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Incompressible Euler Equations

Abstract of A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain, Besse N., Frisch U., 2017, Communications in Mathematical Physics 351, 689

The 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, high-accuracy and well-resolved 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 Hölder-continuous. The latter has been known for about twenty years (Serfati 1995), but the combination of the two, which makes use of recursion relations among time-Taylor coefficients to obtain constructively the time-Taylor 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. Non-constructive 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 time-analyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very high-order Cauchy–Lagrangian method.

Abstract of Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces, Besse N., Frisch U., 2017, Journal of Fluid Mechanics 825, 412


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 volume-preserving 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 Lie-advection 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 Lie-advection invariant p-form 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 finite-time blow up. Our main result is also used for new derivations — and several new results — concerning local helicity-type invariants for fluids and MHD flow in flat or curved spaces of arbitrary dimension.

Bibliography

  1. Arnold, V. I., 1966, Ann. Inst. Fourier 16, 319-361
  2. Zheligovsky, V., Frisch, U., 2014, J. Fluid Mech. 749, 404
  3. Glass, O., Sueur, F., Takahashi, T., 2012, Ann. Scient. Ec. Norm. Sup. 4e serie 45, 1-51
  4. Lingam, M., Milosevich, G., Morrison, P. 2016, Phys. Lett. A 380, 2400-2406
  5. Podvigina, O., Frisch, U., Zheligovski, V., 2016, J. Comput. Phys. 306, 320
  6. Serfati, P., 1995, J. Math. Pures Appl. 74, 95-104
  7. Tao, T., 2016, Ann. PDE 2, 9