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.