On optimal estimates for the Laplace-Leray commutator in planar domains with corners
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- by Elaine Cozzi and Robert L. Pego
- Proc. Amer. Math. Soc. 139 (2011), 1691-1706
- DOI: https://doi.org/10.1090/S0002-9939-2010-10613-5
- Published electronically: October 18, 2010
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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.References
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Bibliographic 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