By using Monge-Ampère geometry, we study the singular structure of a class of nonlinear Monge-Ampère equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Ampère geometry by examining the role of an induced metric on Lagrangian submanifolds of the cotangent bundle. In particular, we show that the signature of the metric serves as a classification of the Monge-Ampère equation, while singularities and elliptic-hyperbolic transitions are revealed by degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations.

D’Onofrio, R., Ortenzi, G., Roulstone, I., Rubtsov, V. (2023). Solutions and singularities of the semigeostrophic equations via the geometry of Lagrangian submanifolds. PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON. SERIES A, 479(2271), 1-21 [10.1098/rspa.2022.0682].

Solutions and singularities of the semigeostrophic equations via the geometry of Lagrangian submanifolds

D’Onofrio, Roberto
;
Ortenzi, Giovanni;
2023

Abstract

By using Monge-Ampère geometry, we study the singular structure of a class of nonlinear Monge-Ampère equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Ampère geometry by examining the role of an induced metric on Lagrangian submanifolds of the cotangent bundle. In particular, we show that the signature of the metric serves as a classification of the Monge-Ampère equation, while singularities and elliptic-hyperbolic transitions are revealed by degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations.
Articolo in rivista - Articolo scientifico
Lagrangian submanifolds; Monge-Ampère equations; pseudo-Riemannian geometry; semigeostrophic equations; singularities;
English
1-mar-2023
2023
479
2271
1
21
20220682
open
D’Onofrio, R., Ortenzi, G., Roulstone, I., Rubtsov, V. (2023). Solutions and singularities of the semigeostrophic equations via the geometry of Lagrangian submanifolds. PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON. SERIES A, 479(2271), 1-21 [10.1098/rspa.2022.0682].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/404784
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