zbMATH — the first resource for mathematics

Geometry and curvature of diffeomorphism groups with \(H^1\) metric and mean hydrodynamics. (English) Zbl 0933.58010
Let \((M,\langle \cdot ,\cdot \rangle)\) be a closed oriented Riemannian manifold and \(\mathcal D_\mu ^s(M)\) the group of volume preserving diffeomorphisms of \(M\). The weak \(L^2\) right invariant Riemannian metric on \(\mathcal D_\mu ^s(M)\) is given by \[ \langle X,Y\rangle _{L^{2}}=\int_{M}\langle X(x),Y(x)\rangle \mu (x), \] where \(X,Y\in T_e\mathcal D_\mu ^s(M)\) are vector fields on \(M\) and \(\mu \) is the volume element on \(M\). V. I. Arnold, D. Ebin and J. Marsden showed that geodesics \(\eta _t\) of the weak \(L^2\) right invariant Riemannian metric on \(\mathcal D_\mu ^s(M)\) are motions of incompressible ideal fluids. D. D. Holm, J. E. Marsden and T. S. Ratiu [Adv. Math. 137, 1-81 (1998)] derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation \[ \dot{V}(t)+\nabla _{U(t)}V(t)-\alpha ^{2}[ \nabla U(t)] ^{t}\cdot \Delta U(t)=-\text{grad } p(t) \] where \(V=(1-\alpha ^{2}\Delta)U\), \(\text{div} U=0\).
In this paper, the equation of mean motion of an ideal fluid is generalized to the case of a manifold \(M\). This model corresponds to the weak \(H^1\) right invariant Riemannian metric on \(\mathcal D_\mu ^s(M)\) which is expressed as \[ \langle X,Y\rangle _{1}=\langle X,(1+\text{Ric})Y\rangle _{L^{2}}+\langle \nabla X,\nabla Y\rangle _{L^{2}} \] for \(X,Y\in T_e\mathcal D_\mu ^s(M)\). The following problems are investigated: \(H^1\)-covariant derivative and its geodesic flow on \(\mathcal D_\mu ^s(M)\), curvature of the \(H^1\)-metric, existence and uniqueness results for the Jacobi equation, stability and curvature.

58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
76M30 Variational methods applied to problems in fluid mechanics
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
Full Text: DOI arXiv
[1] Arnold, V.I., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluids parfaits, Ann. inst. Grenoble, 16, 319-361, (1966) · Zbl 0148.45301
[2] Arnold, V.I.; Khesin, B., Topological methods in hydrodynamics, (1998), Springer-Verlag New York · Zbl 0902.76001
[3] Arnold, V.I.; Khesin, B., Topological methods in hydrodynamics, Annual review of fluid mechanics, (1992), Annual Reviews Palo Alto, p. 145-166 · Zbl 0743.76019
[4] Bao, D.; Lafontaine, J.; Ratiu, T., On a nonlinear equation related to the geometry of the diffeomorphism group, Pacific J. math., 158, 223-242, (1993) · Zbl 0739.58066
[5] Ebin, D., The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. of math., 105, 141-200, (1977) · Zbl 0373.76007
[6] Ebin, D.; Marsden, J., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of math., 92, 102-163, (1970) · Zbl 0211.57401
[7] D. Ebin, G. Misiolek, The exponential map on \(D\)^{s}μ · Zbl 0954.58007
[8] C. Foias, D. D. Holm, E. S. Titi, The three dimensional viscous Cammassa-Holm Equations and their relation to the Navier-Stokes equations and turbulence theory · Zbl 0995.35051
[9] Holm, D.D.; Kouranbaeva, S.; Marsden, J.E.; Ratiu, T.S.; Shkoller, S., A nonlinear analysis of the averaged Euler equations, Fields inst. comm., 2, (1998)
[10] Holm, D.D.; Marsden, J.E.; Ratiu, T.S., Euler – poincaré equations and semidirect products with applications to continuum theories, Adv. in math., 137, 1-81, (1998) · Zbl 0951.37020
[11] Holm, D.D.; Marsden, J.E.; Ratiu, T.S., Euler – poincaré models of ideal fluids with nonlinear Dispersian, Phys. rev. lett., 80, 4273-4277, (1998)
[12] Lang, S., Differentiable and Riemannian manifolds, (1995), Springer-Verlag New York
[13] Lukatskii, A.M., On the curvature of the group of measure-preserving diffeomorphisms of ann, Comm. Moscow math. soc., 179-180, (1980)
[14] Lukatskii, A.M., Curvature of the group of measure-preserving diffeomorphisms of ann, Sibirskii math. J., 25, 76-88, (1984) · Zbl 0582.58007
[15] Lukatskii, A.M., Structure of the curvature tensor of the group of measure-preserving diffeomorphisms of a compact two-dimensional manifold, Sibirskii math. J., 29, 95-99, (1988) · Zbl 0674.58011
[16] Marsen, J.E.; Ebin, D.G.; Fischer, A.E., Diffeomorphism groups, hydrodynamics and relativity, Proceedings of the thirteenth biennial seminar of the Canadian mathematical congress, (1972) · Zbl 0284.58002
[17] Marsden, J.E.; Ratiu, T.S., Introduction to mechanics and symmetry, (1995), Springer-Verlag Berlin/New York
[18] Misiolek, G., Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms, Indiana univ. math. J., 2, 215-235, (1993) · Zbl 0799.58019
[19] Misiolek, G., Conjugate points in \(D\)_{μ}(\(T\)2, Proc. amer. math. soc., 124, 977-982, (1996) · Zbl 0849.58004
[20] Misiolek, G., A shallow water equation as a geodesic flow on the bott – viasoro group, J. geom. phys., 24, 203-208, (1998) · Zbl 0901.58022
[21] Misiolek, G., The exponential map on the free loop space is Fredholm, Geom. funct. anal., 7, 954-969, (1997) · Zbl 0906.58002
[22] Morrey, C.B., Multiple integrals in the calculus of variations, (1966), Springer-Verlag Berlin/New York · Zbl 0142.38701
[23] Rosenberg, S., The Laplacian on a Riemannian manifold, London math. soc., (1997)
[24] Shnirelman, A.I., The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. sb. (N.S.), 128, 82-109, (1985)
[25] Shnirelman, A.I., Generalized fluid flows, their approximation and applications, Geom. funct. anal., 4, 586-620, (1994) · Zbl 0851.76003
[26] Treves, F., Introduction to pseudodifferential and Fourier integral operators, (1982), Plenum New York
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.