Persistence of conormal analyticity for the Euler equations in the half-space

Abstract

We prove the persistence of conormal analyticity through the $H^r$ norm, with $r \geq 4$, for the 3D incompressible Euler equations in the half-space $\Omega = \mathbb{R}_+^3$. We also establish a lower bound on the radius of conormal analyticity which depends on $\|u\|_{H^r}$ and initial data. In particular, persistence is established in both of the spaces $X_\tau^{\rm co}$ and $A_\tau^{\rm co}$, the former being based on the conormal operator $Z_k = \phi^k \partial_3^k$ and the latter being based on $L^k = (\phi \partial_3)^k$. Finally, we show that the norms $\|\cdot \|_{X_\tau^{\rm co}}$ and $\|\cdot \|_{A_\tau^{\rm co}}$ are equivalent up to a small loss in radius, the proof of which uses explicit identities relating the two conormal operators.