An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces (Q6568733)

The authors discussed Yudovich's well-posedness results for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular open set \(\Omega\subset\mathbb{R}^2\) or on the torus \(\Omega=\mathbb{T}^2\). They constructed global-in-time weak solutions with vorticity in \(L^1\cap L_{ul}^p\) and \(L^1\cap Y_{ul}^\Theta\). Next they proved uniqueness of weak solutions in \(L^1\cap Y_{ul}^\Theta\) under the assumption that \(\Theta\) grows moderately at infinity. They applied a Lagrangian strategy to show uniqueness different from Yudovich's energy method. In the process, their methods are different from the classical methods by using C-Z operator and L-P decomposition theory. The paper is delicate.