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

Blow-up rate estimates for weak solutions of the Navier-Stokes equations
Blow-up rate estimates for weak solutions of the Navier-Stokes equations
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→∞    Q1/k(x0,t0)Q1/k (x0,t0)Q1/k (x0,t0)Q1/k (x0,t0)Q1/k (x0,t0)Q1/k (x0,t0)
with Q1/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 EkiQ1/ki (x0,t0) for i = 1,2,..., such that the blow-up rate estimate |u(x,t)|  ≥  ε|tt0|-θ/2|x – x 0|-1+θ, (x,t)  ∈ Eki   holds.
Navier-Stokes equations, weak, solutions, interior, regularity, Lorentz, spaces
1364-5021
2625-2642
Chen, Zhi-Min
e4f81e6e-5304-4fd6-afb2-350ec8d1e90f
Price, W.G.
b7888f47-e3fc-46f4-9fb9-7839052ff17c
Chen, Zhi-Min
e4f81e6e-5304-4fd6-afb2-350ec8d1e90f
Price, W.G.
b7888f47-e3fc-46f4-9fb9-7839052ff17c

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).

Record type: Article

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→∞    Q1/k(x0,t0)Q1/k (x0,t0)Q1/k (x0,t0)Q1/k (x0,t0)Q1/k (x0,t0)Q1/k (x0,t0)
with Q1/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 EkiQ1/ki (x0,t0) for i = 1,2,..., such that the blow-up rate estimate |u(x,t)|  ≥  ε|tt0|-θ/2|x – x 0|-1+θ, (x,t)  ∈ Eki   holds.

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More information

Published date: 2001
Keywords: Navier-Stokes equations, weak, solutions, interior, regularity, Lorentz, spaces

Identifiers

Local EPrints ID: 21786
URI: http://eprints.soton.ac.uk/id/eprint/21786
ISSN: 1364-5021
PURE UUID: d8043d49-08fc-4fbb-b0fe-a8ea4f959f12

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Date deposited: 16 Mar 2006
Last modified: 08 Jan 2022 03:47

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Contributors

Author: Zhi-Min Chen
Author: W.G. Price

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