nonlinear Partial Differential Equations for Future Applications, PDEFA 2017, Oct 2, 2017 - Oct 6, 2017, Sendai, Japan

Cite This

ISO 690

DENK, Robert, 2021. An Introduction to Maximal Regularity for Parabolic Evolution Equations. International workshop on Nonlinear Partial Differential Equations for Future Applications, PDEFA 2017. Sendai, Japan, Oct 2, 2017 - Oct 6, 2017. In: KOIKE, Shigeaki, ed., Hideo KOZONO, ed., Takayoshi OGAWA, ed. and others. Nonlinear Partial Differential Equations for Future Applications. Singapore:Springer, pp. 1-70. ISSN 2194-1009. eISSN 2194-1017. ISBN 978-981-3348-21-9. Available under: doi: 10.1007/978-981-33-4822-6_1

BibTex

@inproceedings{Denk2021Intro-55632,
  year={2021},
  doi={10.1007/978-981-33-4822-6_1},
  title={An Introduction to Maximal Regularity for Parabolic Evolution Equations},
  isbn={978-981-3348-21-9},
  issn={2194-1009},
  publisher={Springer},
  address={Singapore},
  booktitle={Nonlinear Partial Differential Equations for Future Applications},
  pages={1--70},
  editor={Koike, Shigeaki and Kozono, Hideo and Ogawa, Takayoshi},
  author={Denk, Robert}
}

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