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| Author(s): |
Ricardo Martins Mendes Guimarães
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | Campinas, SP. |
| Institution: | Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica |
| Defense date: | 2024-03-26 |
| Examining board members: |
Anne Caroline Bronzi;
Lucas Catão de Freitas Ferreira;
Gabriela Del Valle Planas;
Cilon Valdez Ferreira Perusato
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| Advisor: | Anne Caroline Bronzi; Cecilia Freire Mondaini |
| Abstract | |
Questions related to the existence of global smooth solutions over time or the possibility of developing singularities in finite time for certain equations in fluid dynamics can be challenging and complex. In this work, we analyze possible scenarios of self-similar singularity formation for the generalized surface quasi-geostrophic equation (gSQG) in two dimensions. We show that, under a condition on the growth of the $L^r$ norm of the self-similar profile and its gradient, the self-similar profile is identically zero or its asymptotic $L^p$ behavior can be characterized, for suitable $p$ and $r$, in appropriate intervals of the self-similar parameter. This result generalizes and improves the analogous result proven for the SQG equation in \cite{Xue16}, and recovers the results proven in \cite{CX15}, related to globally self-similar solutions of the gSQG equation. We also analyze the gSQG equation with fractional dissipation, generalizing the result proven in \cite{Chae11}, which excludes the possibility of globally self-similar singularity in finite time for the dissipative SQG equation, under certain conditions on the profile. More precisely, assuming that the gradient of the self-similar profile decays to zero at infinity and the symmetric part of the self-similar velocity gradient is bounded at the maximum points of the self-similar profile, we prove that the self-similar profile is identically zero in $\R^2.$ (AU) | |
| FAPESP's process: | 18/22385-5 - On the global well-posedness of hydrodynamic models |
| Grantee: | Ricardo Martins Mendes Guimarães |
| Support Opportunities: | Scholarships in Brazil - Doctorate |