A maximal regularity estimate for the non-stationary Stokes equation in the strip

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• 作者：Antoine Choffrut ; Camilla Nobili ; ^{camilla.nobili@unibas.ch" class="auth_mail" title="E-mail the corresponding author} ; Felix Otto
• 关键词：Non-stationary Stokes equations ; No-slip boundary condition ; Maximal regularity ; Real interpolation
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 April 2016
• 年：2016
• 卷：260
• 期：7
• 页码：5589-5626
• 全文大小：401 K

文摘

In a d  -dimensional strip with 17ca064e7fd03a42652f873" title="Click to view the MathML source">d≥2$d\ge 2$, we study the non-stationary Stokes equation with no-slip boundary condition in the lower and upper plates and periodic boundary condition in the horizontal directions. In this paper we establish a new maximal regularity estimate in the real interpolation norm where the brackets 〈⋅〉$〈\cdot 〉$ denote the horizontal-space and time average. The norms involved in the definition of ‖⋅‖_(0,1)${\Vert \cdot \Vert }_{(0,1)}$ are critical for two reasons: the exponents are borderline for the Calderón–Zygmund theory and the weight 1/z$1/z$ just fails to be Muckenhoupt. Therefore, the estimate is only true under horizontal bandedness condition (i.e. a restriction to a packet of wave numbers in Fourier space). The motivation to express the maximal regularity in such a norm comes from an application to the Rayleigh–Bénard problem (see [5]).