Well-posedness threshold of the 2D Euler equations in Hölder spaces on infinite sectors

Abstract: We study existence and uniqueness for the 2D incompressible Euler equations on infinite sectors of acute angle $β$, focusing on vorticity in Hölder-type spaces. In the inifinite sectors, for every $β$, H"older regularity cannot be expected without decay at infinity. Moreover, the corner causes Schauder-type estimates to fail at sufficiently high regularity. To address these difficulties, we introduce weighted H"older spaces and establish well-posedness for every acute angle, without any decay assumption at infinity. These spaces include scale-invariant functions, allowing us to study their regularity and dynamics. We then apply this theory to standard $C^{k,α}$ spaces and show that $k+α=π/β-1$ is the well-posedness threshold. Above this threshold, we construct initial data that exhibit instantaneous loss of regularity. At the threshold, the problem is well-posed if and only if $π/β$ is not an integer. For well-posedness in $C^{k,α}$, we assume that the initial vorticity belongs to a weighted H"older space rather than imposing decay at infinity. This class includes compactly supported initial vorticities.