Steady Compressible 3d Euler Flows in Toroidal Volumes Without Continuous Euclidean Isometries

21 Pages Posted: 3 Sep 2024

See all articles by Naoki Sato

Naoki Sato

National Institute for Fusion Science

Michio Yamada

Kyoto University

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.

Keywords: Euler equations, Magnetohydrodynamics, Grad's conjecture, Stellarator

Suggested Citation

Sato, Naoki and Yamada, Michio, Steady Compressible 3d Euler Flows in Toroidal Volumes Without Continuous Euclidean Isometries. Available at SSRN: https://ssrn.com/abstract=4944718 or http://dx.doi.org/10.2139/ssrn.4944718

Naoki Sato (Contact Author)

National Institute for Fusion Science ( email )

Toki
Japan

Michio Yamada

Kyoto University ( email )

Yoshida-Honmachi
Sakyo-ku
Kyoto, 606-8501
Japan

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