L2 decay of weak solutions for the Navier–Stokes equations with supercritical 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 $L^2\left(R^3\right)$ and ${\dot{H}}^s\left(R^3\right)$ ($s\le 0$). More precisely, we prove that $u$ satisfies the following upper bound: ${\Vert u\left(t\right)\Vert }_2^2\le C{(1+t)}^{-\frac{3-2p}{2\alpha }},\forall t>0.$ This estimate leads us to show the next inequality: ${\Vert u\left(t\right)\Vert }_{{\dot{H}}^{-\delta }}^2\le C{(1+t)}^{-\frac{3-2\delta -2p}{2\alpha }},\forall t>0.$ 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 $u_0\in L^2\left(R^3\right)\cap Y^p\left(R^3\right)$ (and also $u_0\in L^1\left(R^3\right)$ for $p=-1$ and a certain finite set of values of $\alpha$).