Steady Compressible 3D Euler Flows in Toroidal Volumes without Continuous Euclidean Isometries

TL;DR

This paper constructs smooth 3D steady Euler flows in toroidal shapes lacking continuous symmetries, highlighting challenges in magnetohydrodynamic equilibrium conjectures due to incompressibility constraints.

Contribution

It demonstrates the existence of asymmetric steady Euler flows with toroidal level sets, challenging assumptions about symmetry in such flows.

Findings

Existence of smooth asymmetric toroidal Euler flows.

Flows can be influenced by external potential or density sources.

Highlights obstacles in magnetohydrodynamic equilibrium theories.

Abstract

We demonstrate the existence of smooth three-dimensional vector fields where the cross product between the vector field and its curl is balanced by the gradient of a smooth function, with toroidal level sets that are not invariant under continuous Euclidean isometries. This finding indicates the existence of steady compressible Euler flows, either influenced by an external potential energy or maintained by a density source in the continuity equation, that are foliated by asymmetric nested toroidal surfaces. Our analysis suggests that the primary obstacle in resolving Grad's conjecture regarding the existence of nontrivial magnetohydrodynamic equilibria arises from the incompressibility constraint imposed on the magnetic field.