On time-local solvability of the Navier-Stokes equations in Besov spaces

Abstract

A time--local solution is constructed for the Cauchy problem of the $n$-dimensional Navier--Stokes equations when the initial velocity belongs to Besov spaces of nonpositive order. The space contains $L^\infty$ in some exponents, so our solution may not decay at space infinity. In order to use iteration scheme we have to establish the Hölder type inequality for estimating bilinear term by dividing the sum of Besov norm with respect to levels of frequency. Moreover, by regularizing effect our solutions belong to $L^\infty$ for any positive time.