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

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 $$ or on the torus $$ . We construct global-in-time weak solutions with vorticity in $$ and in $$ , where $$ and $$ are suitable uniformly-localized versions of the Lebesgue space $$ and of the Yudovich space $$ respectively, with no condition at infinity for the growth function  $$ . We also provide an explicit modulus of continuity for the velocity depending on the growth function  $$ . We prove uniqueness of weak solutions in $$ under the assumption that  $$ 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.