Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations |
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| Authors: | Yeping Li Zhen Luo |
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| Affiliation: | 1. Department of Mathematics, East China University of Science Technology, Shanghai, China;2. Department of Mathematics, Xiamen University, Xiamen, China |
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| Abstract: | In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd. |
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| Keywords: | rarefaction wave zero‐viscosity‐capillarity limit compressible Navier– Stokes– Korteweg equations energy method |
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