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

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

2006-08-23

arxiv.org/abs/math/0608578v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

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 $\bn=(0,\infty)\times\rn$.

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