Lr-variational inequality for vector fields and the helmholtz-weyl decomposition in bounded domains

Hideo Kozono, Taku Yanagisawa

Research output: Contribution to journalArticle

37 Citations (Scopus)

Abstract

We show that every Lr-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 ℝ3 with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u-v ∂Ω = 0 and u × ∂Ω = 0, where v 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 Lr-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 Lr for higher derivatives are given by means of rot u and div u.

Original languageEnglish
Pages (from-to)1853-1920
Number of pages68
JournalIndiana University Mathematics Journal
Volume58
Issue number4
DOIs
Publication statusPublished - 2009
Externally publishedYes

Fingerprint

Hermann Von Helmholtz
Variational Inequalities
Bounded Domain
Vector Field
Hodge Decomposition
Unit normal vector
Decompose
Vector Potential
Incompressible Fluid
Compact Manifold
Vector space
Riemannian Manifold
Harmonic
Scalar
Denote
Boundary conditions
Derivative
Operator
Form

Keywords

  • Betti number
  • Div-curl lemma
  • Harmonic vector fields
  • L-vector fields

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Lr-variational inequality for vector fields and the helmholtz-weyl decomposition in bounded domains. / Kozono, Hideo; Yanagisawa, Taku.

In: Indiana University Mathematics Journal, Vol. 58, No. 4, 2009, p. 1853-1920.

Research output: Contribution to journalArticle

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