Forward Self-Similar Solutions to the 2D Hypodissipative Navier-Stokes Equations

We study the forward self-similar solutions to the $2$D hypodissipative Navier-Stokes equation with fractional diffusion $(-\mathrm{\Delta }{)}^{\alpha }$ for $\frac{1}{2}<\alpha <1$. We first show that for arbitrarily large $(1-2\alpha )$-homogeneous initial data which are locally Lipschitz, there exists at least one weak solution whose profile differs from the self-similar profile of the fractional heat equation by an element of $H^{\alpha }({\text{reall}}^2)$. Moreover, when $\alpha \in (\frac{2}{3},1)$ we show that any such weak solution is actually smooth, hence a strong solution, and satisfies certain far field decay estimates. Finally, we provide numerical evidence for the nonuniqueness of the related $2$D Navier-Stokes equation with time-dependent viscosity.