Periodic Reporting for period 1 - CAMINFLOW (Computer-assisted Analysis and Applications of Moving Interfaces in Incompressible Flows)
Reporting period: 2021-09-01 to 2023-08-31
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.
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.
The project has resulted in three papers:
1. E. Garcia-Juarez, J. Gómez-Serrano, H. Q. Nguyen, B. Pausader. Self-similar solutions for the Muskat equation. Advances in Mathematics, 399, (2022)
2. F. Gancedo, E. Garcia-Juarez. Quantitative Hölder Estimates for Even Singular Integral Operators on Patches. Journal of Functional Analysis, 283 (9), (2022).
3. F. Gancedo, E. Garcia-Juarez. Global Regularity of 2D N-S Free Boundary with Small Viscosity Contrast. To appear in Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, (2023),
and two more papers will be submitted to journals in the following month. Two additional works are expected to be completed and submitted to journals within six months.
Paper 1 is directly related to the objective about the evolution of fluids in a porous medium. The objective proposed (proof of curvature blow-up in the Muskat equation) turned out to be false, but the work on this point has resulted in the proof of the opposite fact (desingularization of corners). The ER has been invited to give talks about this result at seminars in ICERM, Brown University, Chicago and at the RSME bi-annual conference. In addition to the aforementioned paper, the ER in collaboration with J. Gomez-Serrano, S. Haziot and B. Pausader, has extended the methods to deal with multiple moving corners (result under preparation;) the co-author S. Haziot has been invited to give a talk about it at the prestigious BIRS center at Banff. Moreover, the technique in Paper 1 is now being used to obtain the first proof of finite-time singularities for a closely related equation.
The ER, in collaboration with P.C. Kuo, Y. Mori, and R. Strain, has proved the local-in-time existence for the problem of an inextensible string immersed in a Stokes fluid, and this result is now under preparation. Moreover, the ER and collaborators have obtained the local well-posedness for the problem of a three-dimensional elastic membrane immersed in a fluid, which should serve as a first step to extend the inextensible string result to 3D models for vesicles.
1. E. Garcia-Juarez, J. Gómez-Serrano, H. Q. Nguyen, B. Pausader. Self-similar solutions for the Muskat equation. Advances in Mathematics, 399, (2022)
2. F. Gancedo, E. Garcia-Juarez. Quantitative Hölder Estimates for Even Singular Integral Operators on Patches. Journal of Functional Analysis, 283 (9), (2022).
3. F. Gancedo, E. Garcia-Juarez. Global Regularity of 2D N-S Free Boundary with Small Viscosity Contrast. To appear in Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, (2023),
and two more papers will be submitted to journals in the following month. Two additional works are expected to be completed and submitted to journals within six months.
Paper 1 is directly related to the objective about the evolution of fluids in a porous medium. The objective proposed (proof of curvature blow-up in the Muskat equation) turned out to be false, but the work on this point has resulted in the proof of the opposite fact (desingularization of corners). The ER has been invited to give talks about this result at seminars in ICERM, Brown University, Chicago and at the RSME bi-annual conference. In addition to the aforementioned paper, the ER in collaboration with J. Gomez-Serrano, S. Haziot and B. Pausader, has extended the methods to deal with multiple moving corners (result under preparation;) the co-author S. Haziot has been invited to give a talk about it at the prestigious BIRS center at Banff. Moreover, the technique in Paper 1 is now being used to obtain the first proof of finite-time singularities for a closely related equation.
The ER, in collaboration with P.C. Kuo, Y. Mori, and R. Strain, has proved the local-in-time existence for the problem of an inextensible string immersed in a Stokes fluid, and this result is now under preparation. Moreover, the ER and collaborators have obtained the local well-posedness for the problem of a three-dimensional elastic membrane immersed in a fluid, which should serve as a first step to extend the inextensible string result to 3D models for vesicles.
The ideas introduced to obtain the first self-similar solutions to the Muskat problem about fluids in a porous medium are now being used to find a finite-time singularity in a related equation, which would provide a major result in the field. Moreover, the techniques developed to deal with multiple moving corners are very flexible and could be used to study more general settings, including those with more real effects such as surface tension or fluids with different viscosities.
The works on elastic immersed membranes and inextensible strings, to be submitted, will each of them open the way to study further questions such as global in time behavior, appearance of bifurcation phenomena, convergence of numerical methods, to name a few.
The works on elastic immersed membranes and inextensible strings, to be submitted, will each of them open the way to study further questions such as global in time behavior, appearance of bifurcation phenomena, convergence of numerical methods, to name a few.