Uniqueness of the 2D Euler equation on rough domains

We consider the 2D incompressible Euler equation on a bounded simply connected domain Ω . We give sufficient conditions on the domain Ω so that for any initial vorticity ω₀ ∈ L∞(Ω) , the weak solutions are unique. Our sufficient condition is slightly more general than the condition that Ω is a C1,α domain for some α>0 , with its boundary belonging to H3/2(S¹) . As a corollary, we prove uniqueness for C1,α domains for α >1/2 and for convex domains which are also C1,α domains for some α >0 . Previously, uniqueness for general initial vorticity in L∞(Ω) was only known for C1,1 domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below the C1,1 regularity is the fact that for less regular domains, the velocity near the boundary is no longer log-Lipschitz. We overcome this barrier by defining a new change of variable which we then use to define a novel energy functional.