A biinvariant metric on the group of symplectic diffeomorphisms and the equation \((\partial/\partial t)\Delta F = \{\Delta F,F\}\) (Q1822109)

The author considers a chain of Hilbert spaces \(\{E,E^ s\); \(s\in N(d)\}\), where \(d\in N\) is some natural number, \(N(d)=\{s\in N\); \(s\geq d\}\) and the Hilbert space \(E^{s+1}\) is linear, densely embedded in \(E^ s\) and \(E=\cap E^ s\) has the inverse limit topology. The notion of an ILH-Lie group modeled on the chain \(\{E,E^ s\); \(s\in N(d)\}\) is introduced if there exists a system \(\{G^ s\); \(s\in N(d)\}\) of topological groups \(G^ s\) satisfying seven conditions. Some examples of ILH-Lie groups are given. Let M be a compact almost Kähler manifold without boundary, dim M\(=2n\), with fundamental 2-form \(\omega\) on M. The form \(\omega\) is closed and nondegenerate and therefore defines a symplectic structure on M. The author assumes that the cohomology class defined by the form \(\omega\) is integer, i.e., \([\omega]=H^ 2(H,Z)\). Then it follows that \(\omega^ n=\mu\) is the Riemannian volume element on M. Let \({\mathcal D}_{\omega}\) be the ILH-Lie group of smooth symplectic diffeomorphisms of the manifold M. The Lie algebra of the group \({\mathcal D}_{\omega}\) consists of locally Hamiltonian vector fields X on M. For a Hamiltonian function H to be unambiguously defined on its vector field \(X_ H\), the author requires that \(\int_{M}H(x)\mu (x)=0\). The space of such functions on M of Sobolev class \(H^ s\) is denoted by \(H^ s_ 0(M).\) The Lie algebra \({\mathcal H}\) of smooth Hamiltonian vector fields on M is considered. By \textit{T. Ratiu} and \textit{R. Schmid} in their paper [Math. Z. 177, 81-100 (1981; Zbl 0441.58004)] it is shown that there exists an ILH-Lie group \({\mathcal G}\) whose Lie algebra is \({\mathcal H}.\) For group \({\mathcal G}\) there exists a system \(\{{\mathcal G}^ s\); \(s\geq 2n+5\}\) of topological groups satisfying the seven conditions mentioned above. Each of the groups \({\mathcal G}^ s\) is a smooth Hilbert manifold modeled on the space \({\mathcal H}^ s\) of Hamiltonian vector fields of Sobolev class \(H^ s.\) A scalar product on the Lie algebra is defined by the formula \[ <X_ F,X_ H>=\int_{M}F(x)H(x)\mu (x), \] where \(X_ F,X_ H\in {\mathcal H}\), and \(\mu\) is the Riemannian volume element on M. This formula defines a scalar product also on each space \({\mathcal H}^ s\) of Hamiltonian vector fields on M of class \(H^ s\), \(s\geq 2n+5\). It is possible to identify \(T_ e{\mathcal G}^ s={\mathcal H}^ s\). By conditions placed on \({\mathcal G}^ s\), the right translation \(R_ g: {\mathcal G}^ s\to {\mathcal G}^ s\) is a smooth map for each \(g\in {\mathcal G}^ s\). The author then obtains from the defined scalar product on \(T_ e{\mathcal G}^ s={\mathcal H}^ s\) a right-invariant weak Riemannian structure on \({\mathcal G}^ s.\) For any function \(F_ 0(x)\in H^ s_ 0(M)\), \(s\geq 2n+5\), there exists a unique continuous solution F(t,x) of Euler's equation \(\partial \Delta F/\partial t = \{\Delta F,F\}\), defined on \((-\xi,\xi)\times M\) for some \(\xi >0\) with the properties: \(F(0,x)=F_ 0(x)\); \(F(t,x)\in H^ s_ 0(M)\) for any fixed \(t\in (-\xi,\xi)\); if \(F_ 0(x)\in H_ 0^{s+\ell}(M)\), \(\ell >0\), then \(F(t,x)\in H_ 0^{s+\ell}(M).\) If \(F=F(t,x)\) is a solution of Euler's equation, then the flow \(g_ t\) on M generated by the Hamiltonian vector field \(X_ F\) is geodesic on the manifold \({\mathcal G}^ s\) of metric \[ (V,W)_ g=\int_{M}<V(x),W(x)>_{g(x)}\mu (x), \] where V,W belong to the tangent space \(T_ g{\mathcal D}\) to the group \({\mathcal D}\) of smooth diffeomorphisms g of a compact Riemannian manifold M without boundary, \(<.,.>_ y\) is the scalar product on M in the tangent space \(T_ yM\), and \(\mu\) is the Riemannian volume element on the manifold M. Conversely, if \(g_ t\) is a geodesic on \({\mathcal G}^ s\), then the velocity vector field \(X_ F=dR^{-1}_{g_ t} \times (dg_ t/dt)\) has a Hamiltonian F satisfying Euler's equation.