Abstract
In this paper we consider the Cauchy problem for the 3D navier-Stokes equations for incompressible flows. The initial data are assume d to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solution can develop singularities in finite time. Assuming the maximal interval of existence to be finite, we give a unified discussion of various known solution properties as time approaches the blow-up time.
Highlights
The Navier–Stokes equations are of fundamental importance in continuum mechanics
The proof of Theorem 1.2 is completed by method (III) after we consider, in Section 4, the important solution norms u(·, t) Lq, 3 < q ≤ ∞. If any of these norms stays bounded in some interval 0 ≤ t < T, we show that Du(·, t) will be bounded in that interval
Using the lower bound on the blow–up of u(·, t) Lr provided by Theorem 1.3, one obtains (5.1) with an algebraic lower bound, as described
Summary
We extend the proof of Theorem 3.1 by using Hölder’s inequality instead of the Cauchy–Schwarz inequality and the Hausdorff–Young inequality (see e.g. [50], p. 104) instead of Parseval’s relation. The lemma shows how nonlinear differential inequalities such as (3.8) above can be used to derive lower bound estimates in case of finite–time blow–up. We will need here the following well known estimate for solutions of the heat equation: given any 1 ≤ r ≤ q ≤ ∞, and any multi–index α, we have, for all t > 0: Dα e∆tv Lq ≤ C. for all v ∈ Lr(IR3), with C > 0 a constant depending only on the values of q, r and | α |. This shows (1.12), (1.15) for q = 3, and using (1.17), the bound (1.4) for q = 3/2, completing the proof of Theorems 1.2 and 1.3 above. It implies, by (1.13) and Gronwall’s lemma, that.
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