American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 495-504. doi: 10.3934/proc.2011.2011.495

$L^1$ 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  July 2010 Revised  March 2011 Published  October 2011

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.
Keywords: Radon measures, strong solutions, Navier-Stokes equations, Coriolis force, Maximal regularity.
Mathematics Subject Classification: Primary: 35B65, 28B05, 76U05; Secondary: 35Q30, 28C05.
Citation: Yoshikazu Giga, Jürgen Saal. $L^1$ maximal regularity for the laplacian and applications. Conference Publications, 2011, 2011 (Special) : 495-504. doi: 10.3934/proc.2011.2011.495
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