Forward self-similar solutions of the fractional Navier-Stokes Equations

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion $(-\Delta)^{\alpha}.$ First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with $5/6<\alpha\leq1$ for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in $\mathbb R^3\times (0,+\infty)$. In particular, when $\alpha=1$, we prove that the solution constructed by Korobkov-Tsai [Anal. PDE 9 (2016), 1811-1827] satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in [Jia and \v{S}ver\'{a}k, Invent. Math. 196 (2014), 233-265].