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DCDS
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.
KRM
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.
DCDS
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.
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