Abstract
We give sufficient conditions for asymptotic stability of a stationary solution to a flowing problem of a homogeneous incompressible fluid through a given planar domain. We consider a planar problem for the Euler equation and boundary conditions for the curl and the normal component of the velocity; moreover, the latter is given on the whole boundary of the flow domain and the curl is given only on the inlet part of the boundary. We establish asymptotic stability of a stationary flow (in linear approximation), assuming it to have no rest points and to satisfy some smallness condition which means that the perturbations leave the flow domain before they become to affect the main flow. In particular, we prove asymptotic stability for an arbitrary stationary flow in a rectangular canal close to the Couette flow without rest points. Moreover, we show that stability of the main flow in the L 2-norm under curl perturbations implies its stability in higher-order norms depending, for example, on the derivatives of the curl.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kochin N. E., “On an existence theorem in hydrodynamics,” Prikl. Mat. Mekh., 20, No. 10, 5–20 (1957).
Yudovich V. I., “A two–dimensional nonstationary flowing problem for an ideal incompressible fluid through a given domain,” Mat. Sb., 64, No. 4, 562–588 (1964).
Yudovich V. I., “A flowing problem for an ideal incompressible fluid through a given domain,” Dokl. Akad. Nauk SSSR, 146, No. 3, 561–564 (1962).
Kazhikhov A. V., “A remark on the statement of the flowing problem for the equation of an ideal fluid,” Prikl. Mat. Mekh., 44, No. 5, 947–949 (1980).
Antontsev S. N., Kazhikhov A. V., and Monakhov V. N., Boundary Value Problems of Inhomogeneous Fluid Mechanics [in Russian], Nauka, Novosibirsk (1983).
Solopenko V. M., “Solvability properties of time–dependent Euler's equations in terms of H, ψ,” Dokl. Akad. Nauk SSSR, 305, No. 3, 567–571 (1989).
Alekseev G. V., “On vanishing viscosity in the two–dimensional stationary problems of the hydrodynamics of an incompressible fluid,” Dinamika Sploshn. Sredy (Novosibirsk), No. 10, 5–28 (1972).
Alekseev G. V., “On uniqueness and smoothness of plane vortex flows of an ideal fluid,” Dinamika Sploshn. Sredy (Novosibirsk), No. 15, 7–18 (1973).
Morgulis A. B., “Solvability of three dimensional stationary flowing problem,” Siberian Math. J., 40, No. 1, 121–135 (1999).
Yudovich V. I., The Linearization Method in Hydrodynamical Stability Theory, Amer. Math. Soc., Providence, Rodeisland (1989). (Transl. Math. Monogr., 74.)
Yudovich V. I., “On the loss of smoothness of solutions to the Euler equations during time,” Dinamika Sploshn. Sredy, 16, 71–78 (1974).
Yudovich V. I., “On the loss of smoothness of the solutions to the Euler equations and the inherent instability of an ideal fluid flows,” Chaos, 10, No. 3, 705–719 (2000).
Arnol′d V. I., “Remarks on the behavior of flows of a three dimensional ideal fluid for a small perturbation of the initial velocity field,” Prikl. Mat. Mekh., 36, No. 5, 255–262 (1972).
Kra \(\imath\)ko A. N., “On the behavior of two–dimensional non–plane–parallel vortex flows of a nonviscous gas,” Prikl. Mat. Mekh., 46, No. 6, 972–978 (1982).
Hille E. and Phillips R. S., Functional Analysis and Semi–Groups [Russian translation], Izdat. Inostr. Lit., Moscow (1962).
Gilbarg D. and Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer–Verlag, Berlin etc. (1983).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Morgulis, A.B., Yudovich, V.I. Asymptotic Stability of a Stationary Flowing Regime of an Ideal Incompressible Fluid. Siberian Mathematical Journal 43, 674–688 (2002). https://doi.org/10.1023/A:1016376319707
Issue Date:
DOI: https://doi.org/10.1023/A:1016376319707