Periodic Reporting for period 4 - NONFLU (Non-local dynamics in incompressible fluids)
Okres sprawozdawczy: 2023-03-01 do 2024-08-31
The search for singularities in incompressible flows has become a major challenge in the area of non-linear partial differential equations and is relevant in applied mathematics, physics and engineering. The understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open scientific problems of classical physics. Before this project we have made a breakthrough in the formation of singularities and the creation of a mixing zone for the evolution of an interface generated between two immiscible incompressible fluids. These contour dynamics problems are given by basic fluid mechanics systems, such as Euler´s equation, Darcy´s law and the Quasi-geostrophic equation. These give rise to problems such as internal wave, Muskat, two-phase Hele-Shaw and evolution of sharp fronts. The fundamental questions to address are local-existence, uniqueness, global-existence of solutions or on the other hand, formation of singularities on the free boundary or ill-posedness.
The goal of this project is to pursue new methods in the mathematical analysis of non-local and non-linear partial differential equations. For this purpose, several physical scenarios of interest in the context of incompressible fluids are presented from a mathematical point of view as well as for its applications: both from the standpoint of global well-posedness, existence and uniqueness of solutions and as candidates for blow-up.
The equations we consider are the incompressible Euler equations, incompressible porous media equation and the generalized Quasi-geostrophic equation. This research will lead to a deeper understanding of the nature of the set of initial data that develops finite time singularities as well as those solutions that exist for all time for incompressible flows.
The goal of this project is to pursue new methods in the mathematical analysis of non-local and non-linear partial differential equations. For this purpose, several physical scenarios of interest in the context of incompressible fluids are presented from a mathematical point of view as well as for its applications: both from the standpoint of global well-posedness, existence and uniqueness of solutions and as candidates for blow-up.
The equations we consider are the incompressible Euler equations, incompressible porous media equation and the generalized Quasi-geostrophic equation. This research will lead to a deeper understanding of the nature of the set of initial data that develops finite time singularities as well as those solutions that exist for all time for incompressible flows.
The main results of this project are categorized into three main areas: interface dynamics (water waves, Muskat problem, and SQG patches), classical solutions with no boundary (finite energy), and generalized quasi-geostrophic equations.
1. Incompressible Euler Equations
Stationary solutions and splash singularities: We demonstrated the existence of stationary solutions with a self-intersecting interface for the 2D incompressible free boundary Euler equations with two fluids, using weighted estimates for self-intersecting interfaces.
Reference: Cordoba, Enciso & Grubic. Annals of PDE, 2021.
Local well-posedness with interfaces featuring corners: We established local well-posedness for water waves with cornered interfaces without symmetry requirements, proving that initial cornered interfaces generically change angle, potentially leading to finite-time singularities.
References: Cordoba, Enciso & Grubic. Advances in Math, 2023; Preprint arXiv:2303.00027.
Stability of traveling waves in Burgers-Hilbert equation: We used energy methods to prove stability and existence times for solutions close to periodic traveling waves, with applications to incompressible Euler equations.
Reference: Castro, Cordoba & Zheng. Anal. PDE, 2023.
Instantaneous Sobolev regularity loss: Solutions to the 2D incompressible Euler equations in certain super-critical Sobolev spaces lose regularity instantly but still exist globally in time.
Reference: Cordoba, Martínez-Zoroa & Ozanski. Duke Math Journal, 2024.
Blow-up mechanisms for 3D incompressible Euler equations: We introduced a new approach that demonstrates finite-time singularity formation, providing novel insights into blow-up scenarios for the 3D Euler and hypodissipative Navier-Stokes equations.
References: Cordoba, Martinez-Zoroa & Zheng, arXiv:2407.06776; arXiv:2309.08495; arXiv:2308.12197.
2. Incompressible Porous Media Equation
Mixing solutions in Muskat problem: We established mixing solutions for the incompressible porous media equation in the unstable regime with Muskat-type initial data, combining convex integration and contour dynamics.
Reference: Castro, Cordoba & Faraco. Inventiones Mathematicae, 2021.
Global well-posedness with large slopes: For the 2D Muskat problem, we showed global existence for strong solutions with finite energy and large interface slopes using a novel formulation involving oscillatory terms.
Reference: Cordoba & Lazar. Annales Scientifiques de l'École Normale Supérieure, 2021.
Confined IPM equation with stratified solutions: We proved the global existence of smooth solutions for the inviscid incompressible porous media equation under specific initial stratification, removing boundary terms in the energy estimates.
Reference: Castro, Cordoba & Lear. Arch. Ration. Mech. Anal, 2019.
Strong ill-posedness in H²: We constructed perturbations in the incompressible porous media equation that lead to strong ill-posedness in H², neutralizing stability near the origin.
Reference: Bianchini, Cordoba & Martínez-Zoroa. Preprint arXiv:2410.01297.
Finite-time singularities in IPM equation: We demonstrated finite-time singularity formation in smooth solutions of the 2D incompressible porous media equation with a compact smooth source.
Reference: Cordoba & Martínez-Zoroa. Preprint arXiv:2410.22920.
3. Generalized Quasi-Geostrophic (SQG) Equations
Lifespan of classical solutions: For solutions close to radial stationary solutions in the inviscid SQG equation, we analyzed the existence time, establishing global solutions via bifurcation.
