×

zbMATH — the first resource for mathematics

Local structure of the set of steady-state solutions to the 2D incompressible Euler equations. (English) Zbl 1256.35076
Author’s abstract: “It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler’s equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.”

MSC:
35Q35 PDEs in connection with fluid mechanics
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
46T05 Infinite-dimensional manifolds
58C15 Implicit function theorems; global Newton methods on manifolds
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arnold V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 316–361 (1966) · Zbl 0148.45301 · doi:10.5802/aif.233
[2] V.I. Arnold, B. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences 125, Springer–Verlag (1998). · Zbl 0902.76001
[3] Ebin D.G., Marsden J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. (2) 92, 102–163 (1970) · Zbl 0211.57401 · doi:10.2307/1970699
[4] Ebin D.G., Misiołek G., Preston S.C.: Singularities of the exponential map on the volume-preserving diffeomorphism group. Geom. Funct. Anal. 16(4), 850–868 (2006) · Zbl 1105.35070 · doi:10.1007/s00039-006-0573-8
[5] H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer–Verlag (1969). · Zbl 0176.00801
[6] Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7(1), 65–222 (1982) · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2
[7] A.A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics 64, American Mathematical Society (2004). · Zbl 1229.22003
[8] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12, American Mathematical Society (1996). · Zbl 0865.35001
[9] de la Llave R., Obaya R.: Regularity of the composition operator in spaces of Hölder functions. Discrete Contin. Dynam. Systems 5(1), 157–184 (1999) · Zbl 0956.47029
[10] C. Marchioro, M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences 96, Springer–Verlag (1994). · Zbl 0789.76002
[11] Marsden J.E., Weinstein A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Order in Chaos (Los Alamos, N.M., (1982). Phys. D 7(1–3), 305–323 (1983) · Zbl 0576.58008
[12] Miller J., Weichman P., Cross M.C.: Statistical mechanics, Euler’s equation, and Jupiter’s Red Spot. Phys. Rev. A 45, 2328–2359 (1992) · doi:10.1103/PhysRevA.45.2328
[13] Misiołlek G., Preston S.C.: Fredholm properties of Riemannian exponential maps on diffeomorphism groups,. Invent. Math 179(1), 191–227 (2010) · Zbl 1183.58006 · doi:10.1007/s00222-009-0217-3
[14] Moser J.: A new technique for the construction of solutions of nonlinear differential equations. Proc. Nat. Acad. Sci. USA 47, 1824–1831 (1961) · Zbl 0104.30503 · doi:10.1073/pnas.47.11.1824
[15] Nash J.: The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63, 20–63 (1956) · Zbl 0070.38603 · doi:10.2307/1969989
[16] Preston S.C.: The WKB method for conjugate points in the volumorphism group. Indiana Univ. Math. J. 57(7), 3303–3327 (2008) · Zbl 1167.37035 · doi:10.1512/iumj.2008.57.3413
[17] Ratiu T.S., Tudoran R., Sbano L., Sousa Dias E., Terra G.: A Crash Course in Geometric Mechanics, Notes of the Courses Given by Ratiu, London Math. Soc. Lecture Note Ser. 306. Cambridge Univ. Press, Cambridge (2005) · Zbl 1159.70001
[18] Robert R.: A maximum–entropy principle for two–dimensional perfect fluid dynamics. J. Statist. Phys. 65(3–4), 531–553 (1991) · Zbl 0935.76530 · doi:10.1007/BF01053743
[19] Sergeraert F.: Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications. Ann. Sci. École Norm. Sup. (4) 5, 599–660 (1972) · Zbl 0246.58006
[20] Shnirelman A.I.: Lattice theory and flows of ideal incompressible fluid. Russian J. Math. Phys. 1(1), 105–114 (1993) · Zbl 0874.35096
[21] Turkington B.: Statistical equilibrium measures and coherent states in twodimensional turbulence. Comm. Pure Appl. Math. 52(7), 781–809 (1999) · Zbl 0990.76029 · doi:10.1002/(SICI)1097-0312(199907)52:7<781::AID-CPA1>3.0.CO;2-C
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.