Well-Posedness and $$L^2$$ -Decay Estimates for the Navier–Stokes Equations with Fractional Dissipation and Damping

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.