Taiwanese Journal of Mathematics

Singularity Formation of the Non-barotropic Compressible Magnetohydrodynamic Equations Without Heat Conductivity

Xin Zhong

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Abstract

We study the singularity formation of strong solutions to the three-dimensional full compressible magnetohydrodynamic equations with zero heat conduction in a bounded domain. We show that for the initial density allowing vacuum, the strong solution exists globally if the density $\rho$, the magnetic field $\mathbf{b}$, and the pressure $P$ satisfy $\|\rho\|_{L^{\infty}(0,T;L^{\infty})} + \|\mathbf{b}\|_{L^{\infty}(0,T;L^6)} + \|P\|_{L^{\infty}(0,T;L^{\infty})} \lt \infty$ and the coefficients of viscosity verity $3\mu \gt \lambda$. This extends the corresponding results in Duan (2017), Fan et al. (2018) [1,2] where a blow-up criterion in terms of the upper bounds of the density, the magnetic field and the temperature was obtained under the condition $2\mu \gt \lambda$. Our proof relies on some delicate energy estimates.

Article information

Source
Taiwanese J. Math., Volume 24, Number 3 (2020), 603-628.

Dates
Received: 24 February 2019
Accepted: 30 June 2019
First available in Project Euclid: 19 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1589875221

Digital Object Identifier
doi:10.11650/tjm/190701

Subjects
Primary: 76W05: Magnetohydrodynamics and electrohydrodynamics 35B65: Smoothness and regularity of solutions

Keywords
non-barotropic compressible magnetohydrodynamic equations strong solutions blow-up criterion zero heat conduction

Citation

Zhong, Xin. Singularity Formation of the Non-barotropic Compressible Magnetohydrodynamic Equations Without Heat Conductivity. Taiwanese J. Math. 24 (2020), no. 3, 603--628. doi:10.11650/tjm/190701. https://projecteuclid.org/euclid.twjm/1589875221


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