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

We prove the persistence of conormal analyticity through the $H^r$ norm, with $r\ge 4$, for the 3D incompressible Euler equations in the half-space $\mathrm{\Omega }={\mathbb{R}}_{+}^3$. We also establish a lower bound on the radius of conormal analyticity which depends on $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^r}$ and initial data. In particular, persistence is established in both of the spaces $X_{\tau }^{\mathrm{c}\mathrm{o}}$ and $A_{\tau }^{\mathrm{c}\mathrm{o}}$, the former being based on the conormal operator $Z_k={\phi }^k{\mathrm{\partial }}_3^k$ and the latter being based on $L^k=(\phi {\mathrm{\partial }}_3{)}^k$. Finally, we show that the norms $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{X_{\tau }^{\mathrm{c}\mathrm{o}}}$ and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{A_{\tau }^{\mathrm{c}\mathrm{o}}}$ are equivalent up to a small loss in radius, the proof of which uses explicit identities relating the two conormal operators.