article

Title: Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted $l_p$ initial data

Authors: Animikh Biswas and David Swanson

Issue: Volume 56 (2007), Issue 3, 1157-1188

Abstract:

$We prove the existence and Gevrey regularity of local solutions of the three dimensional periodic Navier-Stokes equations in case the sequence of Fourier coefficients of the initial data lies in an appropriate weighted $\ell_{p}$ space. Our work is motivated by that of Foias and Temam (Ciprian Foias and Roger Temam, \emph{Gevrey class regularity for the solutions of the Navier-Stokes equations}, J. Funct. Anal. \textbf{87} (1989), 359--369) and we obtain a generalization of their result. In particular, our analysis allows for initial data that are less smooth than in \emph{op. cit.} and can also have infinite energy. Our main tool is a variant of the Young convolution inequality enabling us to estimate the nonlinear term in weighted $\ell_{p}$ norm.$