New Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System

TL;DR

This paper characterizes the asymptotic behavior of non-stationary solutions to the Navier-Stokes equations in erent dimensions, revealing their potential field structure at infinity and providing new decay bounds for localized data.

Contribution

It introduces new asymptotic profiles for solutions of the Navier-Stokes system, extending understanding of their behavior at large distances and times, especially for small times and localized initial data.

Findings

Solutions behave as a potential field at infinity.

Derived bounds for solution decay at large distances.

Extended decay estimates for weighted-L^p norms over time.

Abstract

We show that solutions $u(x,t)$ of the non-stationnary incompressible Navier--Stokes system in $\R^d$ ($d\geq2$) starting from mild decaying data $a$ behave as $|x|\to\infty$ as a potential field: u(x,t) = e^{t\Delta}a(x) + \gamma_d\nabla_x(\sum_{h,k} \frac{\delta_{h,k}|x|^2 - d x_h x_k}{d|x|^{d+2}} K_{h,k}(t))+\mathfrak{o}(\frac{1}{|x|^{d+1}}) where $\gamma_d$ is a constant and $K_{h,k}=\int_0^t(u_h| u_k)_{L^2}$ is the energy matrix of the flow. We deduce that, for well localized data, and for small $t$ and large enough $|x|$, c t |x|^{-(d+1)} \le |u(x,t)|\le c' t |x|^{-(d+1)}, where the lower bound holds on the complementary of a set of directions, of arbitrary small measure on $\mathbb{S}^{d-1}$. We also obtain new lower bounds for the large time decay of the weighted-$L^p$ norms, extending previous results of Schonbek, Miyakawa, Bae and Jin.