Responsable: Francisco Gancedo García
Tipo de Proyecto/Ayuda: Plan Estatal 2017-2020 - Europa Excelencia
Referencia: EUR2020-112271
Fecha de Inicio: 01-12-2020
Fecha de Finalización: 30-11-2023

Empresa/Organismo financiador/es:

• Ministerio de Ciencia e Innovación

Equipo:

• Equipo de Trabajo:
  • Elena Salguero Quirós

Contratados:

• Investigadores:
  • Francisco José Mengual Bretón
• Técnicos/Personal Administrativo:
  • María Teresa Ayuga de la Flor

Resumen del proyecto:

This research proposal is about global in time regularity scenarios in  fluid dynamics. For fully consolidated models such as the Euler and Navier-Stokes equations, this important question still remains open in the incompressible case. To face this huge challenge, we propose several classical scenarios, considering the dynamics of an interface separating two immiscible  fluids.

These problems involve Navier-Stokes free boundary, the Muskat problem, sharp temperature front dynamics, etc. All of them modeled by well-established  fluid mechanical PDEs, extensively studied for their importance in mathematical physics as well as for their interest within mathematics. The main goals will be to prove global in time regularity.

New techniques, introduced in this area by the candidate, have allowed the analysis of several types of singularity formations for water waves, the Muskat problem, Navier-Stokes free boundary and surface quasi-geostrophic sharp temperature fronts. Furthermore, they have resulted in global existence results for the Muskat problem in critical spaces and the Navier-Stokes problem for twodimensional fluids of different densities (confirming a conjecture of P.L. Lions from 1996). Likewise, these methods have yielded conditional lack of blow-up theorems in important scenarios for those problems. These results are the  first analytical proofs of  finite-time blow-up for incompressible fluids in well-posed problems with  finite energy. In this project, I intend to further advance these achievements, reaching new milestones for the state-of-the-art in these central problems for incompressible  fluids.