On a Generalized System with Applications to Ideal Magnetohydrodynamics

Abstract

Finite-time blowup of solutions $(u(x,t),b(x,t))$ to a generalized system of equations with applications to ideal Magnetohydrodynamics (MHD) and one-dimensional fluid convection and stretching, among other areas, is investigated. The system is parameter-dependent, our spatial domain is the unit interval or the circle, and the initial data $(u_0(x),b_0(x))$ is assumed to be smooth. Among other results, we derive precise blowup criteria for specific values of the parameters by tracking the evolution of $u_x$ along Lagrangian trajectories that originate at a point $x_0$ at which $b_0(x)$ and $b_0'(x)$ vanish. We employ concavity arguments, energy estimates, and ODE comparison methods. We also show that for some values of the parameters, a non-vanishing $b_0'(x_0)$ suppresses finite-time blowup.