Navier–Stokes Cauchy Problem with |<i>v</i>₀(<i>x</i>)|² Lying in the Kato Class <i>K</i>₃

Abstract

We investigate the 3D Navier–Stokes Cauchy problem. We assume the initial datum v0 is weakly divergence free, supR3∫R3|v0(y)|2|x−y|dy∞ and |v0(y)|2∈K3, where K3 denotes the Kato class. The existence is local for arbitrary data and global if supR3∫R3|v0(y)|2|x−y|dy is small. Regularity and uniqueness also hold.

Keywords

• Navier–Stokes equations
• existence
• regular solutions