Weak limits of solutions to the steady incompressible two-dimensional Euler equation in a bounded domain

Abstract

In this work we show that any sequence u^ε of smooth solutions to the steady incompressible two-dimensional Euler equation in a bounded domain Ω, which converges weakly in L²(Ω) as ε goes to zero, converges to a weak solution of this equation provided curl u^ε remains bounded in L¹(Ω).