# zbMATH — the first resource for mathematics

The exponential map near conjugate points in 2D hydrodynamics. (English) Zbl 1329.58003
Let $$M^n$$ be a compact Riemannian manifold. Arnold has shown that motions of the ideal incompressible fluid in $$M$$ are geodesics on the group $$\mathcal{D}_\mu^s = \mathcal{D}_\mu^s(M)$$ of Sobolev class volume-preserving diffeomorphisms. The right-invariant metric on $$\mathcal{D}_\mu^s$$ is given at the identity diffeomorphism $$e$$ by the $$L^2$$ inner product $$\langle v,w \rangle_{L^2}=\int_M \langle v(x),w(x) \rangle d\mu$$, where $$v, w \in T_e \mathcal{D}_\mu^s$$ are divergence free Sobolev $$H^s$$ vector fields on $$M$$. This metric is only weak-Riemannian in that the tangent spaces to $$\mathcal{D}_\mu^s$$ with the induced inner products are (incomplete) pre-Hilbert spaces. It is known [D. G. Ebin et al., Geom. Funct. Anal. 16, No. 4, 850–868 (2006; Zbl 1105.35070)] that the weak-Riemannian exponential map in $$2D$$ hydrodynamics is a nonlinear Fredholm map. In this paper the author proves the following result. Let $$M$$ be a smooth closed Riemannian manifold of dimension $$2$$ and assume $$s>2$$. Consider a geodesic $$\eta(t)$$ in $$\mathcal{D}_\mu^s$$ of the $$L^2$$ metric starting from the identity $$e$$ with velocity $$v_0$$ and let $$\eta(t_c)$$ be the first conjugate point to $$e$$. Then the weak-Riemannian $$L^2$$ exponential map is not injective at $$t_cv_0$$.

##### MSC:
 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 58B25 Group structures and generalizations on infinite-dimensional manifolds 46T05 Infinite-dimensional manifolds 37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics 35Q35 PDEs in connection with fluid mechanics
Full Text:
##### References:
 [1] Arnold, V, Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16, 319-361, (1966) · Zbl 0148.45301 [2] Arnold, V.: Arnold’s Problems. Springer/PHASIS, Berlin/Moscow (2004) · Zbl 1051.00002 [3] Arnold, V., Khesin, B.: Topological Methods in Hydrodynamics. Springer, New York (1998) · Zbl 0902.76001 [4] Benn, J.: PhD Dissertation, University of Notre Dame (2014) · Zbl 0129.36002 [5] Benn, J.: Conjugate points on the symplectomorphism group. Ann. Glob. Anal. Geom. (2015). doi:10.1007/s10455-015-9461-5 · Zbl 1329.35228 [6] Biliotti, L; Exel, R; Piccione, P; Tausk, D, On the singularities of the exponential map in infinite dimensional Riemannian manifolds, Math. Ann., 336, 247-267, (2006) · Zbl 1108.58004 [7] Cheeger, J., Ebin, D.: Comparison Theorems in Riemannian Geometry. North Holland, New York (1975) · Zbl 0309.53035 [8] Ebin, D; Marsden, J, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92, 102-163, (1970) · Zbl 0211.57401 [9] Ebin, D; Misiołek, G; Preston, S, Singularities of the exponential map on the volume-preserving diffeomorphism group, Geom. Funct. Anal., 16, 850-868, (2006) · Zbl 1105.35070 [10] Gohberg, I; Kaashoek, M; Lay, D, Equivalence, linearization and decomposition of holomorphic operator functions, J. Funct. Anal., 28, 102-144, (1978) · Zbl 0384.47018 [11] Grossman, N, Hilbert manifolds without epiconjugate points, Proc. Am. Math. Soc., 16, 1365-1371, (1965) · Zbl 0135.40204 [12] Inci, H.: On the Lagrangian formulation of the incompressible Euler equation. arXiv:1301.5994 [math.AP] · Zbl 1389.35251 [13] Krasnoselskii, M.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, Oxford (1964) [14] Misiołek, G, Stability of ideal fluids and the geometry of the group of diffeomorphisms, Indiana Univ. Math. J., 42, 215-235, (1993) · Zbl 0799.58019 [15] Misiołek, G, Conjugate points in $${\cal D}_μ (\mathbb{T}^2)$$, Proc. Am. Math. Soc., 124, 977-982, (1996) · Zbl 0849.58004 [16] Misiołek, G; Preston, S, Fredholm properties of Riemannian exponential maps on diffeomorphism groups, Invent. Math., 179, 191-227, (2010) · Zbl 1183.58006 [17] Morse, M; Littauer, S, A characterization of fields in the calculus of variations, Proc. Natl. Acad. Sci. USA, 18, 724-730, (1932) · Zbl 0006.35003 [18] Piccione, P; Portaluri, P; Tausk, V, Spectral flow, Maslov index and bifurcation of semi-Riemannian geodesics, Ann. Glob. Anal. Geom., 25, 121-149, (2004) · Zbl 1050.58015 [19] Preston, S, On the volumorphism group, the first conjugate point is always the hardest, Commun. Math. Phys., 267, 493-513, (2006) · Zbl 1113.37062 [20] Savage, L, On the crossings of extremals at focal points, Bull. Am. Math. Soc., 49, 467-469, (1943) · Zbl 0063.06748 [21] Shnirelman, A, Generalized fluid flows, their approximation and applications, Geom. Funct. Anal., 4, 586-620, (1994) · Zbl 0851.76003 [22] Shnirelman, A.: On the analyticity of particle trajectories in the ideal incompressible fluid (2012, preprint). arXiv:1205.5837 [math.AP] · Zbl 1296.35133 [23] Tromba, A, Some theorems on Fredholm maps, Proc. Am. Math. Soc., 34, 578-585, (1972) · Zbl 0256.58001 [24] Warner, F, The conjugate locus of a Riemannian manifold, Am. J. Math., 87, 575-604, (1965) · Zbl 0129.36002 [25] Wolibner, W, Un theoréme sur l’existence du mouvement plan d’un fluide parfait, homogéne, incompressible, pendant un temps infiniment long, Math. Z., 37, 698-726, (1933) · Zbl 0008.06901
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.