On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem

Abstract

We prove space-time decay estimates of suitable weak solutions to the Navier-Stokes Cauchy problem, corresponding to a given asymptotic behavior of the initial data of the same order of decay. We use two main tools. The first is a result obtained by the authors in the paper "A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem" (submitted), on the behavior of the solution in a neighborhood of $t=0$ in the $L_{loc}^{\infty }$-norm, which enables us to furnish a representation formula for a suitable weak solution. The second is the asymptotic behavior of the $L^2({\mathbb{R}}^3\setminus B_R)$ norm of $u(t)$ for $R\to \infty$. Following a Leray's point of view, roughly speaking our result proves that a possible space-time turbulence does not perturb the asymptotic spatial behavior of the initial data of a suitable weak solution.