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

  • Hideo Kozono (Tohoku University)
A3 01 (Sophus-Lie room)


We show that every $L^r$-vector field on $\Omega$ can be uniquely decomposed into two spaces with scalar and vector potentials and the harmonic vector space via $rot$ and $div$, where $\Omega$ is a bounded domain in $R3$. As an application, the generalized Biot-Savard law for the invisid incompressible fluids in $\Omega$ 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 $\Omega$.

Anne Dornfeld

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