Lr-variational Inequality for Vector Fields and the Helmholtz-Weyl Decomposition in Bounded Domains

被引:72
作者
Kozono, Hideo [1 ]
Yanagisawa, Taku [2 ]
机构
[1] Tohoku Univ, Math Inst, Sendai, Miyagi 9808578, Japan
[2] Nara Womens Univ, Dept Math, Nara 6308506, Japan
关键词
L-r-vector fields; harmonic vector fields; Betti number; div-curl lemma; 3-D EULER EQUATIONS; BLOW-UP; SOBOLEV INEQUALITY; SMOOTH SOLUTIONS; OPERATOR ROT; BESOV-SPACES; CRITERION;
D O I
10.1512/iumj.2009.58.3605
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We show that every L-r-vector field on Omega can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Omega is a bounded domain in R-3 with the smooth boundary partial derivative Omega. Our decomposition consists of two kinds of boundary conditions such as u . v|(partial derivative Omega) = 0 and u x v |(partial derivative Omega) = 0, where v denotes the unit outward normal to partial derivative Omega. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of C-infinity-forms on compact Riemannian manifolds into L-r-vector fields on Omega. As an application, the generalized Blot-Savart law for the incompressible fluids in Omega is obtained. Furthermore, various bounds of u in L-r for higher derivatives are given by means of rot u and div u.
引用
收藏
页码:1853 / 1920
页数:68
相关论文
共 36 条
[1]   ESTIMATES NEAR BOUNDARY FOR SOLUTIONS OF ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS SATISFYING GENERAL BOUNDARY CONDITIONS .2. [J].
AGMON, S ;
DOUGLIS, A ;
NIRENBERG, L .
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1964, 17 (01) :35-&
[2]  
[Anonymous], 1966, GRUNDLEHREN MATH WIS
[3]  
[Anonymous], 1971, Trudy Mat. Inst. Steklov., V116, P237
[4]  
[Anonymous], SER ADV MATH APPL SC
[5]   REMARKS ON THE BREAKDOWN OF SMOOTH SOLUTIONS FOR THE 3-D EULER EQUATIONS [J].
BEALE, JT ;
KATO, T ;
MAJDA, A .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1984, 94 (01) :61-66
[6]   A VARIATIONAL APPROACH FOR THE VECTOR POTENTIAL FORMULATION OF THE STOKES AND NAVIER-STOKES PROBLEMS IN 3-DIMENSIONAL DOMAINS [J].
BENDALI, A ;
DOMINGUEZ, JM ;
GALLIC, S .
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 1985, 107 (02) :537-560
[7]  
Bourguignon J. P., 1974, J. Funct. Anal, V15, P341
[8]  
Chae D, 2004, ASYMPTOTIC ANAL, V38, P339
[9]   On the Euler equations in the critical Triebel-Lizorkin spaces [J].
Chae, D .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2003, 170 (03) :185-210
[10]  
Duvaut G., 1976, GRUNDLEHREN MATH WIS, V219