Asymptotics toward rarefaction wave for an inflow problem of the compressible Navier–Stokes–Korteweg equation

In this article, we are concerned with the large-time behavior of solutions to an inflow problem in one-dimensional case for the Navier–Stokes–Korteweg equation, which models compressible fluids with internal capillarity. We first investigate that the asymptotic state is the rarefaction wave under the proper condition of the far fields and boundary values. The asymptotic stability of the rarefaction wave under some smallness conditions is shown. The proof is completed by the energy method with the help of time-decay estimate for the rarefaction wave.

Keywords

compressible Navier–Stokes–Korteweg equation, inflow problem, rarefaction wave, asymptotics, energy method

2010 Mathematics Subject Classification

Primary 76W05. Secondary 35B40.

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The research is supported in part by the National Natural Science Foundation of China (Grant No. 12171258).

Received 25 February 2022

Published 23 May 2022