The solution of the Cauchy problem for the two-dimensional transport equation on a rotating plane

TL;DR

This paper analyzes the two-dimensional transport equation on a rotating plane, deriving explicit solutions and showing that the Coriolis force inhibits the formation of singularities in the solution.

Contribution

It provides an explicit asymptotic solution for the transport equation with Coriolis force and demonstrates its role in preventing singularities.

Findings

Explicit asymptotic representation of solutions

Coriolis force prevents singularity formation

Analysis of singularity development process

Abstract

The limiting case of the system of equations of two-dimensional gas dynamics in the presence of the Coriolis force, which can be obtained under the assumption of a small pressure, is considered. With this approach, the equation for the velocity vector (transport equation) is split off from the system and can be solved separately. Using the method of stochastic perturbation along characteristics, we obtain an explicit asymptotic representation of a smooth solution of transport equations and analyze the process of formation of singularities of solution using a specific example. It is concluded that the presence of the Coriolis force prevents the singularities formation.