Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations

We present a very simple proof of the global existence of a C∞ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has C∞ dependence on initial data u0 in the class of Hs divergence-free vector fields for s > 2. 1. Incompressible Euler equations Let (M, g) be a C compact oriented Riemannian 2-manifold with smooth boundary ∂M , let ∇ denote the Levi-Civita covariant derivative, and let μ denote the Riemannian volume form. The incompressible Euler equations are given by ∂tu+∇uu = gradp div u = 0, u(0) = u0, g(u, n) = 0 on ∂M, (1.1) where p(t, x) is the pressure function, determined (modulo constants) by solving the Neumann problem −△p = div∇uu with boundary condition g(gradp, n) = Sn(u), Sn denoting the second-fundamental form of ∂M . The now standard global existence result for two-dimensional classical solutions states that for initial data u0 ∈ χ s ≡ {v ∈ H(TM) | div u = 0, g(u, n) = 0}, s > 2, the solution u is in C(R, χ) and has C dependence on u0 (see, for example, Taylor’s book [8]). Equation (1.1) gives the Eulerian or spatial representation of the dynamics of the fluid. The Lagrangian representation which is in terms of the volume-preserving fluid particle motion or flow map η(t, x) is obtained by solving ∂tη(t, x) = u(t, η(t, x)), η(0, x) = x. (1.2) This is an ordinary differential equation on the infinite dimensional volume-preserving diffeomorphism group D μ, the set of H s class bijective maps of M into itself with H inverses which leave ∂M invariant. Ebin & Marsden [3] proved that D μ is a C manifold whenever s > 2. They also showed that for an interval I, whenever u ∈ C(I, χ) and s > 3, there exists a unique solution η ∈ C(I,D μ) to (1.2). Thus, for s > 3 the existence of a global C flow map immediately follows from the fact that u remains bounded inH for all time. It is often essential, however, for the Euler flow to depend smoothly on the initial data; in the case of vortex methods, for example, Hald in Assumption 3 of [5] requires this as a necessary condition to establish convergence. Theorem 1.1. For u0 ∈ χ , s > 2, there exists a unique global solution to (1.3) which is in C(R, TD μ) and has C ∞ dependence on u0. Date: April 20, 2000; current version May 29, 2000.