On L^3,∞-stability of the Navier-Stokes system in exterior domains

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• 作者：Hajime Koba ^{iti@sigmath.es.osaka-u.ac.jp}
• 关键词：Asymptotic stability ; Weak L3L3-space ; Lorentz space ; Oseen semigroup ; L3 ; &infin ; L3 ; &infin ; -asymptotic stability ; L&infin ; L&infin ; -decay
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：262
• 期：3
• 页码：2618-2683
• 全文大小：743 K
• 卷排序：262

文摘

This paper studies the stability of a stationary solution of the Navier–Stokes system with a constant velocity at infinity in an exterior domain. More precisely, this paper considers the stability of the Navier–Stokes system governing the stationary solution which belongs to the weak L3L3-space L3,∞L3,∞. Under the condition that the initial datum belongs to a solenoidal L3,∞L3,∞-space, we prove that if both the L3,∞L3,∞-norm of the initial datum and the L3,∞L3,∞-norm of the stationary solution are sufficiently small then the system admits a unique global-in-time strong L3,∞L3,∞-solution satisfying both L3,∞L3,∞-asymptotic stability and L∞L∞-asymptotic stability. Moreover, we investigate L3,rL3,r-asymptotic stability of the global-in-time solution. Using LpLp–LqLq type estimates for the Oseen semigroup and applying an equivalent norm on the Lorentz space are key ideas to establish both the existence of a unique global-in-time strong (or mild) solution of our system and the stability of our solution.