Semigeostrophic equations in physical space with free upper boundary

TL;DR

This paper establishes the existence of various Lagrangian solutions for the 3D incompressible geostrophic system with a free upper boundary, using minimization techniques and measure-valued solutions in physical space.

Contribution

It introduces new notions of Lagrangian solutions for the geostrophic equations with free boundary conditions and proves their existence under different initial data scenarios.

Findings

Existence of Lagrangian solutions in physical space for the system.

Construction of measure-valued dual space solutions from initial measures.

Application of minimization methods to prove solution existence.

Abstract

We define various notions of Lagrangian solution in physical space for 3-d incompressible geostrophic system with free upper boudary under different conditions for initial data,then prove their existence via the minimization with respect to a geostrophic functional.As a byproduct of our proof,we obtain the existence of measure-valued dual space solutions when the initial measure $\nu_0\in\mathcal{P}_2(\mathbf{R}^3)$ and is supported on $\{-\frac{1}{\delta}\leq x_3\leq-\delta\}$