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DCDS

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.

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.

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