Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case
Luigi Ambrosio, Maria Colombo, Guido De Philippis, Alessio Figalli
TL;DR
This paper proves the global existence of distributional solutions to the 2D semigeostrophic equations on a torus using advanced regularity estimates for Monge-Ampere equations, connecting Eulerian and Lagrangian frameworks.
Contribution
It introduces new regularity and stability estimates for Alexandrov solutions, enabling the proof of existence of Eulerian solutions in physical space for the semigeostrophic equations.
Findings
Global in time existence of solutions established
Utilizes new regularity estimates for Monge-Ampere equations
Discusses the link between Eulerian and Lagrangian solutions
Abstract
In this paper we use the new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations estabilished by G.De Philippis and A.Figalli to provide a global in time existence of distributional solutions to a semigeostrophic equation on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Nonlinear Partial Differential Equations