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 only. But we don't need the bound of norm. Moreover, the – 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.
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.