Incompressible Euler Equations and the Effect of Changes at a Distance

TL;DR

This paper investigates how localized changes in initial velocity affect solutions to the incompressible Euler equations, demonstrating that stability persists even without spatial decay, though in a weaker form.

Contribution

It extends the understanding of stability in incompressible Euler solutions to include those without spatial decay, revealing stability properties in more general cases.

Findings

Localized velocity changes influence solutions globally

Stability persists without spatial decay, but weaker

Global pressure determination causes immediate effects

Abstract

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. Indeed, this is the case for solutions having spatial decay, as can be easily shown. We consider the more difficult case of solutions lacking spatial decay, and show that such stability still holds, albeit in a somewhat weaker form.