Charles Fefferman Laboratory

Charles Fefferman (Princeton University, USA)

Charles Fefferman is Professor at Princeton University Mathematics Department. He received the Fields Medal at the International Congress of Mathematicians celebrated in Helsinki in 1978. Fefferman’s research has had a great impact in many fields: mathematical analysis, partial differential equations, Fourier analysis, mathematical physics, fluid dynamics, neural networks and differential geometry.

This Laboratory is a continuation of the first (2012-20215), second (2016-2019) and third (2020-2023) ones.

Research field

Fluid mechanics, and, more concretely, Navier-Stokes problem.

Research project

The laboratory’s research will focus on solving problems in fluid mechanics, particularly the potential formation and propagation of singularities, commonly associated with the Navier-Stokes equations and Euler equations—a set of nonlinear partial differential equations that describe fluid motion. These equations govern phenomena such as atmospheric dynamics, ocean currents, and the flow around vehicles or projectiles, as well as any system involving Newtonian fluids. Despite their wide applicability, no general solution exists for these equations. The search for finite time singularities have become one of the major challenges in the area of mathematical analysis of non-linear partial differential equations. This remains one of the longstanding unsolved problems in mathematics and physics, persisting for over 250 years.

In 2023, the group presented a novel approach (arXiv:2308.12197 and arXiv:2309.08495) to establish a blow-up mechanism for the 3D incompressible Euler equations. They constructed solutions in two different scenarios within the local existence regime, both of which develop finite-time singularities with finite energy. Moreover, they establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations (arXiv: 2407.06776).

In this next phase, the lab will continue investigating singularities within the Euler and Navier-Stokes frameworks while expanding the new techniques developed. These will be applied to address a broader range of singularities and complex phenomena, such as turbulence.

Members

• Charles Fefferman(Princeton University) – chair
• Diego Córdoba (ICMAT-CSIC) – coordinator
• Oscar Domínguez (CUNEF)
• Runan He (ICMAT-CSIC)
• Andrés Lain Sanclemente (ICMAT-CSIC)
• José Antonio Lucas Manchón (ICMAT-CSIC)
• Luis Martínez-Zoroa (Basel University, Switzerland)
• Fan Zheng (ICMAT-CSIC)

Job Opportunities

• ICMAT POSTDOC SO 2024 – Analysis and PDEs

Activities

Papers

Preprints:

1. “Non Existence and Strong Ill-Posedness in $H^2$ for the Stable IPM Equation” (with R. Bianchini & L. Martínez-Zoroa) Preprint arXiv:2410.01297.
2. “Instantaneous continuous loss of regularity for the SQG equation” (with L. Martínez-Zoroa & Wojciech S. Ożański) Preprint arXiv:2409.18900.
3. “Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L_t^1C^{1, epsilon}cup L^2$ force” (with L. Martinez-Zoroa & F. Zheng) Preprint arxiv:2407.06776.
4. “Blow-up for the incompressible 3D-Euler equations with uniform $C^{1, 1/2-epsilon}cup L^2$ force” (with L. Martinez-Zoroa) Preprint arxiv:2309.08495.
5. “Finite time singularities to the 3D incompressible Euler equations for solutions in $C^{infty}(mathbb{R}^3 setminus {0})cap C^{1,alpha}cap L^2$” (with L. Martinez-Zoroa & F. Zheng) Preprint arxiv:2308.12197.
6. “Finite-time singularity formation for angled-crested water waves” (with Alberto Enciso & Nastasia Grubic) Preprint arXiv:2303.00027.

Publications (2024)

1. “Global unique solutions with instantaneous loss of regularity for SQG with fractional diffusion” (with Luis Martínez-Zoroa). Annals of PDE, 10 (21), 2024.
2. “Non-existence and strong ill-posedness in $C^{k,β}$ for the generalized Surface Quasi-geostrophic equation” (with Luis Martínez-Zoroa). Comm. Math. Phys. 405 (7), 2024.
3. “Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations” (with Luis Martínez-Zoroa & Wojciech Ozanski). Duke Math Journal. 173 (10), 2024.