Reference: Cordoba, Castro & Zheng. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2021.
Stable global solutions in gSQG patch equation: We achieved global stability for patch solutions under the generalized SQG equation with α ∈ (1,2), demonstrating the first construction of non-trivial global solutions.
Reference: Cordoba, Gómez-Serrano & Ionescu. Arch. Ration. Mech. Anal, 2019.
Global smooth solutions in SQG: We identified the first non-trivial family of global smooth solutions for the inviscid SQG equation.
Reference: Castro, Cordoba & Gómez-Serrano. Mem. Amer. Math. Soc., 2020.
Instantaneous loss of regularity: Our research proved global solutions exhibiting instant loss of critical and super-critical regularity for gSQG equations.
References: Cordoba & Martínez-Zoroa, Annals of PDE, 2024; Comm. Math. Phys., 2024; Adv. Math., 2022; Preprint arXiv:2409.18900.
These results provide significant advancements in fluid dynamics, PDEs, and singularity formation, establishing theoretical foundations and new approaches in both compressible and incompressible fluid scenarios.
1. Incompressible Euler Equations
Stationary solutions and splash singularities: We demonstrated the existence of stationary solutions with a self-intersecting interface for the 2D incompressible free boundary Euler equations with two fluids, using weighted estimates for self-intersecting interfaces.
Reference: Cordoba, Enciso & Grubic. Annals of PDE, 2021.
Local well-posedness with interfaces featuring corners: We established local well-posedness for water waves with cornered interfaces without symmetry requirements, proving that initial cornered interfaces generically change angle, potentially leading to finite-time singularities.
References: Cordoba, Enciso & Grubic. Advances in Math, 2023; Preprint arXiv:2303.00027.
Stability of traveling waves in Burgers-Hilbert equation: We used energy methods to prove stability and existence times for solutions close to periodic traveling waves, with applications to incompressible Euler equations.
Reference: Castro, Cordoba & Zheng. Anal. PDE, 2023.
Instantaneous Sobolev regularity loss: Solutions to the 2D incompressible Euler equations in certain super-critical Sobolev spaces lose regularity instantly but still exist globally in time.
Reference: Cordoba, Martínez-Zoroa & Ozanski. Duke Math Journal, 2024.
Blow-up mechanisms for 3D incompressible Euler equations: We introduced a new approach that demonstrates finite-time singularity formation, providing novel insights into blow-up scenarios for the 3D Euler and hypodissipative Navier-Stokes equations.
References: Cordoba, Martinez-Zoroa & Zheng, arXiv:2407.06776; arXiv:2309.08495; arXiv:2308.12197.
2. Incompressible Porous Media Equation
Mixing solutions in Muskat problem: We established mixing solutions for the incompressible porous media equation in the unstable regime with Muskat-type initial data, combining convex integration and contour dynamics.
Reference: Castro, Cordoba & Faraco. Inventiones Mathematicae, 2021.
Global well-posedness with large slopes: For the 2D Muskat problem, we showed global existence for strong solutions with finite energy and large interface slopes using a novel formulation involving oscillatory terms.
Reference: Cordoba & Lazar. Annales Scientifiques de l'École Normale Supérieure, 2021.
Confined IPM equation with stratified solutions: We proved the global existence of smooth solutions for the inviscid incompressible porous media equation under specific initial stratification, removing boundary terms in the energy estimates.
Reference: Castro, Cordoba & Lear. Arch. Ration. Mech. Anal, 2019.
Strong ill-posedness in H²: We constructed perturbations in the incompressible porous media equation that lead to strong ill-posedness in H², neutralizing stability near the origin.
Reference: Bianchini, Cordoba & Martínez-Zoroa. Preprint arXiv:2410.01297.
Finite-time singularities in IPM equation: We demonstrated finite-time singularity formation in smooth solutions of the 2D incompressible porous media equation with a compact smooth source.
Reference: Cordoba & Martínez-Zoroa. Preprint arXiv:2410.22920.
3. Generalized Quasi-Geostrophic (SQG) Equations
Lifespan of classical solutions: For solutions close to radial stationary solutions in the inviscid SQG equation, we analyzed the existence time, establishing global solutions via bifurcation.
Reference: Cordoba, Castro & Zheng. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2021.
Stable global solutions in gSQG patch equation: We achieved global stability for patch solutions under the generalized SQG equation with α ∈ (1,2), demonstrating the first construction of non-trivial global solutions.
Reference: Cordoba, Gómez-Serrano & Ionescu. Arch. Ration. Mech. Anal, 2019.
Global smooth solutions in SQG: We identified the first non-trivial family of global smooth solutions for the inviscid SQG equation.
Reference: Castro, Cordoba & Gómez-Serrano. Mem. Amer. Math. Soc., 2020.
Instantaneous loss of regularity: Our research proved global solutions exhibiting instant loss of critical and super-critical regularity for gSQG equations.
References: Cordoba & Martínez-Zoroa, Annals of PDE, 2024; Comm. Math. Phys., 2024; Adv. Math., 2022; Preprint arXiv:2409.18900.
These results provide significant advancements in fluid dynamics, PDEs, and singularity formation, establishing theoretical foundations and new approaches in both compressible and incompressible fluid scenarios.
Progress beyond the state of the art: see above