Two-Dimensional Fluids Via Matrix Hydrodynamics

Two-dimensional (2-D) incompressible, inviscid fluids produce fascinating patterns of swirling motion. How and why the patterns emerge are long-standing questions, first addressed in the 19th century by Helmholtz, Kirchhoff, and Kelvin. Countless researchers have since contributed to innovative techniques and results. However, the overarching problem of swirling 2-D motion and its long-time behavior remains largely open. Here we shed light on this problem via a link to isospectral matrix flows. The link is established through V. Zeitlin’s beautiful model for the numerical discretization of Euler’s equations in 2-D. When considered on the sphere, Zeitlin’s model offers deep connections between 2-D hydrodynamics and unitary representations of the rotation group; consequently, it provides a dictionary that maps hydrodynamical concepts to matrix Lie theory, which in turn gives connections to matrix factorizations, random matrices, and integrability theory, for example. Results about finite-dimensional matrices can then be transferred to infinite-dimensional fluids via quantization theory, which is here used as an analysis tool (albeit traditionally describing the limit between quantum and classical physics). We demonstrate how the dictionary is constructed and how it unveils techniques for 2-D hydrodynamics. We also give accompanying convergence results for Zeitlin’s model on the sphere.