On almost axisymmetric incompressible magnetohydrodynamics in three dimensions

In this paper, we study the Cauchy problem of three-dimensional incompressible magnetohydrodynamics with almost symmetrical initial values in the cylindrical coordinates. Here the almost axisymmetric means that (∂_θu ₀ ^r , ∂_θu ₀ ^θ , ∂_θu ₀ ^z ) is small. With additional small-ness assumption on ( \(u^\theta_0,b^\theta_0\) $$u_0^{\theta },b_0^{\theta }$$ ), we prove the global existence of a unique strong solution (u, b), which keeps close to some axisymmetric vector fields. Moreover, we give the initial data with some special symmetric structures that will persist for all time.