Navier–Stokes Cauchy Problem with |v0(x)|2 Lying in the Kato Class K3 - Francesca Crispo; Paolo Maremonti (Mathematics, 2021)
We investigate the 3D Navier–Stokes Cauchy problem. We assume the initial datum <semantics>v0</semantics> is weakly divergence free, <semantics><mstyle scriptlevel="0" displaystyle="true"><munder>supR3</munder></mstyle></semantics><semantics><munder>∫R3</munder>|v0(y)|2<mspace width="-2.84544pt"></mspace>|x−y|<mspace width="1.42271pt"></mspace>dy<∞</semantics> and <semantics>|v0(y)|2∈K3</semantics>, where <semantics>K3</semantics> denotes the Kato class. The existence is local for arbitrary data and global if <semantics><mstyle scriptlevel="0" displaystyle="true"><munder>supR3</munder></mstyle></semantics><semantics><munder>∫R3</munder>|v0(y)|2<mspace width="-2.84544pt"></mspace>|x−y|<mspace width="1.42271pt"></mspace>dy</semantics> is small. Regularity and uniqueness also hold.