Global existence and convergence rates of solutions for the 3D compressible magnetohydrodynamic equations without heat conductivity

ABSTRACT

The Cauchy problem for the 3D compressible Navier–Stokes and magnetohydrodynamic equations without heat conductivity is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data is small perturbations of some given constant state in the framework of Sobolev space $H^3\left({\mathbb{R}}^3\right)$ only. But we don't need the bound of $L^1$ norm. Moreover, the $L^2$–$L^2$ convergence rates are also obtained for the solution. This is different from the works of Duan and Ma [Global existence and convergence rates for the 3-D compressible Navier–Stokes equations without heat conductivity. Indiana Univ Math J. 2008;57:2299–2319] and Tan and Wang [On hyperbolic-dissipative systems of composite type. J Diff Eq. 2016;260:1091–1125]. Our proof is based on the benefit of the low-frequency and high-frequency decompositions, here, we just need spectral analysis of the low-frequency part of the Green function to the linearized system, so that we succeed to avoid some complicate analysis.

KEYWORDS:

• Navier–Stokes equations
• magnetohydrodynamics
• global existence
• convergence rates

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Q30
• 76N15
• 76P05
• 82C40

Acknowledgments

We would like to express our sincere thanks to Professor Zheng-an Yao for their fruitful help and discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China (Grant Nos.11501373, 11701380, and 11271381), the Natural Science Foundation of Guangdong Province (Nos. 2017A030307022, 2016A030310019, and 2016A030307042), the Guangdong Provincial Culture of Seedling of China (No. 2013LYM0081), the Education Research Platform Project of Guangdong Province (No. 2014KQNCX208), and the Education Reform Project of Guangdong Province (No. 2015558).