论文标题
用$ l _ {\ infty} $对称椭圆系数张量以及应用于Stokes和Navier-Stokes边界问题的应用程序问题的变化方法的层势方法。
Variational approach for layer potentials of the Stokes system with $L_{\infty }$ symmetrically elliptic coefficient tensor and applications to Stokes and Navier-Stokes boundary problems
论文作者
论文摘要
本文的第一个目的是在Lipschitz的基于$ l_2 $的加权Sobolev空间中开发一层潜在理论，用于$ {\ Mathbb r}^n $，$ n \ geq 3 $，用于$ l_ {矩阵。为此，我们探索了等效的混合变异配方，并证明了$ {\ Mathbb r}^n $的各向异性Stokes System的某些传输问题的适当性，并在$ l_2 $ b的加权Sobolev空间中具有给定数据。这些结果用于定义牛顿和层势并获得其特性。然后，我们分析了外部Dirichlet，Neumann的良好性，以及使用$ L _ {\ infty} $对称椭圆系数张量的Stokes系统的混合问题。其中一些问题的解决方案也以各向异性的牛顿和层电位为代表。最后，我们证明了在$ {\ Mathbb r}^3 $中的互补Lipschitz域中存在的薄弱解决方案，用于各向异性Navier-Stokes System，其中包含$ L_2 $的加权Sobolev Space的常规数据。该分析依赖于有界域家族中各向异性Navier-Stokes系统的Dirichlet问题以及Leray-Schauder固定点定理的存在结果。
The first aim of this paper is to develop a layer potential theory in $L_2$-based weighted Sobolev spaces on Lipschitz bounded and exterior domains of ${\mathbb R}^n$, $n\geq 3$, for the anisotropic Stokes system with $L_{\infty }$ viscosity coefficient tensor satisfying an ellipticity condition for symmetric matrices. To do this, we explore equivalent mixed variational formulations and prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of ${\mathbb R}^n$, with the given data in $L_2$-based weighted Sobolev spaces. These results are used to define the Newtonian and layer potentials and to obtain their properties. Then we analyze well-posedness of the exterior Dirichlet, Neumann and mixed problems for the Stokes system with $L_{\infty }$ symmetrically elliptic coefficient tensor. Solutions of some of these problems are also represented in terms of the anisotropic Stokes Newtonian and layer potentials. Finally, we prove the existence of a weak solution for a transmission problem in complementary Lipschitz domains in ${\mathbb R}^3$ for the anisotropic Navier-Stokes system with general data in $L_2$-based weighted Sobolev spaces. The analysis relies on an existence result for a Dirichlet problem for the anisotropic Navier-Stokes system in a family of bounded domains, and on the Leray-Schauder fixed point theorem.
