Affine Variant of Fractional Sobolev Space with Application to Navier-Stokes System

Abstract

It is proved that for $\alpha\in (0,1)$, $Q_\alpha(\rn)$, not only as an intermediate space of $W^{1,n}(\rn)$ and $BMO(\rn)$ but also as an affine variant of Sobolev space $\dot{L}^{2}_\alpha(\rn)$ which is sharply imbedded in $L^{\frac{2n}{n-2\alpha}}(\rn)$, is isomorphic to a quadratic Morrey space under fractional differentiation. At the same time, the dot product $\nabla\cdot\big(Q_\alpha(\rn)\big)^n$ is applied to derive the well-posedness of the scaling invariant mild solutions of the incompressible Navier-Stokes system in $\bn=(0,\infty)\times\rn$.