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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
DOI: https://doi.org/10.1090/S0002-9939-2010-10613-5
Published electronically: October 18, 2010
MathSciNet review: 2763758
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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
Email: rpego@andrew.cmu.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10613-5
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.