Behavior of mild solutions for the Navier-Stokes equations with fractional dissipation in Lei-Lin-Gevrey spaces

This work presents new decay rates for mild solutions of the Navier-Stokes equations, with fractional dissipation of order $\alpha \in [\frac{1}{2},1]$, by considering Lei-Lin-Gevrey spaces. More precisely, our unique solution u satisfies the following result:$\underset{t\to \infty }{\lim \sup }{t}^{\frac{2k+3}{2s+3}\min \{\frac{5-4\alpha }{2\alpha },\frac{2s+3}{2\alpha }\}}{\Vert u\left(t\right)\Vert }_{X_{a,\sigma }^k}=0,\text{with }s\in [-1,0],k\in (-\frac{3}{2},s).$ This is a consequence of the fact that ${\Vert u\left(t\right)\Vert }_{{\dot{H}}_{a,\sigma }^0}\to 0$, as $t\to \infty$. Moreover, our main estimate is given by the inequality below:${\Vert u\left(t\right)\Vert }_{X_{a,\sigma }^k}\le C{(1+t)}^{-\frac{k+3-p}{2\alpha }},\text{with }k\ge -1,p\in [-1,k+3),$ where $t>0$ and C is a positive constant.