Gakuto International Series Mathematical Sciences and Applications, Vol.35(2011) Mathematical Analysis on the Navier-Stokes Equations and Related Topics,Past and Future –In memory of Professor Tetsuro Miyakawa, pp.17–30 SLOWLY DECAYING SOLUTIONS TO INCOMPRESSIBLE NAVIER-STOKES SYSTEM To the memory of Tetsuro Miyakawa, our teacher and friend Marco Cannone Universit´ e Paris-Est, Laboratoire d’Analyse et de Math´ ematiques Appliqu´ ees, UMR 8050 CNRS, 5 boulevard Descartes, Cit´ e Descartes Champs-sur-Marne, 77454 Marne-la-Vall´ ee cedex 2, France (E-mail : marco.cannone@univ-mlv.fr) Cheng He Division of Mathematics, Department of Mathematical & Physical Sciences, National Natural Science Foundation of China, 100085, P. R. China (E-mail : hecheng@nsfc.gov.cn) and Grzegorz Karch Instytut Mathematyczny, Uniwersytet Wroclawski pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland (E-mail : grzegorz.karch@math.uni.wroc.pl) Abstract. We study the large time asymptotics of solutions to the Cauchy problem in R n for the incompressible Navier-Stokes system. Imposed assumptions on initial data imply that, at the first approximation, solutions look as solutions to the linear heat equa- tion. The main goal of this work is to derive second terms of the asymptotic expansions of solutions and to extend, in this way, the results by Fujigaki & Miyakawa (SIAM J. Math. Anal. 33 (2001), 523–544) and Gallay & Wayne (R. Soc. London Philos. Trans. Ser. A Math. Phys. Eng. Sci. 360 (2002), 2155–2188). Communicated by Editors; Received July 30, 2010, Accepted Oct. 25, 2010. Keywords: incompressible Navier-Stokes system; the Cauchy problem; asymptotic expansions, decay estimates. 17