ETD

• desingularization
• Euler equation
• nonlinear stability
• semilinear elliptic equations
• traveling vortex pairs
• uniqueness
• vorticity method

In this work, we present the desingularization of a pair of point vortices with opposite signs by constructing a family of traveling solutions for the 2D incompressible Euler equations associated with a general vorticity function, through a variational technique. Desingularization refers to the process of finding solutions to the Euler equations with vorticity concentrated around specific points, such that the support shrinks to these points. This process establishes a rigorous connection between the Euler equations and the classical point vortex model described by Helmholtz and Kirchhoff in the late 19th century.
A key contribution of this thesis is the analysis of the problem under weaker assumptions, which allow us to satisfy the Euler equations up to a small error.
Finally, we present the main ideas underlying the uniqueness and nonlinear stability results of these structures for the vortex patch case, recently solved by D. Cao, W. Zhan, G. Qin and C. Zou (2025). Their approach appears to be the most promising direction for establishing uniqueness in the general case, which remains an open problem even when the vorticity function is highly regular.