On a Generalized System with Applications to Ideal Magnetohydrodynamics
Alejandro Sarria
TL;DR
The paper analyzes a generalized stagnation-point form system linking one-dimensional convection-stretching dynamics with magnetic coupling, as an infinite-energy surrogate for ideal MHD and related models. Through a Lagrangian framework, energy methods, and reductions to Euler-type equations, it derives sharp finite-time blowup criteria in multiple parameter regimes (notably $-1\le\lambda<0$, $\kappa\le-\lambda$, and $\kappa=-\lambda$ with $\lambda\neq0$), as well as mechanisms for blowup suppression and global existence in special cases (notably $(-\tfrac12,0)$ and $(0,0)$). The analysis hinges on the behavior of zeros of the initial magnetic field and its derivative, the preservation of zero-order structures along Lagrangian paths, and nonlocal energy contributions; Sturm comparison and exact solution frameworks for associated ODEs clarify the role of the Euler analogue and pressure terms. By connecting to classical models like Proudman–Johnson, Hunter–Saxton, and the 2D Boussinesq/Euler context, the work illuminates how one-dimensional convection, stretching, and coupling govern singularity formation and regulation in MHD-like systems with broad physical relevance.
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.