Remarks on a paper by J. T. Beale, T. Kato, and A. Majda (Remarks on the breakdown of smooth solutions for the 3-dimensional Euler equations). (English) Zbl 0589.76040
Summary: See the article, [ibid. 94, 61-66 (1984; Zbl 0573.76029).]
We prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3-dimensional Euler equations. More precisely, the loss of regularity in a local smooth solution of the Euler equations implies the growth without bound of the deformation tensor as the critical time approaches; equivalently, if the deformation tensor remains bounded the existence of a smooth solution is guaranteed.
We prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3-dimensional Euler equations. More precisely, the loss of regularity in a local smooth solution of the Euler equations implies the growth without bound of the deformation tensor as the critical time approaches; equivalently, if the deformation tensor remains bounded the existence of a smooth solution is guaranteed.
MSC:
| 76Bxx | Incompressible inviscid fluids |
| 35Q99 | Partial differential equations of mathematical physics and other areas of application |
Keywords:
maximum norm of the deformation tensor; breakdown of smooth solutions; Euler equations; loss of regularity; local smooth solution; growth without bound of the deformation tensor; critical time; existenceCitations:
Zbl 0573.76029References:
| [1] | Beale, J. T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations (preprint) · Zbl 0573.76029 |
| [2] | Stein, E. M., Singular integrals and differentiability properties of functions. Princeton: Princeton University Press, 1970 · Zbl 0207.13501 |
| [3] | Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. In: Applied Mathematical Sciences Series. Berlin, Heidelberg, New York: Springer, 1983 |
| [4] | Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math.34, 481-524 (1981) · Zbl 0476.76068 · doi:10.1002/cpa.3160340405 |
| [5] | Moser, J.: ?A rapidly convergent iteration method and nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa20, (1966), 265-315 (1966) · Zbl 0144.18202 |
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