CAMINFLOW · Computer-assisted Analysis and Applications of Moving Interfaces in Incompressible Flows
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2021-09-01 → 2023-08-31
- EU contribution
- €160,932
- Participants
- 1
- Scheme
- MSCA-IF
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Results in brief
Computer-assisted Analysis and Applications of Moving Interfaces in Incompressible Flows
Most physical phenomena, from fluids to biology, passing through economics or general relativity, are governed by Partial Differential Equations (PDEs). The lack of a rigorous understanding of these equations forces engineers and physicists to rely on heuristics, approximations and available data, which has undesirable implications in terms of efficiency and cost. Despite the diverse fields that give rise to PDEs, there is one conspicuous and fundamental question that unifies all of them: do solutions break down? If the model is correct, one expects that, given certain initial conditions, the evolution of the system is determined and predictable. However, whether solutions corresponding to smooth initial conditions propagate their regularity for all time or, on the contrary, form finite-time singularities is yet unknown for the majority of nonlinear PDE. In particular, among classical physics, fluid mechanics and the mystery of turbulence is an outstanding example, belonging to the exclusive list of Clay Millenium Problems. To delve into this regularity versus finite-time singularity issues, the evolution of fluid interfaces provides very rich scenarios. One could think for example in the smooth steady translation of small waves in the sea versus the turbulent wave breaking process near the seashore. Moreover, the techniques needed for their study are the basis to study the more complex case of fluid-structure moving interfaces, with strong implications in the modeling of bio-structures such as cells in the blood. In this project, we proposed to answer these questions for several scenarios: surface diffusion for a solid; movement of fluids in a porous medium (such as petroleum); and elastic immersed interfaces (vesicles). In addition to state-of-the-art mathematical methods to analyze these problems, this proposal includes the possibility of using computer-assisted proofs whenever traditional methods are not sufficient.
Data: CORDIS, © European Union
Project objective
The CAMINFLOW project aims to further explore the question of global regularity versus finite-time singularity formation in mathematical fluid mechanics. It proposes three horizons: 1) Modulated self-similar finite-time singularities in degenerate parabolic equations, 2) Fluid-interface finite-time singularities, 3) Rigorous analysis of fluid-structure moving interfaces.Module 1 is organized in two Work Packages: 1.1) Finite-time self-similar pinchoff for the axisymmetric surface diffusion equation (local, 1d), 1.2) Self-similar finite-time singularity in incompressible porous medium (nonlocal, 2d).Module 2 focuses on the blowup of the curvature of the Muskat problem (also known as Hele-Shaw).Module 3 contains two Work Packages: 3.1) Local and global well-posedness theory for the inextensible membrane problem. 3.2) Rigorous proof of the tumbling/tank-treading transition for inextensible membranes in a shear flow.A central and unifying method in this action is Computer-Assisted Proofs (CAP). Due to the highly demanding technical level of the analysis involved, new interval arithmetic libraries for singular integrals will be developed in Arb. Moreover, new modules in the framework Dedalus will be developed as well to perform accurate numerical simulations (that will help deciding whether a singularity is forming or not). These techniques will be applied complementing the methods from contour dynamics, harmonic analysis, and energy methods, needed to obtain results in the mathematical analysis of fluid interface problems.The CAMINFLOW project will be carried out by the experienced researcher, who worked during his PhD thesis on the global regularity question for incompressible fluid interfaces coming from nonlinear, nonlocal parabolic partial differential equations, and then as a postdoc moved on to fluid-structure elastic interfaces. The ER will collaborate with a Supervisor who is a prominent expert in CAP and their application to the fluid mechanics.
Original text from CORDIS.
Participants
- UNIVERSITAT DE BARCELONA · BarcelonaCoordinatorSpain
Links
Data: CORDIS, © European Union