Global and singular solution to a nonlocal model of three-dimensional incompressible Navier–Stokes equations

We in this paper study the singularity formation and global well-posedness of a nonlocal model for some initial boundary condition with a real parameter, which is a one dimensional weak advection model for the three dimensional incompressible Navier–Stokes equations. Based on the Lyapunov functional and contradiction argument, we can prove that the inviscid nonlocal model develops a finite time blowup solution with some even initial data. But, for some special positive parameter and initial data with the given symbol, the inviscid model also has a global smooth solution by the characteristic’ method. Furthermore, by the energy estimations and Gagliardo–Nirenberg inequality, we also obtain that the viscous nonlocal model has a unique global solution with some initial data with the given symbol for all nonnegative parameter. More specially, there is a particular model to the nonlocal model such that the global solution to this model exists for some negative parameter.