Existence, uniqueness and Lq-estimates for the stokes problem in an exterior domain
Existence, uniqueness and Lq-estimates for the stokes problem in an exterior domain
复制标题
外域斯托克斯问题的存在性、唯一性和Lq估计
DOI：
10.1007/bf02384076
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发表时间：
1990
影响因子：
2.5
通讯作者：
C. Simader
中科院分区：
文献类型：
--
作者：
G. Galdi;C. Simader
众所周知，斯托克斯系统解的性质研究在纳维-斯托克斯方程控制的粘性流体流动的数学理论中起着至关重要的作用。毫不夸张地说，斯托克斯系统的索博列夫空间中的存在性、唯一性和先验估计是与纳维-斯托克斯方程相关的每个基本问题的基础，例如存在性、正则性、时间渐近行为等，参见，例如，LADYZHENSKAYA (1969)、SOLONNI~:OV (1977) 和 TE~AM (1977)。然而到目前为止，斯托克斯系统仅针对有界域进行了全面分析，参见 SOBOLEVSKn (1960)、LADVZnENSKAYA (1959)、SOLONNU~OV (1960)、VOROVlCH & Yot:oowcr~ (1961)、CATTABRmA (1961)。特别是，在 Cattabriga 的论文中，对于空间维度 n = 3 ，在 Sobotev 空间 W ~'~, l ~ -1, 1 < q < c ~ 中给出了完整的处理。 Cattabriga 的结果本质上表明，如果 .(2 是 C 类 R a 中的有界域 m, m = max (l + 2, 2), l ~ --1，则给定 f E [wt'q(Y2)] 3, gC Wt+l'q(O), q~ E [Wl+2-1!q'q(8o)] a, 1 < q < ~ ，则存在一个且仅有一个（分布式）解 v C [W1+Z'q(-Q)] 3, p E Wt+l'q(o)，到斯托克斯系统
As is well known, the study of the properties of solutions to the Stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the Navier-Stokes equations. It is not art overstatement to say that existence, uniqueness and a priori estimates in Sobolev spaces for the Stokes system are fundamental in every basic question related to the Navier-Stokes equations, such as existence, regularity, asymptotic behavior in time, etc., see, e.g., LADYZHENSKAYA (1969), SOLONNI~:OV (1977) and TE~AM (1977). So far however the Stokes system has been fully analyzed only for bounded domains, see SOBOLEVSKn (1960), LADVZnENSKAYA (1959), SOLONNU~OV (1960), VOROVlCH & Yot:oowcr~ (1961), CATTABRmA (1961). In particular, in the paper of Cattabriga a complete treatment is given in the Sobotev spaces W ~'~, l ~ -1, 1 < q < c ~ for space dimension n = 3 . Cattabriga's results essentially state that if .(2 is a bounded domain in R a of class C m, m = max (l + 2, 2), l ~ --1, then given f E [wt'q(Y2)] 3, gC Wt+l'q(O), q~ E [Wl+2-1!q'q(8o)] a, 1 < q < ~ , there exists one and only one (distributional) solution v C [W1+Z'q(-Q)] 3, p E Wt+l'q(o), to the Stokes system