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)

 
 

 

A steady-state exterior Navier-Stokes problem that is not well-posed


Author: Giovanni P. Galdi
Journal: Proc. Amer. Math. Soc. 137 (2009), 679-684
MSC (2000): Primary 76D05, 76D03; Secondary 76D07
DOI: https://doi.org/10.1090/S0002-9939-08-09658-5
Published electronically: August 15, 2008
MathSciNet review: 2448590
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the exterior Navier-Stokes problem with zero velocity at infinity is not well-posed in homogeneous Sobolev spaces. This result complements and clarifies well-known previous results obtained by various authors.


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

  • 1. N. Aronszajn and E. Gagliardo, Interpolation Spaces and Interpolation Methods, Ann. Mat. Pura Appl., 68, 1965, 51-117. MR 0226361 (37:1951)
  • 2. W. Borchers and T. Miyakawa, On Stability of Exterior Stationary Navier-Stokes Flows, Acta Math., 174, 1995, 311-382. MR 1351321 (96j:35186)
  • 3. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems. Springer Tracts in Natural Philosophy, vol. 38. Springer-Verlag, New York, 1998 (Revised Edition). MR 1284205 (95i:35216a)
  • 4. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems. Springer Tracts in Natural Philosophy, vol. 39. Springer-Verlag, New York, 1998 (Revised Edition). MR 1284206 (95i:35216b)
  • 5. G. P. Galdi and M. Padula, Existence of Steady Incompressible Flows Past an Obstacle, Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics, RIMS Kokyuroku, Kyoto, vol. 745, 1991, 87-101.
  • 6. G. P. Galdi and C.G. Simader, New Estimates for the Steady-State Stokes Problem in Exterior Domains with Applications to the Navier-Stokes Problem, Differential Integral Equations, 7, 1994, 847-861. MR 1270107 (95c:35192)
  • 7. H. Kozono and M. Yamazaki, Exterior Problem for the Stationary Navier-Stokes Equations in the Lorentz Space, Math. Ann., 310, 1998, 279-305. MR 1602012 (98m:35159)
  • 8. H. Kozono and H. Sohr, On Stationary Navier-Stokes Equations in Unbounded Domains, Ricerche Mat., 42, 1993, 69-86. MR 1283806 (95d:35128)
  • 9. H. Kozono, H. Sohr, and M. Yamazaki, Representation Formula, Net Force and Energy Relation to the Stationary Navier-Stokes Equations in 3-Dimensional Exterior Domains, Kyushu J. Math., 51, 1997, 239-260. MR 1437320 (98g:35163)
  • 10. J. Leray, Étude de Diverses Équations Intégrales non Linéaires et de Quelques Problèmes que Pose l'Hydrodynamique, J. Math. Pures Appl., 12, 1933, 1-82.
  • 11. T. Miyakawa, On Uniqueness of Steady Navier-Stokes Flows in an Exterior Domain, Adv. Math. Sci. Appl., 5, 1995, 411-420. MR 1360998 (97f:35164)
  • 12. S. Smale, An Infinite Dimensional Version of Sard's Theorem, Amer. J. Math., 87, 1965, 861-866. MR 0185604 (32:3067)
  • 13. H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2001. MR 1928881 (2004b:35265)
  • 14. E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Applied Math. Sci., vol. 109, Springer-Verlag, 1995. MR 1347692 (96i:00006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 76D05, 76D03, 76D07

Retrieve articles in all journals with MSC (2000): 76D05, 76D03, 76D07


Additional Information

Giovanni P. Galdi
Affiliation: Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15261
Email: galdi@engr.pitt.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09658-5
Keywords: Navier-Stokes equations, exterior problem, homogeneous Sobolev spaces.
Received by editor(s): January 9, 2008
Published electronically: August 15, 2008
Additional Notes: This work was supported in part by NSF Grant #DMS-0707281.
Communicated by: Walter Craig
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society