Codes and methods
Theory and analysis Past events 
Incompressible Euler EquationsThe 3D incompressible Euler equations is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or absence of boundaries. For a good understanding, it is crucial to carry out, besides mathematical studies, highaccuracy and wellresolved numerical exploration. Such studies can be very demanding in computational resources, but recently it has been shown that very substantial gains can be achieved, first, by using Cauchy’s Lagrangian formulation of the Euler equations and second, by taking advantages of analyticity results of the Lagrangian trajectories for flows whose initial vorticity is Höldercontinuous. The latter has been known for about twenty years (Serfati 1995), but the combination of the two, which makes use of recursion relations among timeTaylor coefficients to obtain constructively the timeTaylor series of the Lagrangian map, has been achieved only recently (Zheligovsky & Frisch 2014; Podvigina et al. 2016). Here we extend this methodology to incompressible Euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity. Nonconstructive regularity results for these cases have already been obtained by Glass et al. (2012). Using the invariance of the boundary under the Lagrangian flow, we establish novel recursion relations that include contributions from the boundary. This leads to a constructive proof of timeanalyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very highorder Cauchy–Lagrangian method. Figure 1: Symbolic representation of a Lagrangian map. Figure extracted from Besse & Frisch (2017). Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three dimensional (3D) ideal flow. Looking at such invariants with the modern tools of differential geometry and of geodesic flow on the space SDiff of volumepreserving transformations (Arnold 1966), all manners of generalisations are here derived. The Cauchy invariants equation and the Cauchy formula, relating the vorticity and the Jacobian of the Lagrangian map, are shown to be two expressions of this Lieadvection invariance, which are duals of each other (specifically, Hodge dual). Actually, this is shown to be an instance of a general result which holds for flow both in flat (Euclidean) space and in a curved Riemannian space: any Lieadvection invariant pform which is exact (i.e. is a differential of a (p−1)form) has an associated Cauchy invariants equation and a Cauchy formula. This constitutes a new fondamental result in linear transport theory, providing a Lagrangian formulation of Lie advection for some classes of differential forms. The result has a broad applicability: examples include the magnetohydrodynamics (MHD) equations and various extensions thereof, discussed by Lingam, Milosevich & Morrison (2016) and include also the equations of Tao (2016), Euler equations with modified Biot–Savart law, displaying finitetime blow up. Our main result is also used for new derivations — and several new results — concerning local helicitytype invariants for fluids and MHD flow in flat or curved spaces of arbitrary dimension. Bibliography
