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An Introduction to Maximal Regularity for Parabolic Evolution Equations

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2021

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https://doi.org/10.1007/978-981-33-4822-6_1
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KOIKE, Shigeaki, ed., Hideo KOZONO, ed., Takayoshi OGAWA, ed. and others. Nonlinear Partial Differential Equations for Future Applications. Singapore: Springer, 2021, 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

Zusammenfassung

In this note, we give an introduction to the concept of maximal Lp-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.

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Fachgebiet (DDC)
510 Mathematik

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

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International workshop on Nonlinear Partial Differential Equations for Future Applications, PDEFA 2017, 2. Okt. 2017 - 6. Okt. 2017, Sendai, Japan
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ISO 690DENK, 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, 2. Okt. 2017 - 6. Okt. 2017. In: KOIKE, Shigeaki, ed., Hideo KOZONO, ed., Takayoshi OGAWA, ed. and others. Nonlinear Partial Differential Equations for Future Applications. Singapore: Springer, 2021, 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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