Abstract
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.
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Beale, J. T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations (preprint)
Stein, E. M., Singular integrals and differentiability properties of functions. Princeton: Princeton University Press, 1970
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
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)
Moser, J.: ‘A rapidly convergent iteration method and nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa20, (1966), 265–315 (1966)
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Communicated by L. Nirenberg
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Ponce, G. Remarks on a paper by J. T. Beale, T. Kato, and A. Majda. Commun.Math. Phys. 98, 349–353 (1985). https://doi.org/10.1007/BF01205787
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DOI: https://doi.org/10.1007/BF01205787