A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W−1/q,q
A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W−1/q,q
复制标题
边界数据为 W−1/q,q 的平稳斯托克斯和纳维-斯托克斯方程的一类解
DOI：
10.1007/s00208-004-0573-7
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发表时间：
2005
影响因子：
1.4
通讯作者：
H. Sohr
中科院分区：
文献类型：
--
作者：
G. Galdi;C. Simader;H. Sohr
本文建立了有界域Ω中定常Stokes方程和N-S方程的一类很弱解的一般理论，其边界∂Ω为C2，1，对应于分布空间W−1/Q，Q(∂Ω)，1<；q<；∞中的边界数据。这些解在其存在类中存在且是唯一的(对于小数据，在非线性情况下)，并且满足关于数据的相应估计。此外，如果数据是规则的，它们就会变得规则。据我们所知，对于Stokes情形，得到这种边界数据的解的唯一存在结果是由于Giga，[16]，命题2.2。然而，本文所用的方法和途径不同于Giga的方法和途径，它涵盖了更一般的问题，包括非线性的Navier-Stokes方程和通过解获得边界数据的精确方式。在最后一节中，我们还介绍了Solution类的进一步推广。
Abstract.We develop a theory for a general class of very weak solutions to stationary Stokes and Navier-Stokes equations in a bounded domain Ω with boundary ∂Ω of class C2,1, corresponding to boundary data in the distribution space W−1/q,q(∂Ω), 1<q<∞. These solutions exist and are unique (for small data, in the nonlinear case) in their class of existence, and satisfy a correponding estimate in terms of the data. Moreover, they become regular if the data are regular. To our knowledge, the only existence result for solutions attaining such boundary data is due to Giga, [16], Proposition 2.2, for the Stokes case. However, the methods and the approach used in the present paper are different than Giga’s and cover more general issues, including the nonlinear Navier-Stokes equations and the precise way in which the boundary data are attained by the solutions. We also introduce, in the last section, a further generalization of the solution class.