Incompressible Euler Equations and the Effect of Changes at a Distance

Abstract

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.

Well-posedness of the 2D Euler equations when velocity grows at infinity

“…The usual way to establish well-posedness of Eulerian solutions to the 2D Euler equations is to construct Lagrangian solutions (which are automatically Eulerian) and then prove uniqueness using the Eulerian formulation only. Such an approach works for bounded vorticity, bounded velocity solutions, as uniqueness using the Eulerian formulation was shown in [16] (see also [7]). Whether this can be extended to the solutions we study here is a subject for future work.…”

“…In [7], the focus was not on continuous dependence on initial data, per se, but rather on understanding the effect at a distance. Hence, we used a function, ζ(r) = (1 + r) α for any α ∈ (0, 1), in place of h and obtained a bound on (u 1 (t) − u 2 (t))/ζ in terms of (u 0 1 − u 0 2 )/ζ, each in the L ∞ norm.…”

“…One approach, motivated in part by this vortex patch example, is to make some assumption on the closeness of the initial vorticities locally uniformly in an L p norm for p < ∞, as was done in [16,1]. This assumption is, however, unnecessary (and in our setting somewhat artificial) as shown for the special case of bounded velocity (h ≡ 1) in [7].…”