Reduction from the compressible Euler equation

Determine whether an analog of the binormal equation for augmented triples can be obtained by a reduction or ansatz from the compressible Euler equation on the ambient manifold $Q$, analogous to the localized induction approximation in the incompressible setting.

Background

The paper identifies the localized induction approximation as the mechanism relating incompressible Euler dynamics to the binormal or skew-mean-curvature evolution of vortex membranes. It then asks whether a comparable relationship exists for the generalized symplectic structure associated with compressible-fluid singular vorticities.

The unresolved issue is specifically whether the desired compressible binormal-type dynamics can arise from the compressible Euler equation through a reduction or ansatz on QQ.

References

Here are several open questions related to the introduction of symplectic structures on the augmented triples.

  1. What could be an interesting (and possible integrable in lower dimensions) analog of the binormal equation for membranes in this compressible setting?
  2. Is it possible to obtain such an analog of the binormal equation by means of a reduction or ansatz from the compressible Euler equation on $Q$, similar to the localized induction approximation in the incompressible setting?
— Generalized Marsden-Weinstein symplectic structures  (2610.01401 - Khesin et al., 1 Oct 2026) in Section "Dynamics on membranes and open questions", subsection "The binormal equation and its analogs" (Section 5.1)