Low Mach number limit of the three-dimensional full compressible Navier–Stokes–Korteweg equations

  • Kaijian Sha
  • Yeping LiEmail author


In this paper, we justify the low Mach number limit for the three-dimensional full compressible Navier–Stokes–Korteweg equations rigorously within the framework of smooth solution. Under the assumptions of small density and temperature perturbation, we show that for sufficiently small Mach number, the initial-value problem of the three-dimensional full compressible Navier–Stokes–Korteweg equations admits a unique smooth solution on the time interval where the smooth solution of the corresponding incompressible Navier–Stokes equations exists. Moreover, we obtain the convergence of smooth solutions for the full compressible Navier–Stokes–Korteweg equations toward those for the incompressible Navier–Stokes equations with a convergence rate.


Full compressible Navier–Stokes–Korteweg equations Low Mach number limit Incompressible Navier–Stokes equations Error estimate 

Mathematics Subject Classification

35B35 35B40 76N15 



We are grateful to two anonymous referees for valuable comments which greatly improved our original manuscript. The research is supported in partial by the National Science Foundation of China (Grant No. 11671134).


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsEast China University of Science and TechnologyShanghaiPeople’s Republic of China

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