DCDS
Asymptotic behavior of Navier-Stokes-Korteweg with friction in $\mathbb{R}^{3}$
Zhong Tan Xu Zhang Huaqiao Wang
We consider the compressible barotropic Navier-Stokes-Korteweg system with friction in this paper. The global solutions and optimal convergence rates are obtained by pure energy method provided the initial perturbation around a constant state is small enough. In particular, the decay rates of the higher-order spatial derivatives of the solution are obtained. Our proof is based on a family of scaled energy estimates and interpolations among them without linear decay analysis.
keywords: energy method Sobolev interpolation. optimal decay rates Korteweg Compressible Navier-Stokes equations optimal decay rates
DCDS
Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity
Zhong Tan Qiuju Xu Huaqiao Wang
In this paper, the compressible magnetohydrodynamic equations without heat conductivity are considered in $\mathbb{R}^3$. The global solution is obtained by combining the local existence and a priori estimates under the smallness assumption on the initial perturbation in $H^l (l>3)$. But we don't need the bound of $L^1$ norm. This is different from the work [5]. Our proof is based on pure estimates to get the time decay estimates on the pressure, velocity and magnet field. In particular, we use a fast decay of velocity gradient to get the uniform bound of the non-dissipative entropy, which is sufficient to close the priori estimates. In addition, we study the optimal convergence rates of the global solution.
keywords: The compressible magnetohydrodynamic equations without heat conductivity energy estimates. global solution optimal convergence rates

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