Abstract
Current proofs of time independent energy bounds for solutions of the time dependent Navier–Stokes equations, and of bounds for the Dirichlet norms of steady solutions, are dependent upon the construction of an extension of the prescribed boundary values into the domain that satisfies the inequality (1.1) below, for a value of κ less than the kinematic viscosity. It is known from the papers of Leray (J Math Pure Appl 12:1–82, 1993), Hopf (Math Ann 117:764–775, 1941) and Finn (Acta Math 105:197–244, 1961) that such a construction is always possible if the net flux of the boundary values across each individual component of the boundary is zero. On the other hand, the nonexistence of such an extension, for small values of κ, has been shown by Takeshita (Pac J Math 157:151–158, 1993) for any two or three-dimensional annular domain, when the boundary values have a net inflow toward the origin across each component of the boundary. Here, we prove a similar result for boundary values that have a net outflow away from the origin across each component of the boundary. The proof utilizes a class of test functions that can detect and measure deformation. It appears likely that much of our reasoning can be applied to other multiply connected domains.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Leray J.: Etude de diverses équations intégrales nonlinear et de quelques problèmes que pose l’hydrodynamique. J. Math. Pure Appl. 12, 1–82 (1993)
Hopf E.: Ein allgemeiner Endlichkeitssatz der Hydrodynamik. Math. Ann. 117, 764–775 (1941)
Finn R.: On the steady-state solutions of the Navier–Stokes equations. III. Acta Math. 105, 197–244 (1961)
Takeshita A.: A remark on Leray’s inequality. Pac. J. Math. 157, 151–158 (1993)
Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations Volume II. Springer, Berlin (1994)
Fujita, H.: On stationary solutions to Navier-Stokes equations in symmetric plane domains under general out-flow condition. In: Proceedings of International Conference on Navier–Stokes Equations, Theory and Numerical Methods, June 1997, Varenna Italy, Pitman Research Notes in Mathematics, vol. 388, pp. 16–30
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Heywood, J.G. On the Impossibility, in Some Cases, of the Leray–Hopf Condition for Energy Estimates. J. Math. Fluid Mech. 13, 449–457 (2011). https://doi.org/10.1007/s00021-010-0028-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-010-0028-8