On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay

Abstract

We study the energy decay of the turbulent solutions to the Navier–Stokes equations in the whole three-dimensional space. We show as the main result that the solutions with the energy decreasing at the rate ${O(t^{-\alpha}), t \rightarrow \infty, \alpha \in [0, 5/2]}$, are exactly characterized by their initial conditions belonging into the homogeneous Besov space ${\dot{B}^{-\alpha}_{2, \infty}}$. Similarly, for a solution u and ${p \in [1, \infty]}$ the integral ${\int_{0}^{\infty} \|t^{\alpha/2} u(t)\|^p \frac{1}{t} dt}$ is finite if and only if the initial condition of u belongs to the homogeneous Besov space ${\dot{B}_{2, p}^{-\alpha}}$. For the case ${\alpha \in (5/2, 9/2]}$ we present analogical results for some subclasses of turbulent solutions.