Finite time analyticity for the two- and three-dimensional Rayleigh-Taylor instability

Sulem, C.; Sulem, P.-L.

doi:10.1090/S0002-9947-1985-0766210-5

Finite time analyticity for the two- and three-dimensional Rayleigh-Taylor instability
HTML articles powered by AMS MathViewer

by C. Sulem and P.-L. Sulem PDF

Trans. Amer. Math. Soc. 287 (1985), 127-160 Request permission

Abstract:

The Rayleigh-Taylor instability refers to the dynamics of the interface between two ideal irrotational fluids of different densities superposed one over the other and in relative motion. The well-posedness of this problem is considered for two- and three-dimensional flows in the entire space and in the presence of a horizontal bottom. In the entire space, finite time analyticity of the interface is proven when the initial interface has sufficiently small gradients and is flat at infinity. In the presence of a horizontal bottom, the initial interface corrugations has also to be small initially but it is not required to vanish at infinity.

References

S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, International Series of Monographs on Physics, Clarendon Press, Oxford, 1961. MR 0128226
Garrett Birkhoff, Helmholtz and Taylor instability, Proc. Sympos. Appl. Math., Vol. XIII, American Mathematical Society, Providence, R.I., 1962, pp. 55–76. MR 0137423
L. V. Ovsjannikov, To the shallow water theory foundation, Arch. Mech. (Arch. Mech. Stos.) 26 (1974), 407–422 (English, with Russian and Polish summaries). MR 347218
Tadayoshi Kano and Takaaki Nishida, Sur les ondes de surface de l’eau avec une justification mathématique des équations des ondes en eau peu profonde, J. Math. Kyoto Univ. 19 (1979), no. 2, 335–370 (French). MR 545714, DOI 10.1215/kjm/1250522437
V. I. Nalimov, A priori estimates of solutions of elliptic equations in the class of analytic functions, and their applications to the Cauchy-Poisson problem, Dokl. Akad. Nauk SSSR 189 (1969), 45–48 (Russian). MR 0262645

Water waves; analytic solutions, uniqueness and continuous dependence on the data

V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy Vyp. 18 Dinamika Židkost. so Svobod. Granicami (1974), 104–210, 254 (Russian). MR 0609882
Hideaki Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci. 18 (1982), no. 1, 49–96. MR 660822, DOI 10.2977/prims/1195184016
Bui An Ton, On a free boundary problem for an inviscid incompressible fluid, Nonlinear Anal. 6 (1982), no. 4, 335–347. MR 654810, DOI 10.1016/0362-546X(82)90020-7

24

200

12

C. Sulem, P.-L. Sulem, C. Bardos, and U. Frisch, Finite time analyticity for the two- and three-dimensional Kelvin-Helmholtz instability, Comm. Math. Phys. 80 (1981), no. 4, 485–516. MR 628507
Takaaki Nishida, A note on a theorem of Nirenberg, J. Differential Geometry 12 (1977), no. 4, 629–633 (1978). MR 512931
M. S. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kovalevsky theorems, Comm. Partial Differential Equations 2 (1977), no. 11, 1151–1162. MR 481322, DOI 10.1080/03605307708820057

6

W. Wolibner, Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z. 37 (1933), no. 1, 698–726 (French). MR 1545430, DOI 10.1007/BF01474610
Tosio Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188–200. MR 211057, DOI 10.1007/BF00251588
C. Bardos and S. Benachour, Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de $R^{n}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 647–687 (French). MR 454413

287

7

9

Gregory R. Baker, Daniel I. Meiron, and Steven A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech. 123 (1982), 477–501. MR 687014, DOI 10.1017/S0022112082003164
G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954