Maximal L1 -regularity of the heat equation and application to a free boundary problem of the Navier-Stokes equations near the half-space

Takayoshi Ogawa; Senjo Shimizu

doi:10.1007/s41808-021-00133-w

Maximal L¹ -regularity of the heat equation and application to a free boundary problem of the Navier-Stokes equations near the half-space

Takayoshi Ogawa, Senjo Shimizu

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

This is a survey of recent results concerning on maximal L¹-regularity of the heat equation with the Naumann boundary condition in the half Euclidian space Ogawa and Shimizu (Proc Jpn Acad A, 96:57–62, 2020). It also includes maximal L¹-regularity for the initial boundary value of the Stokes system in the half-space under the stress free boundary condition. As an application, we introduce the time global well-posedness for the free boundary problem of the incompressible Navier-Stokes equations under the small initial data in the half Euclidean spaces R+n developed in Danchin-Hieber-Mucha-Tolksdorf (arXiv:2011.07918) and Ogawa and Shimizu (2021).

Original language	English
Pages (from-to)	509-535
Number of pages	27
Journal	Journal of Elliptic and Parabolic Equations
Volume	7
Issue number	2
DOIs	https://doi.org/10.1007/s41808-021-00133-w
Publication status	Published - 2021 Dec

Keywords

End-point estimate
Free boundary problems
Heat equations
Initial-boundary value problems
Maximal L-regularity
The Neumann boundary condition

ASJC Scopus subject areas

Analysis
Numerical Analysis
Applied Mathematics

Access to Document

10.1007/s41808-021-00133-w

Cite this

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title = "Maximal L1 -regularity of the heat equation and application to a free boundary problem of the Navier-Stokes equations near the half-space",

abstract = "This is a survey of recent results concerning on maximal L1-regularity of the heat equation with the Naumann boundary condition in the half Euclidian space Ogawa and Shimizu (Proc Jpn Acad A, 96:57–62, 2020). It also includes maximal L1-regularity for the initial boundary value of the Stokes system in the half-space under the stress free boundary condition. As an application, we introduce the time global well-posedness for the free boundary problem of the incompressible Navier-Stokes equations under the small initial data in the half Euclidean spaces R+n developed in Danchin-Hieber-Mucha-Tolksdorf (arXiv:2011.07918) and Ogawa and Shimizu (2021).",

keywords = "End-point estimate, Free boundary problems, Heat equations, Initial-boundary value problems, Maximal L-regularity, The Neumann boundary condition",

author = "Takayoshi Ogawa and Senjo Shimizu",

note = "Funding Information: The first author is partially supported by JSPS grant-in-aid for Scientific Research (S) #19H05597 and Challenging Research (Pioneering) #20K20284. The second author is partially supported by JSPS grant-in-aid for Scientific Research (B) #16H03945, (B) #21H00992 and by JSPS Fostering Joint Research Program (B) #18KK0072. Publisher Copyright: {\textcopyright} 2021, Orthogonal Publisher and Springer Nature Switzerland AG.",

year = "2021",

month = dec,

doi = "10.1007/s41808-021-00133-w",

language = "English",

volume = "7",

pages = "509--535",

journal = "Journal of Elliptic and Parabolic Equations",

issn = "2296-9020",

publisher = "Springer International Publishing AG",

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AU - Ogawa, Takayoshi

AU - Shimizu, Senjo

N1 - Funding Information: The first author is partially supported by JSPS grant-in-aid for Scientific Research (S) #19H05597 and Challenging Research (Pioneering) #20K20284. The second author is partially supported by JSPS grant-in-aid for Scientific Research (B) #16H03945, (B) #21H00992 and by JSPS Fostering Joint Research Program (B) #18KK0072. Publisher Copyright: © 2021, Orthogonal Publisher and Springer Nature Switzerland AG.

PY - 2021/12

Y1 - 2021/12

N2 - This is a survey of recent results concerning on maximal L1-regularity of the heat equation with the Naumann boundary condition in the half Euclidian space Ogawa and Shimizu (Proc Jpn Acad A, 96:57–62, 2020). It also includes maximal L1-regularity for the initial boundary value of the Stokes system in the half-space under the stress free boundary condition. As an application, we introduce the time global well-posedness for the free boundary problem of the incompressible Navier-Stokes equations under the small initial data in the half Euclidean spaces R+n developed in Danchin-Hieber-Mucha-Tolksdorf (arXiv:2011.07918) and Ogawa and Shimizu (2021).

AB - This is a survey of recent results concerning on maximal L1-regularity of the heat equation with the Naumann boundary condition in the half Euclidian space Ogawa and Shimizu (Proc Jpn Acad A, 96:57–62, 2020). It also includes maximal L1-regularity for the initial boundary value of the Stokes system in the half-space under the stress free boundary condition. As an application, we introduce the time global well-posedness for the free boundary problem of the incompressible Navier-Stokes equations under the small initial data in the half Euclidean spaces R+n developed in Danchin-Hieber-Mucha-Tolksdorf (arXiv:2011.07918) and Ogawa and Shimizu (2021).

KW - End-point estimate

KW - Free boundary problems

KW - Heat equations

KW - Initial-boundary value problems

KW - Maximal L-regularity

KW - The Neumann boundary condition

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