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

TL;DR

This paper establishes decay estimates for suitable weak solutions to the Navier-Stokes equations, showing that space-time turbulence does not alter the asymptotic spatial decay of initial data, using advanced regularity and asymptotic analysis tools.

Contribution

It provides new space-time decay estimates for weak solutions, linking initial data decay to long-term behavior without turbulence interference.

Findings

Decay estimates match initial data asymptotics

Turbulence does not affect spatial decay at infinity

Uses novel representation formulas and asymptotic norms

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 $L^\infty_{loc}$-norm, which enables us to furnish a representation formula for a suitable weak solution. The second is the asymptotic behavior of the $L^2(\R^3\setminus B_R)$ 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.