Gevrey instability in the inviscid inflow-outflow problem

We consider the 2D incompressible Euler equations on a periodic channel $\mathbb{T}\times (0,1)$ with inflow-outflow boundary condition $u=(0,1)$ on $\mathbb{T}\times \{0,1\}$. We also impose the incoming vorticity boundary condition $\omega =\eta$ on $\mathbb{T}\times \{0\}$, where $\eta$ is prescribed. We show that the problem is globally well-posed in Gevrey spaces (for any value of the Gevrey exponent $s>1$) as long as $\eta$ remains Gevrey. This proves that the inflow-outflow velocity boundary condition determines the solution locally in time if and only if the solution is considered in an analytic class. In particular, leaving the analytic class, nonuniquness of solutions occurs already in any Gevrey class, by prescribing $\eta$. Hence, prescribing an analytic inflow-outflow velocity leads to precisely one analytic and continuum $s$-Gevrey solutions for every $s>1$. Furthermore, the result implies that if $\eta$ is analytic, then the unique global solution can lose analyticity in space for all $t>0$, but remain $s$-Gevrey regular for all $s$.