2020/02/03 by Yuuki Shimizu, Shimizu, Yuuki
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2002.00626
openalex publication_date 2020/02/03 · openalex created_date 2020/08/21 · openalex updated_date 2026/07/28
The point vortex dynamics in background fields on surfaces is justified as an Euler-Arnold flow in the sense of de Rham currents. We formulate a current-valued solution of the Euler-Arnold equation with a regular-singular decomposition. For the solution, we first prove that, if the singular part of the vorticity is given by a linear combination of delta functions centered at qn(t) for n=1,…,N, qn(t) is a solution of the point vortex equation. Conversely, we next prove that, if qn(t) is a solution of the point vortex equation for n=1,…,N, there exists a current-valued solution of the Euler-Arnold equation with a regular-singular decomposition such that the singular part of the vorticity is given by a linear combination of delta functions centered at qn(t). As a corollary, we generalize the Bernoulli law to the case where the flow field is a curved surface and where the presence of point vortices is taken into account. From the viewpoint of the application, the mathematical justification is of a significance since the point vortex dynamics in the rotational vector field on the unit sphere is adapted as a mathematical model of geophysical flow in order to take effect of the Coriolis force on inviscid flows into consideration.