In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Euler–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Euler–Korteweg equations, we establish a convergence-stability principle. Then we show that when the Mach number is sufficiently small, the initial-value problem of the compressible Euler–Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Euler equations have a smooth solution. It is important to remark that when the incompressible Euler equations have a global smooth solution, the existence time of the solution for the compressible Euler–Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Euler–Korteweg equations towards those for the incompressible Euler equations with a convergence rate.

中文翻译：

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可压缩的Euler–Korteweg方程的零马赫数极限

在本文中，我们研究了在光滑解范围内的三维可压缩Euler-Korteweg方程的零马赫数极限。基于可压缩的Euler-Korteweg方程的局部存在理论，我们建立了收敛稳定原理。然后我们表明，当马赫数足够小时，可压缩的Euler–Korteweg方程的初值问题在相应的不可压缩的Euler方程具有平滑解的时间间隔内具有唯一的平滑解。重要的是要指出，当不可压缩的Euler方程具有全局光滑解时，随着Mach数变为零，可压缩的Euler-Korteweg方程的解的存在时间趋于无穷大。此外，