# Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains

Research output: Contribution to journalArticle

10 Citations (Scopus)

### Abstract

Consider the nonstationary Stokes equations in exterior domains Ω ⊂ ℝn (n ≥ 3) with the compact boundary ∂Ω. We show first that the solution u(t) decays like ∥u(t)∥r = O(t-n/2(1-1/r)) for all 1 < r ≤ ∞ as t → ∞. This decay rate n/2(1 - 1/r) is optimal in the sense that ∥u(t)∥r = o(t-n/2(1-1/r)) for some 1 < r ≤ ∞ as t → ∞ occurs if and only if the net force exerted by the fluid on ∂Ω is zero.

Original language English 709-730 22 Mathematische Annalen 320 4 https://doi.org/10.1007/s002080100214 Published - 2001 Yes

Exterior Domain
Stokes Equations
Stokes Flow
Decay Rate
Decay
If and only if
Fluid
Zero

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

In: Mathematische Annalen, Vol. 320, No. 4, 2001, p. 709-730.

Research output: Contribution to journalArticle

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AB - Consider the nonstationary Stokes equations in exterior domains Ω ⊂ ℝn (n ≥ 3) with the compact boundary ∂Ω. We show first that the solution u(t) decays like ∥u(t)∥r = O(t-n/2(1-1/r)) for all 1 < r ≤ ∞ as t → ∞. This decay rate n/2(1 - 1/r) is optimal in the sense that ∥u(t)∥r = o(t-n/2(1-1/r)) for some 1 < r ≤ ∞ as t → ∞ occurs if and only if the net force exerted by the fluid on ∂Ω is zero.

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