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Time periodic solutions to Navier-Stokes-Korteweg system with friction

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  • 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.
    Mathematics Subject Classification: Primary: 35M10, 35Q35; Secondary: 35B10.

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