Long Time Evolution of Concentrated Euler Flows with Planar Symmetry

Abstract

We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss how long this concentration persists, showing that in some cases this happens for quite long times. Moreover, we analyze a toy model that shows a similar behavior and gives some hints on the original problem.2010 Mathematics Subject Classification. 76B47, 37N10, 70K65.

Long time confinement of vorticity around a stable stationary point vortex in a bounded planar domain

“…Let us first mention that the point vortex system was used as a numerical approximation of the Euler system. More precisely, consider a smooth solution of the Euler equations and construct an initial discrete vorticity which is a sum of Dirac masses located on a grid (hj) j∈Z 2 where h ∈ R is the length of the grid, with masses h 2 ω 0 (hj). Solve then the point vortex system with this initial vorticity.…”

Long time localization of modified surface quasi-geostrophic equations

“…The decreasing in time of the Lipschitz constant D t does not depend on α, and it is the same as for the Euler case α = 0. The improvement of the content of Theorems 1.1 and 2.1 is closely related to the decreasing property of the Lipschitz constant, so it is the same as analysed in [4] (to which we address for the proof), leading to T ,β > −ζ0 , ∀ ∈ (0, 0 ), for suitable 0 > 0 and ζ 0 > 0.…”

“…The proof of Theorem 1.1 follows quite immediately from Theorem 2.1, and we address to [4] for the details.…”

“…We neglect the proof of this lemma because it is immediate to see that it coincides to the Euler case proved in [4] (Lemma 2.3). We observe that if F is null, B (t) and I (t) are conserved along the motion in the mSQG case as well as in the Euler case.…”

“…The main aim of this work is to give a rigorous estimate of the lower bound on the first time in which a filament reaches the boundary of a small region containing the support. Essentially, we want to generalize the results proved in [4] (for the Euler equation, i.e. α = 0) to the case of the mSQG equation (0 < α < 1).…”