We prove global existence of finite energy weak solutions to the quantum Navier-Stokes equations in the whole space with non trivial far-field condition in dimensions $d=2,3$. The vacuum regions are included in the weak formulation of the equations. Our method consists in an invading domains approach. More precisely, by using a suitable truncation argument we construct a sequence of approximate solutions. The energy and the BD entropy bounds allow for the passage to the limit in the truncated formulation leading to a finite energy weak solution. Moreover, the result is also valid in the case of compressible Navier-Stokes equations with degenerate viscosity.

我们证明了量子空间Navier-Stokes方程在有限空间内具有非平凡远场条件的有限能量弱解的整体存在性 $d=\mathrm{2个}，3$。真空区域包含在方程的弱公式中。我们的方法包括一种侵入域方法。更准确地说，通过使用适当的截断参数，我们构造了一系列近似解。能量和BD熵范围允许通过截断公式中的极限，从而导致有限的能量弱解。而且，该结果在具有简并粘度的可压缩Navier-Stokes方程的情况下也是有效的。