Abstract
The Stokes operator is a differential-integral operator induced by the Stokes equations. In this paper, we analyze the Stokes operator from the point of view of the Helmholtz minimum dissipation principle. We show that, through the Hodge orthogonal decomposition, a pair of bounded linear operators, a restriction operator and an extension operator, are induced from the divergence-free constraint. As a consequence, we use it to calculate the eigenvalues of the Stokes operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Constantin, P. and Foias, C. The Navier-Stokes Equations, University of Chicago Press, Chicago (1992)
Temam, R. Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd Ed., SIAM, Philadelphia (1995)
Temam, R. Navier-Stokes Equations: Theory and Numerical Analysis, 3rd Ed., North-Holland, Amsterdam (1984)
Abels, H. and Teresawa, Y. On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344(2), 381–429 (2009)
Bae, Hyeong-Ohk. Analyticity for the Stokes operator in Besov spaces. J. Korean Math. Soc. 40(6), 1061–1074 (2003)
Lee, D.-S. and Rummler, B. The eigenfunctions of the Stokes operator in special domains. ZAMM 82(6), 399–407 (2002)
Turinsky, V. V. A lower bound for the principle eigenvalue of the Stokes operator in a random domain. Ann. Inst. H. Poincaré Probab. Statist. 44(1), 1–18 (2008)
Wang, L. Z., Li, D. S., and Li, K. T. Eigenvalue problem of the Stokes operator on the spherical gap region and its application (in Chinese). Applied Mathematics, A Journal of Chinese Universities (Series A) 16(2), 87–96 (2001)
Chen, B., Li, X. W., and Liu, G. L. Application of the Hodge decomposition in the minimum dissipation principle (in Chinese). Mathematics in Practice and Theory 39(8), 197–200 (2009)
Lu, W. D. Variational Methods in Differential Equations (in Chinese), Science Press, Beijing (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contributed by Gao-lian LIU
Project supported by the National Natural Science Foundation of China (No. 10772103) and the Shanghai Leading Academic Discipline Project (No. Y0103)
Rights and permissions
About this article
Cite this article
Chen, B., Li, Xw. & Liu, Gl. Constraint-induced restriction and extension operators with applications. Appl. Math. Mech.-Engl. Ed. 30, 1345–1352 (2009). https://doi.org/10.1007/s10483-009-1101-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-009-1101-x
Key words
- Stokes operator
- induced operators
- restriction and extension
- variational method
- Hodge decomposition
- eigenvalue problem