arXiv 1409.46662014-09-16math.AP

二维有界区域中具有混合边界条件的斯托克斯方程的解

Solutions to the Navier-Stokes Equations with Mixed Boundary Conditions in Two-Dimensional Bounded Domains

Michal Beneš, Petr Kučera

备注：: 19 pages, 3 figures

摘要：

本文考虑具有混合边界条件的非定常斯托克斯方程组。我们研究了该系统解的存在唯一性。我们定义的Banach空间$X$和$Y$，分别是这个问题的“可能”的解决方案和空间的数据。我们定义的运营商$\mathcal{N}:X\rightarrow Y$和制定我们的问题的运营商方程。令$\mathbf{u}\in X$和${{\mathcal G}_{\mathcal P}}_{\mathbf{u}}: X\rightarrow Y$为$\mathcal{N}$在$\mathbf{u}$的弗雷歇导数。我们证明了${{\mathcal G}_{\mathcal P}}_{\mathbf{u}}$是一对一的，并在$Y$上。因此，假设系统可以用一些给定的数据（初速度和右手边）来解。然后，存在一个唯一的解决方案，该系统的数据是小扰动的前。本文附录中的另一个结果是$W^{2,2}$ -充分光滑数据下定常Stokes方程组混合边界条件解的正则性。

英文摘要：

In this paper we consider the system of the non-steady Navier-Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces $X$ and $Y$, respectively, to be the space of "possible" solutions of this problem and the space of its data. We define the operator $\mathcal{N}:X\rightarrow Y$ and formulate our problem in terms of operator equations. Let $\mathbf{u}\in X$ and ${{\mathcal G}_{\mathcal P}}_{\mathbf{u}}: X\rightarrow Y$ be the Frechet derivative of $\mathcal{N}$ at $\mathbf{u}$. We prove that ${{\mathcal G}_{\mathcal P}}_{\mathbf{u}}$ is one-to-one and onto $Y$. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. Next result proved in the Appendix of this paper is $W^{2,2}$- regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.

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摘要：

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