Clustered travelling vortex rings to the axisymmetric three-dimensional incompressible Euler flows

For the three dimensional axisymmetric Euler flow, we construct a family of solutions with multiple travelling vortex rings, with large speed of order O(|lnɛ|), where ɛ>0 is a small parameter. Our construction is based on the analysis of the following nonlinear elliptic equation: ∂rrψ+3r∂rψ+∂zzψ=−F((ψ−α2|lnɛ|)r2),(r,z)∈R2,ψr(0,z)=0forr=0,for some special functions F, where α is a parameter. The location of the vortex rings is governed by some balancing systems, which can be solved by the polynomial method in several special cases. For the non-swirl case, in the core of each vortex ring, our solutions can be regarded as a rescaled finite mass solution of the Liouville equation. The results can be generalized directly to the case with swirl, for which we also construct different types of solutions with multiple vortex rings.