Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such effects to decrease with distance from the localized change, giving the solutions a type of stability. One can easily show that this is the case for sufficiently smooth solutions having spatial decay. In this talk, we consider a broader class of weak solutions with vorticity lacking spatial decay, and we show that such stability still holds. This is joint work with James Kelliher.