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In this paper, I shall construct a family of global smooth solutions with large initial data for the incompressible Navier–Stokes equations, which are allowed to have a large three-dimensional part and a small three-dimensional part. In addition, one also acquires a global well-posedness with low regular data to the incompressible Navier–Stokes equations.

Smooth solution for incompressible Navier–Stokes equations with large initial