Desingularization of a Steady Vortex Pair in the Lake Equation
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Desingularization of a Steady Vortex Pair in the Lake Equation Justin Dekeyser1 Received: 10 November 2019 / Accepted: 9 September 2020 / © Springer Nature B.V. 2020
Abstract We construct a family of steady solutions of the lake model perturbed by some small Coriolis force, that converge to a singular vortex pair. The desingularized solutions are obtained by maximization of the kinetic energy over a class of rearrangements of sign changing functions. The precise localization of the asymptotic singular vortex pair is proved to depend on the depth function and the Coriolis parameter, and it is independent on the geometry of the lake domain. We apply our result to construct a singular rotating vortex pair in a rotation invariant lake. Keywords Lake model · Rearrangements · Vortex pair · Singular vortex · Desingularization · Asymptotic Mathematics Subject Classification (2010) 76C05 (35Q35 · 35C15 · 58E30)
1 Introduction Statement of the problem The lake equations arise from the incompressible 3D Euler equations in a regime where the typical velocity magnitude is small in comparison to the magnitude of gravity waves (small Froude number regime Fr 1, see also [10]). Mathematically, a lake can be modeled as a planar open set D ⊆ R2 together with a positive depth function b. The velocity field v : R × D → R2 and the pressure field p : R × D → R are governed by the system of equations ⎧ ⎪ on R × D, ⎨div bv = 0 ∂t v + (v · ∇)v + f v ⊥ = −∇p on R × D, ⎪ ⎩ bv · ηˆ = ν on R × ∂D. Here ηˆ : ∂D → R2 is the unit outward normal to ∂D, (v1 , v2 )⊥ = (−v2 , v1 ), f : D → R is a Coriolis parameter, and ν : ∂D → R is a penetration condition. When the depth Justin Dekeyser
[email protected] 1
D´epartement de Math´ematiques, Universit´e catholique de Louvain, Louvain-La-Neuve 1348, Belgium
J. Dekeyser
function is constant, b ≡ 1, the system reduces to the incompressible 2D Euler equations. The velocity field v in the lake model may be understood as the horizontal velocity of a column water, whose total mass may vary according to the depth of the lake [10]. Global well-posedness of the lake equations has been studied by Levermore & Oliver & Titi [18], Lacave & Nguyen & Pausader [19], Munteanu [24], Huang & Chaocheng [15]. The problem we study in this article is the asymptotic behavior of steady velocity fields that are constructed as maximizers of the kinetic energy 1 |v |2 dμ, dμ = b dm = b(x) dx, E(v ) = 2 D in the regime where the vorticity shrinks to a Dirac mass. Let us motivate the problem we address by first recalling what is known in the easiest case where the Coriolis force is null: f = 0; and the topography does not vary: b ≡ 1. This situation is known as the 2D Euler’s equations. Introducing the vorticity ω = curl(v) (and up to prescribe circulation conditions on ∂D), one may think of v as v = curl−1 (ω), which allows one to think off the energy as a function of ω only: E(v) ≡ E(ω). Moreover, taking the curl of the evolution equation, we get the transport equation ∂t ω + (v|∇ω)= 0. These observations
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