Royal Society Publishing

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

Zhi–Min Chen, W. G. Price


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) ∈ Omega; × (0,T) if there exist constants 0 ⩽ θ < 1 and small ∈ 0 such that

with Q1/ki (x0,t0) = {xRn; |xx0 < 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 EkiQ1/ki (x0,t0) for i = 1,2,..., such that the blow–up rate estimate


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