Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On optimal estimates for the Laplace-Leray commutator in planar domains with corners

Authors: Elaine Cozzi and Robert L. Pego
Journal: Proc. Amer. Math. Soc. 139 (2011), 1691-1706
MSC (2010): Primary 35-XX; Secondary 76-XX
Published electronically: October 18, 2010
MathSciNet review: 2763758
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For smooth domains, Liu et al. (Comm. Pure Appl. Math. 60: 1443-1487, 2007) used optimal estimates for the commutator of the Laplacian and the Leray projection operator to establish well-posedness of an extended Navier-Stokes dynamics. In their work, the pressure is not determined by incompressibility, but rather by a certain formula involving the Laplace-Leray commutator. A key estimate of Liu et al. controls the commutator strictly by the Laplacian in $ L^2$ norm at leading order. In this paper we show that this strict control fails in a large family of bounded planar domains with corners. However, when the domain is an infinite cone, we find that strict control may be recovered in certain power-law weighted norms.

References [Enhancements On Off] (What's this?)

  • 1. Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
  • 2. V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997. MR 1469972
  • 3. Jian-Guo Liu, Jie Liu, and Robert L. Pego, Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure Appl. Math. 60 (2007), no. 10, 1443–1487. MR 2342954,
  • 4. R. Rostamian and A. M. Soane.
    Variational Problems in Weighted Sobolev Spaces on Non-smooth Domains, Quart. Appl. Math., 68:439-458, 2010.
  • 5. Hermann Sohr, The Navier-Stokes equations, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2001. An elementary functional analytic approach. MR 1928881

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35-XX, 76-XX

Retrieve articles in all journals with MSC (2010): 35-XX, 76-XX

Additional Information

Elaine Cozzi
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890
Address at time of publication: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104

Robert L. Pego
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890

Received by editor(s): December 18, 2009
Received by editor(s) in revised form: May 19, 2010
Published electronically: October 18, 2010
Additional Notes: This material is based upon work supported by the National Science Foundation under Grants No. DMS06-04420 and DMS09-05723 and partially supported by the Center for Nonlinear Analysis (CNA) under National Science Foundation Grant No. DMS06-35983.
Communicated by: Walter Craig
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.