The generalized three dimensional Navier–Stokes equations with damping are considered. Firstly, existence and uniqueness of strong solutions in the periodic domain \({\mathbb {T}}^{3}\) are proved for \(\frac{1}{2}<\alpha <1,~~ \beta +1\ge \frac{6\alpha }{2\alpha -1}\in (6,+\infty )\). Then, in the whole space \(R^3,\) if the critical situation \(\beta +1= \frac{6\alpha }{2\alpha -1}\) and if \(u_{0}\in H^{1}(R^{3}) \bigcap {\dot{H}}^{-s}(R^{3})\) with \(s\in [0,1/2]\), the decay rate of solution has been established. We give proofs of these two results, based on energy estimates and a series of interpolation inequalities, the key of this paper is to give an explanation for that on the premise of increasing damping term, the well-posedness and decay can still preserve at low dissipation \(\alpha <1,\) and the relationship between dissipation and damping is given.

考虑带有阻尼的广义三维纳维-斯托克斯方程。首先，证明了周期域\({\mathbb {T}}^{3}\)中强解的存在唯一性，对于\(\frac{1}{2}<\alpha <1,~~ \beta +1\ge \frac{6\alpha }{2\alpha -1}\in (6,+\infty )\)。那么，在整个空间\(R^3,\)中，如果临界情况\(\beta +1= \frac{6\alpha }{2\alpha -1}\)并且如果\(u_{0}\在 H^{1}(R^{3}) \bigcap {\dot{H}}^{-s}(R^{3})\)和\(s\in [0,1/2]\ )，溶液的衰减率已确定。我们对这两个结果进行了证明，基于能量估计和一系列插值不等式，本文的关键是解释在增加阻尼项的前提下，适定性和衰减仍然可以保持在较低的水平给出了耗散\(\alpha <1,\)以及耗散和阻尼之间的关系。