Global well-posedness and exponential decay for the inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation

We consider the Cauchy problem for the inhomogeneous incompressible logarithmical hyper-dissipative Navier-Stokes equations in higher dimensions. By means of the Littlewood-Paley techniques and new ideas, we establish the existence and uniqueness of the global strong solution with vacuum over the whole space R n \mathbb {R}^{n} . Moreover, we also obtain the exponential decay-in-time of the strong solution. Our result holds without any smallness on the initial data and the initial density is allowed to have vacuum.