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

Abstract

We revisit Yudovich’s well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set \(\Omega \subset \mathbb {R}^2\) or on the torus \(\Omega =\mathbb {T}^2\). We construct global-in-time weak solutions with vorticity in \(L^1\cap L^p_{ul}\) and in \(L^1\cap Y^\Theta _{ul}\), where \(L^p_{ul}\) and \(Y^\Theta _{ul}\) are suitable uniformly-localized versions of the Lebesgue space \(L^p\) and of the Yudovich space \(Y^\Theta \) respectively, with no condition at infinity for the growth function \(\Theta \). We also provide an explicit modulus of continuity for the velocity depending on the growth function \(\Theta \). We prove uniqueness of weak solutions in \(L^1\cap Y^\Theta _{ul}\) under the assumption that \(\Theta \) grows moderately at infinity. In contrast to Yudovich’s energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón–Zygmund theory or Littlewood–Paley decomposition, and actually applies not only to the Biot–Savart law, but also to more general operators whose kernels obey some natural structural assumptions.