L^∞ continuation principle to the non‐baratropic non‐resistive magnetohydrodynamic equations without heat conductivity

Abstract

The aim of this paper is to establish a continuation principle for strong solutions to the full compressible magnetohydrodynamic system without resistivity and heat conductivity. We prove that if the solution loses its regularity in finite time, the dominated part is due to the hyperbolic effect. More precisely, it is essentially shown that the strong solution exists globally if the density, temperature, and magnetic field are bounded from above, where vacuum is allowed to exist. This verifies a problem proposed by D.Serre.

A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity

“…An related weak result was obtained in [6]. Very recently, Huang and Wang [12] establish the following regularity criterion…”

“…By the same calculations as in [12], it is easy to verify that where we used u L ∞ (0,T ;W 1,6 ) ≤ v L ∞ (0,T ;W 1,6 ) + w L ∞ (0,T ;W 1,6 ) ≤ C.…”

“…As in [12], it is easy to prove that ∇ρ L ∞ (0,T ;L q ) + ∇p L ∞ (0,T ;L q ) ≤ C, (3.27) ∇u L 2 (0,T ;L ∞ ) + u L ∞ (0,T ;H 2 ) ≤ C.…”

“…Very recently, Huang and Wang [12] establish the following regularity criterion ρ, θ, b ∈ L ∞ (0, T ; L ∞ ) with 2µ > λ.…”