We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.

中文翻译：

---

针对Korteweg系统的全球大型解决方案和最佳的时间衰减估计

我们证明了Korteweg系统的整体解，而没有对速度不可压缩部分的垂直分量施加小的条件。引入了不可压缩的Navier-Stokes方程式众所周知的加权Chemin-Lerner-norm技术来推导先验估计。作为副产品，我们通过使用纯能量参数（与光谱分析无关）来获得溶液的最佳时间衰减率。与可压缩的Navier-Stokes系统相比，由于Korteweg张量的平滑作用，时间衰减估计更加准确。