Ill-posedness of incompressible Kelvin-Helmholtz problem with transverse magnetic field

In this paper, we prove the linear and nonlinear ill-posedness of the well-known Kelvin-Helmholtz problem of the incompressible ideal magnetohydrodynamics (MHD) equations with transverse magnetic field. Our proof rigorously verifies that “the development of the Kelvin-Helmholtz instability, in the direction of the streaming, is uninfluenced by the presence of the magnetic field in the transverse direction” which was proposed by S. Chandrasekhar’ book named by Hydrodynamic and Hydromagnetic stability.