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 $r\ge 4$, 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 $\Vert u{\Vert }_{H^r}$ and initial data. In particular, persistence is established in both of the spaces $X_{\tau }^{\mathrm{co}}$ and $A_{\tau }^{\mathrm{co}}$, the former being based on the conormal operator $Z_k={\varphi }^k{\partial }_3^k$ and the latter being based on $L^k=(\varphi {\partial }_3{)}^k$. Finally, we show that the norms $\Vert \cdot {\Vert }_{X_{\tau }^{\mathrm{co}}}$ and $\Vert \cdot {\Vert }_{A_{\tau }^{\mathrm{co}}}$ are equivalent up to a small loss in radius, the proof of which uses explicit identities relating the two conormal operators.