DCDS
Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity
Zhong Tan Qiuju Xu Huaqiao Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 5083-5105 doi: 10.3934/dcds.2015.35.5083
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
KRM
Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type
Hong Cai Zhong Tan Qiuju Xu
Kinetic & Related Models 2015, 8(1): 29-51 doi: 10.3934/krm.2015.8.29
In this paper, the non-isentropic compressible Navier-Stokes-Korteweg system with a time periodic external force is considered in $\mathbb{R}^n$. The optimal time decay rates are obtained by spectral analysis. Using the optimal decay estimates, we show that the existence, uniqueness and time-asymptotic stability of time periodic solutions when the space dimension $n\geq 5$. Our proof is based on a combination of the energy method and the contraction mapping theorem.
keywords: optimal time decay rates energy estimates. Non-isentropic compressible Navier-Stokes-Korteweg system time periodic solution
DCDS
Time periodic solutions to Navier-Stokes-Korteweg system with friction
Hong Cai Zhong Tan Qiuju Xu
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 611-629 doi: 10.3934/dcds.2016.36.611
In this paper, the compressible Navier-Stokes-Korteweg system with friction is considered in $\mathbb{R}^3$. Via the linear analysis, we show the existence, uniqueness and time-asymptotic stability of the time periodic solution when a time periodic external force is taken into account. Our proof is based on a combination of the energy method and the contraction mapping theorem. In particular, this is the first paper that a time periodic solution can be obtained in the whole space $\mathbb{R}^3$ only under the suitable smallness condition of $H^{N-1}\cap L^1$--norm$(N\geq5)$ of time periodic external force.
keywords: optimal time decay rates Navier-Stokes-Korteweg system with friction time periodic solution energy estimates.

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