$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains
$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains
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DOI：
10.1512/iumj.2009.58.3605
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发表时间：
2009-10
影响因子：
1.1
通讯作者：
H. Kozono;T. Yanagisawa
中科院分区：
文献类型：
--
作者：
H. Kozono;T. Yanagisawa
证明了Ω上的每个L r-向量场可以唯一地分解为具有标量势和向量势的两个空间以及调和向量场空间，其中Ω是R3中具有光滑边界∂Ω的有界域.我们的分解由两种边界条件组成，如u。V|∂Ω=0和u xν|∂Ω=0，其中ν表示向外垂直于∂Ω的单位。我们的结果可以看作是紧黎曼流形上著名的C-∞形式的De Rham-Hodge-Kodaira分解到Ω上的L r-向量场的推广。作为应用，得到了Ω中不可压缩流体的广义毕奥-萨伐尔定律。在此基础上，利用roto u和div u给出了高阶导数在L r中的各种界。
We show that every L r -vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Ω is a bounded domain in R 3 with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u . v |∂Ω = 0 and u x ν|∂Ω = 0, where ν denotes the unit outward normal to ∂Ω. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of C ∞ -forms on compact Riemannian manifolds into L r -vector fields on Ω. As an application, the generalized Biot-Savart law for the incompressible fluids in Ω is obtained. Furthermore, various bounds of u in L r for higher derivatives are given by means of rot u and div u.