Nonuniqueness of Leray-Hopf solutions to the forced fractional Navier-Stokes Equations in three dimensions, up to the J. L. Lions exponent

In this paper, we show that for $\alpha \in (1\mathrm{/}2,5\mathrm{/}4)$, there exists a force $f$ and two distinct Leray-Hopf flows $u_1,u_2$ solving the forced fractional Navier-Stokes equation starting from rest. This shows that the J.L. Lions exponent is sharp in the class of Leray-Hopf solutions for the forced fractional Navier-Stokes equation.