Abstract
We study the geometry of streamlines and stability properties for steady state solutions of the Euler equations for an ideal fluid.
Similar content being viewed by others
References
Alinhac S., Métivier G.: Propagation de l’analyticité locale pour les solutions de léquation dEuler. Arch. Rational Mech. Anal. 92, 287–296 (1986)
Alpern S., Prasad V.S.: Typical Dynamics of Volume Preserving Homeomorphisms. Cambridge University Press, Cambridge (2000)
Arnold V.: Sur la géométrie differentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier 16, 319–361 (1966)
Arnold V.I.: On an apriori estimate in the theory of hydrodynamical stability. Am. Math. Soc. Transl. 19, 267–269 (1969)
Arnold V.I., Khesin B.A.: Topological Methods in Hydrodynamics. Springer, Berlin (1998)
Bardos C., Benachour S., Zerner M.: Analyticité des solutions périodiques de l’équation d’Euler en deux dimensions. C. R. Acad. Sci. Paris Sér. A-B 282(17), 995–998 (1976)
Bernshtein S.: Démonstration du théorème de M. Hilbert sur la nature analytique des solutions des équations du type elliptique sans l’emploi des séries normales. Math. Zeitschrift 28, 330–348 (1928)
Burton G.R.: Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. Henri Poincaré 6, 295–319 (1989)
Burton G.R.: Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex. Acta Math. 163, 291–309 (1989)
Cabré X., Chanillo S.: Stable solutions of semilinear elliptic problems in convex domains. Selecta Math. (N.S.) 4, 1–10 (1998)
Chemin J-Y.: Perfect Incompressible Fluids. Clarendon Press, Oxford (1998)
Ebin D., Marsden J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970)
Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)
Günter, N.M.: Potential Theory and its Application to Basic Problems of Mathematical Physics. Frederick Ungar Publishing Co., New York, 1967
Hamel, F., Nadirashvili, N., Sire, Y.: Convexity of level sets for elliptic problems in convex domains or convex rings: two counterexamples. Preprint arXiv: 1304.3355 (2013)
Hoffmann-Ostenhof M., Hoffman-Ostenhof T.: Local properties of solutions of Schrodinger equations. Commun. PDE 17, 491–522 (1992)
Hoffmann-Ostenhof M., Hoffman-Ostenhof T., Nadirashvili N.: Interior Hölder estimates for solutions of Schrödinger equations and the regularity of nodal sets. Commun. PDE 20, 1241–1273 (1995)
Hörmander L.: Linear Partial Differential Operators. Springer, New York (1963)
Hörmander L.: Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Berlin (1997)
Hughes T.J.R., Kato T., Marsden J.E.: Well-posed Quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Ration. Mech. Anal. 63, 273–294 (1976)
Kamynin L.I.: The smoothness of heat potentials, part 5. Differ Equat. 4, 185–195 (1968)
Lewy, H.: Über den analytischen Charakter der Lösungen elliptischer Diffe- rentialgleichungen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Math. Phys. KL, 178–186 (1927)
Lichtenstein L.: Über einige Hilfssätze der Potentialtheorie. Math. Zeitschrift 23, 72–78 (1925)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics. v.1. Incompressible Models. Claredon Press, Oxford, 1996
Milnor, J.: Remarks on infinite-dimensional Lie groups. Proceedings Summer School on Quantum Gravity, Held in Amsterdam, pp. 1007–1057, 1984
Miranda C.: Partial Differential Equations of Elliptic Type, 2nd edn. Springer, New York (1970)
Petrowsky I.G.: Sur l’analyticité des solutions des systèmes d’équations diffrentielles. Mat. Sbornik 5(47), 3–70 (1939)
Wiegner M.: Schauder estimates for boundary layer potentials. Math. Methods Appl. Sci. 16, 877–894 (1993)
Yudovich V.I.: Non-stationary flow of an ideal incompressible liquid. Zhurn. Vych. Mat. 3, 1032–1066 (1963)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Šverák
Rights and permissions
About this article
Cite this article
Nadirashvili, N. On Stationary Solutions of Two-Dimensional Euler Equation. Arch Rational Mech Anal 209, 729–745 (2013). https://doi.org/10.1007/s00205-013-0642-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-013-0642-8