Executive Summary : | Vortices are long-lived coherent structures in fluid flow, found in various physical systems and often two-dimensional. They can form geometrical patterns and can be modeled using the incompressible Euler equations of fluid dynamics. The study aims to construct vortex solutions to this equation, considering different models of vortices and the conditions under which they can be solutions of the equations of motion. Relative equilibria are an important class of solutions, where geometrical patterns of vortices do not move relative to one another but the entire pattern may rotate, translate, or be stationary. Two-dimensional vortex models considered are point vortices, vortex sheets, and diffuse smooth vorticity. Point vortices can model coherent vorticity, vortex sheets can model vorticity concentrated in a thin slice of fluid, and smooth vorticity can model background vorticity. The objective is to obtain equilibrium solutions that consist of a combination of these three types of vorticity, capturing the interaction between different vortex models. However, vortex sheet equilibria are not well-studied, with only a few rotationally symmetric equilibria described so far. Equilibria combining different vortex models are also rare. The current proposal aims to fill these gaps in our knowledge by focusing on the mathematical understanding of vortex sheet equilibria as limiting cases of point vortex equilibria. |