On a Generalized System with Applications to Ideal Magnetohydrodynamics

2025/07/20 by Alejandro Sarria, Sarria, Alejandro
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.14772

openalex publication_date 2025/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (u₀(x),b₀(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 x₀ at which b₀(x) and b₀'(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 b₀'(x₀) suppresses finite-time blowup.

Citations

• On the explicit blowup solutions for 3D incompressible Magnetohydrodynamics equations
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• The Role of Initial Curvature in Solutions to the Generalized Inviscid Proudman-Johnson Equation
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• Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations
• The Euler Equations and Non-Local Conservative Riccati Equations
• Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations

Related

• Asymptotic stability of explicite infinite energy blowup solutions for three dimensional incompressible Magnetohydrodynamics equations
• A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface
• Finite Time Blow-up of a 3D Model for Incompressible Euler Equations
• The lifespan of strong solutions to the compressible MHD equations with entropy transport in the presence of vacuum
• Global estimates and blow-up criteria for the generalized Hunter-Saxton system