Uniqueness of the 2D Euler equation on rough domains | CU Experts

; We consider the 2D incompressible Euler equation on a bounded simply connected domain ; ; Omega; ; . We give sufficient conditions on the domain ; ; Omega; ; so that for any initial vorticity ; ; omega_{0} in L^{infty}(Omega); ; , the weak solutions are unique. Our sufficient condition is slightly more general than the condition that ; ; Omega; ; is a ; ; C^{1,alpha}; ; domain for some ; ; alpha>0; ; , with its boundary belonging to ; ; H^{3/2}(mathbb{S}^{1}); ; . As a corollary, we prove uniqueness for ; ; C^{1,alpha}; ; domains for ; ; alpha >1/2; ; and for convex domains which are also ; ; C^{1,alpha}; ; domains for some ; ; alpha >0; ; . Previously, uniqueness for general initial vorticity in ; ; L^{infty}(Omega); ; was only known for ; ; C^{1,1}; ; domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below the ; ; C^{1,1}; ; 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.;

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