Axisymmetric self-similar finite-time singularity solution of the Euler equations

• Published in: Advances in continuous and discrete models 2023-05, Vol.2023 (1), p.30, Article 30
• Main Authors: Cádiz, Rodrigo, Martínez-Argüello, Diego, Rica, Sergio
• Format: Article
• Language: English
• Subjects: Analysis; Approximation; Boundary conditions; Coordinates; Difference and Functional Equations; Eigenvalues; Euler-Lagrange equation; Eulers equations; Fluid flow; Functional Analysis; Incompressible flow; Inviscid flow; Kinetic Theory and Related Topics: A Tribute to the Memory of Jean Ginibre; Mathematics; Mathematics and Statistics; Nonlinear Waves; Ordinary Differential Equations; Partial Differential Equations; Self-similarity; Singularities; Velocity; Vorticity
• ISSN: 2731-4235; 1687-1839; 2731-4235; 1687-1847
• DOI: 10.1186/s13662-023-03774-4

Self-similar finite-time singularity solutions of the axisymmetric Euler equations in an infinite system with a swirl are provided. Using the Elgindi approximation of the Biot–Savart kernel for the velocity in terms of vorticity, we show that an axisymmetric incompressible and inviscid flow presents a self-similar finite-time singularity of second specie, with a critical exponent ν . Contrary to the recent findings by Hou and collaborators, the current singularity solution occurs at the origin of the coordinate system, not at the system’s boundaries or on an annular rim at a finite distance. Finally, assisted by a numerical calculation, we sketch an approximate solution and find the respective values of ν . These solutions may be a starting point for rigorous mathematical proofs.