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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On optimal estimates for the Laplace-Leray commutator in planar domains with corners
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by Elaine Cozzi and Robert L. Pego PDF
Proc. Amer. Math. Soc. 139 (2011), 1691-1706 Request permission

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.
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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
  • Email: ecozzi@andrew.cmu.edu, ecozzi@drexel.edu
  • Robert L. Pego
  • Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890
  • MR Author ID: 137455
  • ORCID: 0000-0001-8502-2820
  • Email: rpego@andrew.cmu.edu
  • 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 1691-1706
  • MSC (2010): Primary 35-XX; Secondary 76-XX
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10613-5
  • MathSciNet review: 2763758