Study of possible scenarios of singularity formation for generalized surface quasi-geostrophic equations

• Grant number: 24/00744-4
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: June 01, 2024
• End date: May 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Lucas Catão de Freitas Ferreira
• Grantee: Ricardo Martins Mendes Guimarães
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated scholarship(s): 25/08196-9 - Stability of solutions for 2D anisotropic gradient flows and Blow-Up Conditions in Some 2D Fluid Equations, BE.EP.PD

Abstract

Our main aim in this project is to study possible singularity formation scenarios for a class of generalized dissipative surface quasi-geostrophic equations, in which global well-posedness or singularity in finite time are still open problems. Thus, our proposal is to obtain results about the exclusion of the self-similar blow-up scenario, and in case of non-exclusion, to obtain results about the asymptotic behavior of the self-similar profile in Lp-spaces. Moreover, we will investigate some geometries of the sets of level curves of the solution, since numerical simulations suggested a singular behavior for the inviscid SQG equations when the geometry of the level curves presents a saddle hyperbolic.