Existence of solutions for nonlocal and nonlinear problems

• Grant number: 23/05762-8
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: January 01, 2024
• End date: October 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Lucas Catão de Freitas Ferreira
• Grantee: Edison Fausto Cuba Huamani
• Supervisor: Peter Markowich
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: King Abdullah University Of Science & Technology,
• Associated to the scholarship: 21/10769-6 - Well-posedness and regularity theory for nonlocal and nonlinear problems, BP.PD

Abstract

The purpose of this project is to study time periodic solutions for the inviscid generalized surface quasi-geostrophic equations and a 1D transport equation with nonlocal velocities. We intend to obtain the existence of non trivial rotating and traveling vortex patches by suitable perturbation of stationary solutions. The construction of those special solutions are done through a combination of a desingularization of the vorticity and bifurcation theory. In general, the spectral problemis very delicate and strongly depends on the shape of the initial stationary solutions.