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

Pages: 495 - 504, Issue Special, September 2011

 Abstract        Full Text (370.0K)              

Yoshikazu Giga - Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914, Japan (email)
J├╝rgen Saal - University of Konstanz, Department of Mathematics and Statistics, Box D 187, 78457 Konstanz, Germany (email)

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.

Keywords:  Maximal regularity, Radon measures, Navier-Stokes equations, Coriolis force, strong solutions
Mathematics Subject Classification:  Primary: 35B65, 28B05, 76U05; Secondary: 35Q30, 28C05

Received: July 2010;      Revised: March 2011;      Published: October 2011.