Kink solutions of the binormal flow

In these pages we shall present some recent work done in collaboration with S. Gutiérrez on the selfsimilar solutions of the binormal flow Xt = Xs ∧Xss (1) Here X(s, t) ∈ R, s ∈ R, t ∈ R and s represents the arclength parameter |Ẋ(s, t)| = 1. Differentiation in (1) gives Xst = Xs ∧Xsss and as long as X is regular enough we get d ds |Xs| = Xs(Xs)t = Xs · (Xs ∧Xsss) = 0. This flow was found for the first time by Da Rios [DaR] in 1906 in his Tesis di Laurea under the direction of Levi-Civita as an approximation of the motion of a vortex tube of infinitesimal cross section (vortex filament) within an inviscid ideal fluid. This amounts to say that the vorticity ω = ∇×u, with u the velocity of the fluid, is a singular vectorial measure with support on a curve in R (precisely X(s, t)). The vectorial identity ∇× (∇× u) = −∆u+∇div u = −∆u allows us to compute the velocity field u from the vorticity w by the Biot-Savart integral u(P ) = − Γ 4π ∫ ∞ −∞ P −X(s) |P −X(s)|3 × T (s) ds (2)