Summary
The goal of this proposal is twofold: on the one hand to pursue methods and ideas developed in recent work in the search for either singularities or global existence in incompressible fluids with finite energy and on the
other transfer the techniques to solve long-standing open problems in spectral geometry. A key ingredient in its success is to have accurate numerics together with a deep understanding of the regularity theory. Therefore, the interdisciplinary nature of this project, which involves numerical computations, computer-assisted proofs, modern PDE methods and harmonic analysis, is an essential ingredient for the successful outcome.
This proposal is divided in three blocks, the first two involving global existence and/or singularities for: the incompressible Euler and Navier-Stokes equations; the surface quasi-geostrophic (SQG), the generalized-SQG equations and related models; and a third one on applications to spectral geometry. There is a strong analogy between the SQG and the 3D Euler equations, and many results that hold for the former also hold for the latter.
A major theme is the interplay between rigorous computer calculations and traditional mathematics. Interval arithmetics are used as part of a proof whenever they are needed. As an evidence of its capabilities, I have pioneered techniques to show singularities in PDE related to fluid mechanics – even in low regularity settings –, developed a way to treat singular integrals, and solved eigenvalue problems using computer-assisted proofs. This is a completely novel approach that can be blended with more classical ones, resulting in very powerful theorems solving problems that can not be treated currently with pen and paper methods.
other transfer the techniques to solve long-standing open problems in spectral geometry. A key ingredient in its success is to have accurate numerics together with a deep understanding of the regularity theory. Therefore, the interdisciplinary nature of this project, which involves numerical computations, computer-assisted proofs, modern PDE methods and harmonic analysis, is an essential ingredient for the successful outcome.
This proposal is divided in three blocks, the first two involving global existence and/or singularities for: the incompressible Euler and Navier-Stokes equations; the surface quasi-geostrophic (SQG), the generalized-SQG equations and related models; and a third one on applications to spectral geometry. There is a strong analogy between the SQG and the 3D Euler equations, and many results that hold for the former also hold for the latter.
A major theme is the interplay between rigorous computer calculations and traditional mathematics. Interval arithmetics are used as part of a proof whenever they are needed. As an evidence of its capabilities, I have pioneered techniques to show singularities in PDE related to fluid mechanics – even in low regularity settings –, developed a way to treat singular integrals, and solved eigenvalue problems using computer-assisted proofs. This is a completely novel approach that can be blended with more classical ones, resulting in very powerful theorems solving problems that can not be treated currently with pen and paper methods.
Unfold all
/
Fold all
More information & hyperlinks
| Web resources: | https://cordis.europa.eu/project/id/852741 |
| Start date: | 01-07-2020 |
| End date: | 30-06-2025 |
| Total budget - Public funding: | 1 483 073,00 Euro - 1 483 073,00 Euro |
Cordis data
Original description
The goal of this proposal is twofold: on the one hand to pursue methods and ideas developed in recent work in the search for either singularities or global existence in incompressible fluids with finite energy and on theother transfer the techniques to solve long-standing open problems in spectral geometry. A key ingredient in its success is to have accurate numerics together with a deep understanding of the regularity theory. Therefore, the interdisciplinary nature of this project, which involves numerical computations, computer-assisted proofs, modern PDE methods and harmonic analysis, is an essential ingredient for the successful outcome.
This proposal is divided in three blocks, the first two involving global existence and/or singularities for: the incompressible Euler and Navier-Stokes equations; the surface quasi-geostrophic (SQG), the generalized-SQG equations and related models; and a third one on applications to spectral geometry. There is a strong analogy between the SQG and the 3D Euler equations, and many results that hold for the former also hold for the latter.
A major theme is the interplay between rigorous computer calculations and traditional mathematics. Interval arithmetics are used as part of a proof whenever they are needed. As an evidence of its capabilities, I have pioneered techniques to show singularities in PDE related to fluid mechanics – even in low regularity settings –, developed a way to treat singular integrals, and solved eigenvalue problems using computer-assisted proofs. This is a completely novel approach that can be blended with more classical ones, resulting in very powerful theorems solving problems that can not be treated currently with pen and paper methods.
Status
TERMINATEDCall topic
ERC-2019-STGUpdate Date
27-04-2024
Images
No images available.
Geographical location(s)
