Temporal decay rates for weak solutions of the Navier-Stokes Equations with supercritical fractional dissipation

Abstract

In this paper, we establish temporal decay for a weak solution $u(x,t)$ (with initial data $u_0$) of the Navier-Stokes equations with supercritical fractional dissipation $\alpha \in (0,\frac{5}{4})$ in $L^2({\mathbb{R}}^3)$ and ${\dot{H}}^s({\mathbb{R}}^3)$ ($s\le 0$). More precisely, we prove that $u$ satisfies the following upper bound: $$\Vert u(t){\Vert }_2^2\le C(1+t{)}^{-\frac{3-2p}{2\alpha }},\forall t>0.$$ This estimate leads us to show the next inequality: $$\Vert u(t){\Vert }_{{\dot{H}}^{-\delta }}^2\le C(1+t{)}^{-\frac{3-2\delta -2p}{2\alpha }},\forall t>0.$$ These results are obtained by applying standard Fourier Analysis and they hold for $\alpha \in (0,\frac{5}{4}),$ $p\in [-1,\frac{3}{2})$, $\delta \in [0,\frac{3-2p}{2})$ and $u_0\in L^2({\mathbb{R}}^3)\cap {\mathcal{Y}}^p({\mathbb{R}}^3)$ (and also $u_0\in L^1({\mathbb{R}}^3)$ for $p=-1$ and a certain finite set of values of $\alpha$).

18 pages