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Geodesics on the symplectomorphism group. (English) Zbl 1252.58006

Let \(M\) be a compact smooth manifold and consider the problem of determining the motion of an incompressible fluid that fills \(M\), which is encoded in the Euler equations \(\partial_t u+\nabla_u u=-\nabla p\), \(\text{div} \enskip {u}=0\), where \(u(x,t)\) represents the velocity field of the fluid at \(x\in M\) at time \(t\) and \(p\) is the pressure function. V. I. Arnol’d [Ann. Inst. Fourier 16, No. 1, 319–361 (1966; Zbl 0148.45301)] established a beautiful correspondence between the configuration space of all possible motions of such a fluid on \(M\) with the infinite-dimensional “Lie” group of volume preserving diffeomorphisms (volumorphisms) of \(M\). This correspondence was explored in the celebrated paper of E. G. Ebin and J. Marsden [Ann. Math. (2) 92, 102–163 (1970; Zbl 0211.57401)], who proved short-time existence of solutions by endowing the space \(\mathcal D_\mu^s(M)\) of Sobolev \(H^s\) volumorphisms of \(M\), \(s>\tfrac{1}{2}\dim M+1\), with a weak right-invariant Riemannian metric (the \(L^2\)-metric) whose geodesics are solutions to the Euler equations. With this at hand, standard ODE techniques (Picard iteration) give not only short-time existence and uniqueness, but also regularity and continuous dependence on initial conditions. It is worth pointing out that similar conclusions were then inferred for the Navier-Stokes equations.
In the paper under review, a related problem is considered. Namely, \(M\) is assumed to carry a symplectic form \(\omega\), and the group \(\mathcal D_\omega^s\) of Sobolev \(H^s\) diffeomorphisms that preserve \(\omega\) (symplectomorphisms) is endowed with a weak right-invariant Riemannian metric. For this, a Riemannian metric and an almost complex structure on \(M\), both compatible with \(\omega\), are used. The geodesics of this \(L^2\)-metric on the symplectomorphism group satisfy an Euler-type conservative PDE, analogously to the above mentioned case of the volumorphism group. Furthermore, they are possibly related to plasma physics, in a similar way in which volumorphism geodesics were related to ideal hydrodynamics, see J. E. Marsden and A. Weinstein [Physica D 4, No. 3, 394–406 (1982; Zbl 1194.35463)]. Of course, in dimension \(2\), the symplectomorphism and volumorphism groups coincide. In this case, global existence of solutions to the Euler equations (hence of \(L^2\)-geodesics) was known since the 1930’s, see W. Wolibner [Math. Z. 37, 698–726 (1933; Zbl 0008.06901)]. The main result of the paper under review is that the same long-term existence holds for the \(L^2\)-geodesics on the symplectomorphism group, for all higher dimensions.
As a side note, B. Khesin [“Dynamics of symplectic fluids and point vortices”, arXiv:1106.1609] announced an extension of this paper’s results for the case where the metric on \(M\) is not necessarily compatible with the symplectic structure. The close relation of this subject with the case of the volumorphism group could potentially shed light on the problem of understanding long-term existence for some solutions of the Euler equations, and, eventually, of the Navier-Stokes equations. Other open questions that may be more at hand are also presented in the last section of the paper. Overall, this paper constitutes an important resource for researchers in the area, and is very clear, concise and elegantly written.

MSC:

58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
53C22 Geodesics in global differential geometry
53D05 Symplectic manifolds (general theory)
53D25 Geodesic flows in symplectic geometry and contact geometry
53D35 Global theory of symplectic and contact manifolds
58B10 Differentiability questions for infinite-dimensional manifolds
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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References:

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