Axi-symmetric solutions for active vector models generalizing 3D Euler and electron–MHD equations

Abstract

We study systems interpolating between the 3D incompressible Euler and electron–MHD equations, given by ∂tB+V⋅∇B=B⋅∇V,V=−∇×(−Δ)−aB,∇⋅B=0,where B is a time-dependent vector field in R3. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range 1/2<a<1. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions.