×

Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations. (English) Zbl 0655.76041

Summary: One of the central problems in the mathematical theory of turbulence is that of breakdown of smooth (indefinitely differentiable) solutions to the equations of motion. In 1934 J. Leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3-D incompressible Navier-Stokes equations. The problem is still open. We show in this report that breakdown of smooth solutions to the 3-D incompressible slightly viscous (i.e. corresponding to high Reynolds numbers, or “highly turbulent”) Navier-Stokes equations cannot occur without breakdown in the corresponding solution of the incompressible Euler (ideal fluid) equation. We prove then that solutions of distorted Euler equations, which are equations closely related to the Euler equations for short term intervals, do breakdown.

MSC:

76Fxx Turbulence
35Q99 Partial differential equations of mathematical physics and other areas of application
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3?D Euler equations. Commun. Math. Phys.94, 61-66 (1984) · Zbl 0573.76029
[2] Constantin, P.: Blow-up for a non-local evolution equation. M.S.R.I. 038-84-6 Berkeley, California, July 1984
[3] Constantin, P., Lax, P.D., Majda, A.: A simple one dimensional model for the three dimensional vorticity equation (submitted to Commun. Pure Appl. Math.) · Zbl 0615.76029
[4] Foias, C.: Unpublished
[5] Kato, T.: Nonstationary flows of viscous and ideal fluids in ?3. J. Funct. Anal.9, 296-305 (1972) · Zbl 0229.76018
[6] Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton, NJ: Princeton University Press 1971 · Zbl 0232.42007
[7] Swann, H.: The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in ?3. Trans. Am. Math. Soc.157, 373-397 (1971) · Zbl 0218.76023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.