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Semi-discrete optimal transport techniques for the compressible semi-geostrophic equations

David P. Bourne, Charlie P. Egan, Théo Lavier, Beatrice Pelloni

TL;DR

The paper addresses global weak solutions for the 3D compressible semi-geostrophic equations with compactly supported, measure-valued initial data. It develops a semi-discrete optimal transport framework, proving existence and uniqueness of discrete geostrophic solutions that conserve energy, and then extends to general compactly supported data via a particle-approximation and limit process, yielding a Lipschitz-in-time global weak solution. The approach hinges on a dual formulation of the energy minimisation, regularity of the centroid map, and a carefully constructed ODE system for particle seeds, providing a rigorous foundation for semi-discrete OT-based numerical schemes in the compressible SG setting. The results generalise previous incompressible and compressible SG work and create a pathway for robust OT-driven numerical methods for 3D atmospheric front modeling within geostrophic coordinates.

Abstract

We prove existence of weak solutions of the 3D compressible semi-geostrophic (SG) equations with compactly supported measure-valued initial data. These equations model large-scale atmospheric flows. Our proof uses a particle discretisation and semi-discrete optimal transport techniques. We show that, if the initial data is a discrete measure, then the compressible SG equations admit a unique, twice continuously differentiable, energy-conserving and global-in-time solution. In general, by discretising the initial measure by particles and sending the number of particles to infinity, we show that for any compactly supported initial measure there exists a global-in-time solution of the compressible SG equations that is Lipschitz in time. This significantly generalises the original results due to Cullen and Maroofi (2003), and it provides the theoretical foundation for the design of numerical schemes using semi-discrete optimal transport to solve the 3D compressible SG equations.

Semi-discrete optimal transport techniques for the compressible semi-geostrophic equations

TL;DR

The paper addresses global weak solutions for the 3D compressible semi-geostrophic equations with compactly supported, measure-valued initial data. It develops a semi-discrete optimal transport framework, proving existence and uniqueness of discrete geostrophic solutions that conserve energy, and then extends to general compactly supported data via a particle-approximation and limit process, yielding a Lipschitz-in-time global weak solution. The approach hinges on a dual formulation of the energy minimisation, regularity of the centroid map, and a carefully constructed ODE system for particle seeds, providing a rigorous foundation for semi-discrete OT-based numerical schemes in the compressible SG setting. The results generalise previous incompressible and compressible SG work and create a pathway for robust OT-driven numerical methods for 3D atmospheric front modeling within geostrophic coordinates.

Abstract

We prove existence of weak solutions of the 3D compressible semi-geostrophic (SG) equations with compactly supported measure-valued initial data. These equations model large-scale atmospheric flows. Our proof uses a particle discretisation and semi-discrete optimal transport techniques. We show that, if the initial data is a discrete measure, then the compressible SG equations admit a unique, twice continuously differentiable, energy-conserving and global-in-time solution. In general, by discretising the initial measure by particles and sending the number of particles to infinity, we show that for any compactly supported initial measure there exists a global-in-time solution of the compressible SG equations that is Lipschitz in time. This significantly generalises the original results due to Cullen and Maroofi (2003), and it provides the theoretical foundation for the design of numerical schemes using semi-discrete optimal transport to solve the 3D compressible SG equations.
Paper Structure (17 sections, 23 theorems, 184 equations, 2 figures)

This paper contains 17 sections, 23 theorems, 184 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Background
  4. Previous results
  5. Outline of the paper
  6. Preliminaries
  7. Notation and assumptions
  8. The compressible SG equations
  9. Semi-discrete optimal transport
  10. The dual problem
  11. Discrete solutions of the compressible SG equations
  12. Existence of weak solutions
  13. Explicit examples
  14. Steady state example
  15. Single seed example
  16. ...and 2 more sections

Key Result

Theorem 1.1

Fix $N\in\mathbb{N}$, an arbitrary final time $\tau>0$, and an initial discrete probability measure $\overline{\alpha}^N = \sum_{i=1}^N m^i \delta_{\overline{\vb{z}}^i} \in\mathscr{P}^N(\mathcal{Y})$ that is well-prepared in the sense of wellprep. Then there exists a unique solution $\vb{z}\in \math Then $\alpha^N$ is the unique weak solution of the compressible SG equations (in the sense of Defin

Figures (2)

  • Figure 1: On the left is the source space $\mathcal{X}$ coloured by the density of the optimal source measure $\sigma_*[\alpha_t^N]$. The boundaries and centroids $\vb{C}^j(\vb{z}(t))$ of the corresponding $c$-Laguerre cells are plotted in black, with the boundary of the $i$-th cell, $L^i_c$, highlighted in red. On the right is the target space $\mathcal{Y}$ with the seeds $\vb{z}^j(t)$ in blue. The $i$-th seed, $\vb{z}^i(t)$, corresponding to the $i$-th cell is highlighted in red. The union of the seeds is the support of the target measure $\alpha_t^N$.
  • Figure 2: $c$-Laguerre tessellations (see Definition \ref{['def:cLagCells']}) in the $(x_1,x_3)$-plane for the cost function $c=c_{\mathrm{2d}}$ (see \ref{['eq:c2d']}). The colours distinguish the cells. For each plot, $\mathcal{X}=[0,1]^2$, $f_{\textrm{cor}}=1$, $g=1$, the seeds $\vb{z}^i$ were sampled uniformly from $\mathcal{X}$, and the weights $w^i$ were chosen so that the cells have equal area (by maximising the dual function as in \ref{['eq:optdiscT']}).

Theorems & Definitions (63)

  • Theorem 1.1: Existence of discrete geostrophic solutions
  • Theorem 1.2: Existence of weak geostrophic solutions
  • Definition 2.1: The cost function
  • Definition 2.2: Optimal transport cost
  • Definition 2.3: Geostrophic energy
  • Lemma 2.4: Existence and uniqueness of minimisers of $E$
  • proof
  • Definition 2.5: Optimal source measure and transport map
  • Definition 2.6: Weak solution
  • Remark 2.7: Equivalent weak formulation
  • ...and 53 more