Alternative Theorem of Navier-Stokes Equations in $\mathbb{R}^3$

Abstract

We consider Cauchy problem of the incompressible Navier-Stokes equations with initial data u0 ∈ L 1 (R 3 ) ∩ L ∞ (R 3 ). There exist a maximum time interval [0, Tmax) and a unique solution u ∈ C [0, Tmax); L 2 (R 3 ) ∩ L p (R 3 ) (∀p > 3). We find one of function class S regular defined by scaling invariant norm pair such that Tmax = ∞ provided u0 ∈ S regular . Especially, u0 L p is arbitrarily large for any u0 ∈ S regular and p > 3. On the other hand, the alternative theorem is proved. It is that either Tmax = ∞ or Tmax ∈ (T l , Tr]. Especially, Tr < Tmax < ∞ is disappearing. Here the explicit expressions of T l and Tr are given. This alternative theorem is one kind of regular criterion which can be verified by computer. If Tmax = ∞, the solution u is regular for any (t, x) ∈ (0, ∞) × R 3 . As t → ∞, the solution is decay. On the other hand, lower bound of blow up rate of u is obtained again provided Tmax ∈ (T l , Tr].