Solvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial Trace of a Vector Field on Sobolev-Class Domains

Solvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial... We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field u are prescribed in an open, bounded, Sobolev-class domain $${\Omega \subseteq \mathbb{R}^{\rm n}}$$ Ω ⊆ R n , and either the normal component $${{\bf u} \cdot {\bf N}}$$ u · N or the tangential components of the vector field $${{\bf u} \times {\bf N}}$$ u × N are prescribed on the boundary $${\partial \Omega}$$ ∂ Ω . For $${{\rm k} > {\rm n}/2}$$ k > n / 2 , we prove that u is in the Sobolev space $${H^{\rm k+1}(\Omega)}$$ H k + 1 ( Ω ) if $${\Omega}$$ Ω is an $${H^{\rm k+1}}$$ H k + 1 -domain, and the divergence, curl, and either the normal or tangential trace of u has sufficient regularity. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients, and with a rather general set of Dirichlet and Neumann boundary conditions. The resulting regularity theory for the vector u is fundamental in the analysis of free-boundary and moving interface problems in fluid dynamics. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

Solvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial Trace of a Vector Field on Sobolev-Class Domains

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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-016-0289-y
Publisher site
See Article on Publisher Site

Abstract

We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field u are prescribed in an open, bounded, Sobolev-class domain $${\Omega \subseteq \mathbb{R}^{\rm n}}$$ Ω ⊆ R n , and either the normal component $${{\bf u} \cdot {\bf N}}$$ u · N or the tangential components of the vector field $${{\bf u} \times {\bf N}}$$ u × N are prescribed on the boundary $${\partial \Omega}$$ ∂ Ω . For $${{\rm k} > {\rm n}/2}$$ k > n / 2 , we prove that u is in the Sobolev space $${H^{\rm k+1}(\Omega)}$$ H k + 1 ( Ω ) if $${\Omega}$$ Ω is an $${H^{\rm k+1}}$$ H k + 1 -domain, and the divergence, curl, and either the normal or tangential trace of u has sufficient regularity. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients, and with a rather general set of Dirichlet and Neumann boundary conditions. The resulting regularity theory for the vector u is fundamental in the analysis of free-boundary and moving interface problems in fluid dynamics.

Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Sep 12, 2016

References

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