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2018-GdT ITER - N. Besse
Mardi 13 Mars 2018 à 11h en 15-16-413 : Nicolas Besse (Observatoire de Nice)
Regularity of the geodesic flow of the incompressible Euler equations on a manifold
In this talk we investigate the time regularity of the volume-preserving geodesic flow, which is associated to the incompressible Euler equations on a compact d-dimensional Riemannian manifold with boundary.
Not only the global regularity for smooth initial data is an open issue, but the behaviour of solutions and many pratical applications also depend on complex geometries. Our result completes the local-in-time well-posedness theory of Ebin and Marsden (Ann. Math. 92 (1970) 102–163).
Indeed, the first result of this talk states roughly that the time smoothness of geodesic curves is only limited by the smoothness of the manifold and its boundary, which is measured in a broad ultradifferentiable class, including the real analytic class.
A second result is that our simple constructive proof can be used to design efficient and very high-order semi-Lagrangian methods for integrating accurately the Euler equations on a manifold. Indeed, the proof makes use of a new Lagrangian formulation of Euler equations on d-dimensional manifolds, which is a generalization of the Cauchy invariants equation in R^3. This Lagrangian formulation, together with the Lagrangian incompressibility condition and the invariance of the boundary under the Lagrangian flow allow us to derive new recursion relations among time-Taylor coefficients of the time-series expansion of the geodesic flow.