DCDS
Time periodic solutions to Navier-Stokes-Korteweg system with friction
Hong Cai Zhong Tan Qiuju Xu
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.
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
KRM
Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type
Hong Cai Zhong Tan Qiuju Xu
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

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