$L^1$ maximal regularity for the laplacian and applications

Yoshikazu Giga; Jürgen Saal

doi:10.3934/proc.2011.2011.495

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2011, Volume 2011: 495-504. Doi: 10.3934/proc.2011.2011.495 open access

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$L^1$ maximal regularity for the laplacian and applications

Yoshikazu Giga^1, and
Jürgen Saal^2,

1.
Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914
2.
University of Konstanz, Department of Mathematics and Statistics, Box D 187, 78457 Konstanz

Received: June 30, 2010

Revised: February 28, 2011

Published: August 2017

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Abstract

Inter alia we prove $L^1$ maximal regularity for the Laplacian in the space of Fourier transformed nite Radon measures FM. This is remarkable, since FM is not a UMD space and by the fact that we obtain $L_p$ maximal regularity for $p$ = 1, which is not even true for the Laplacian in $L^2$. We apply our result in order to construct strong solutions to the Navier-Stokes equations for initial data in FM in a rotating frame. In particular, the obtained results are uniform in the angular velocity of rotation.

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Mathematics Subject Classification: Primary: 35B65, 28B05, 76U05; Secondary: 35Q30, 28C05.

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