We show that t 3/4 u(·, t) L ∞ (R 3 ) → 0 as t → ∞ for all Leray-Hopf's global weak solutions u(·, t) of the incompressible Navier-Stokes equationswhere e ∆t is the heat semigroup, as well as other fundamental new results.In spite of the complexity of the questions, our approach is elementary and is based on standard tools like conventional Fourier and energy methods.

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We strengthen the classic result about the regularity time t * of arbitrary Leray solutions of the Navier-Stokes equations in R n (n = 3, 4), which have the form tif n = 4 (in particular, by reducing the current known values for the constants K 3 , K 4 ). Some related results are also included in our discussion.

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