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.
