Blow-up rate estimates for weak solutions of the Navier-Stokes equations

Chen, Zhi-Min and Price, W.G. (2001) Blow-up rate estimates for weak solutions of the Navier-Stokes equations. Proceedings of the Royal Society A, 457 (2015), 2625-2642. (doi:10.1098/rspa.2001.0854).

Abstract

The interior regularity problem for the Leray weak solutions u of the Navier—Stokes equations in a domain Ω ⊂ Rⁿ with n ≥ 3 is investigated. It is shown that u is regular in a neighbourhood of a point (x₀,t₀) ∈ Ω × (0, T) if there exist constants 0 ≤ θ < 1 and small ε > 0 such that

lim       ess sup     |t – t₀|^θ/2|x – x₀|^1-θ |u(x,t)| < ε
k→∞    Q_1/k(x₀,t₀)Q_1/k (x0,t0)Q_1/k (x0,t0)Q_1/k (x0,t0)Q_1/k (x0,t0)Q_1/k (x0,t0)
with Q_1/k (x₀,t₀) = {x ∈ Rⁿ; |x – x₀| < 1/k} × (t₀ – 1/k²,t⁰ + 1/k²).  
If (x₀,t₀) is an irregular point of u, there exists a sequence of non-zero measure
sets E_ki ⊂ Q_1/ki (x₀,t₀) for i = 1,2,..., such that the blow-up rate estimate |u(x,t)|  ≥  ε|t – t₀|^-θ/2|x – x ₀|^-1+θ, (x,t)  ∈ E_ki   holds.

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