Blowup Phenomenon of Ideal Compressible Non-isentropic Magnetohydrodynamic Equations with Radius Weighted Functional

Abstract

In this paper, we investigate the long-time behaviors of the ideal compressible non-isentropic magnetohydrodynamic (MHD) equations, alternatively named the Lundquist equations with non-constant entropy. To be specific, we show that a finite-time breakdown of the ideal MHD system will occur eventually by deriving a differential inequality of blowup type in terms of a functional weighted by the radius of the spatial variable and given initial data only. Our result complements some existing result, in which the author considered the unweighted radial component of momentum. Moreover, our blowup result is independent of the initial magnetic field, as long as it has compact support, the magnetic permeability constant and the sign of the initial mass functional.