Helmholtz-Weyl decomposition in $L^r$ via $rot$ and $div$

We show that every $L^r$-vector field on $\mathrm{\Omega }$ can be uniquely decomposed into two spaces with scalar and vector potentials and the harmonic vector space via $rot$ and $div$, where $\mathrm{\Omega }$ is a bounded domain in $R3$. As an application, the generalized Biot-Savard law for the invisid incompressible fluids in $\mathrm{\Omega }$ is obtained. Furthermore, we prove a blow-up criterion such as Beale-Kato-Majda type via vorticity in $bmo$ on the classical solution of the Navier-Stokes equations in $\mathrm{\Omega }$.