Monica Musso

Gluing methods for vortices in Euler equations

The Euler equations (1755) define a system of non-linear PDEs that models the dynamics of an inviscid, incompressible fluid. In dimension 2, a classical problem is the desingularized N-vortex problem, namely the existence of true smooth solutions of Euler equations with highly concentrated vorticities around N points. Using gluing methods, for any sufficiently small ε>0 we show the existence of a solution with vorticity of the form

ω_ε(x,t) ≈ ∑_j=1,...,N (κ_j / ε²) W( (x-ξ_j(t)) / ε )

where W(y) = 8 / (1+ |y |²)² is the standard Liouville bubble which has mass 8π, κ_j ∈ ℝ and the centers ξ_j(t) solve the Kirchoff-Routh law of motion. This refines a previous construction by Marchioro and Pulvirenti, giving precise description of the velocity field. In dimension 3, if the initial vorticity is concentrated along a smooth curve in space, a long standing question is whether the associated solution exhibits a vorticity still very concentrated around a curve on finite times. The formal derivation of the motion of the curve was first computed by Da Rios in 1903, and it approximately evolves by the bi-normal flow of curves. Jerrard and Seis used refined energy estimates to prove the validity of the asymptotic law under the assumption that vorticity is indeed concentrated at all time. The big open problem is whether one can find solutions of the Euler equations for which the vorticity remains close for a significant period of time to a filament evolving by binormal flow. We prove that this is the case when the curve is an helix evolving by bi-normal flow. Using helical symmetries and looking for rotating solutions, this problem can be reduced to finding a concentrating solution for a 2-dimensional elliptic problem in divergence form. These results are in collaboration with Juan Dávila, Manuel del Pino and Juncheng Wei.