ERC Consolidator Grant
Stable solutions and nonstandard diffusions:
PDE questions arising in Mathematical Physics
SSNSD
2024 – 2029
The concept of diffusion is ubiquitous in the physical sciences. From the mathematical point of view, its study started in the early 19th century with the development of PDE theory, and has many connections to Physics, Probability, Geometry, and Functional Analysis. This project aims to answer several outstanding questions related to the mathematics of diffusion.
The proposal is divided into two blocks. The first one corresponds to the study of stable solutions to reaction-diffusion PDE, and more precisely the classification of global stable solutions in the physical space for a general class of problems including the Allen-Cahn, the Alt-Phillips, or the thin Alt-Caffarelli equations. We will also investigate the same question for complex-valued solutions in 2D, which arises in the construction of nite-energy travelling waves for the Gross-Pitaevskii equation.
The second block corresponds to nonstandard diffusions. In particular, we will study the Boltzmann equation (a fundamental model in statistical mechanics), nonlocal diffusions (deeply related to Lévy processes and “anomalous diffusions”), as well as the porous medium equation (a classical nonlinear PDE that arises in various physical models in which diffusion is “slow”).
Research Team
Prof. Xavier Ros Oton
Principal Investigator
Xavier is an ICREA Research Professor at the University of Barcelona since 2020. Previously, he has been Assistant Professor at Universität Zürich, as well as R. H. Bing Instructor at the University of Texas at Austin. He is a mathematician who works on Partial Differential Equations (PDEs). Specifically, he studies the regularity of solutions to elliptic and parabolic PDEs, and he is mostly known for his results on free boundary problems and integro-differential equations. He has been the PI of an ERC Starting Grant (2019-2024) and an ERC Consolidator Grant (2024-2029), and has received several awards for young mathematicians in Spain, including the ‘Premio Nacional de Investigación’ for researchers under 40 in Mathematics and Computer Science. He also received the Scientific Research Award from the Fundación Princesa de Girona, as well as the Stampacchia Gold Medal, an international prize awarded every three years in recognition of outstanding contributions to the Calculus of Variations. In 2022, he was elected member of the Spanish Royal Academy of Sciences.
For further information on Xavier’s previous publications and research career, please refer to his personal website.
Marvin Weidner
Marvin Weidner is a postdoctoral researcher in the ERC project “SSNSD” at Universitat de Barcelona, where he previously joined the ERC research team of Prof. Xavier Ros-Oton for the project “EllipticPDE”. Before coming to Barcelona in September 2022, he did his PhD at Bielefeld University (Germany). His research interests lie in partial differential equations with an emphasis on the regularity theory for nonlocal problems.
Joan Domingo
He began his PhD studies in October 2024 under the supervision of Xavier Ros-Oton in the Department of Mathematics at the Universitat de Barcelona.
Alejandro Martínez
He began his PhD studies in October 2024 under the supervision of Xavier Ros-Oton in the Department of Mathematics at the Universitat de Barcelona.
This project has received funding from the European Research Council (ERC) research and innovation programme in the panel ERC Consolidator Grant under Agreement No. 101123223.
The ERC’s mission is to encourage the highest quality research in Europe through competitive funding and to support investigator-driven frontier research across all fields, on the basis of scientific excellence. ERC grants are awarded through open competition to projects headed by starting and established researchers in Europe, irrespective of their origins. The sole criterion for selection is scientific excellence.
In the long term ERC looks to substantially strengthen and shape the European research system.
For more information, please visit ERC’s website.
The University of Barcelona (UB) is the home institution of the SSNSD project. Established in 1450, is one of the top universities in Southern Europe, and it ranks first among Spanish universities according to the QS World University Rankings and the Shanghai Ranking. It has become a European benchmark for research activity, both in terms of the number of research programmes that it conducts and the excellent results achieved.
The Mathematics Department at UB has a long tradition and it has been an important center of research in mathematics for decades. Many of their professors are members of the IMUB, an institute that was created to foster and support research in all areas of Mathematics in 2000. The IMUB hosts conferences, workshops, seminars and advanced courses, and promotes interdisciplinary collaboration.
Publications
Regularity for the Boltzmann equation conditional to pressure and moment bounds,
Xavier Fernandez-Real, Xavier Ros-Oton, Marvin Weidner,
preprint arXiv (2024).
C^\infty regularity in semilinear free boundary problems,
Daniel Restrepo, Xavier Ros-Oton,
preprint arXiv (2024).
L^p estimates for the Laplacian via blow-up,
Jan Lewenstein-Sanpera, Xavier Ros-Oton,
preprint arXiv (2024).
Improvement of flatness for nonlocal free boundary problems,
Xavier Ros-Oton, Marvin Weidner,
preprint arXiv (2024).
Regularity for nonlocal equations with local Neumann boundary conditions,
Xavier Ros-Oton, Marvin Weidner,
preprint arXiv (2024).
Optimal regularity for nonlocal elliptic equations and free boundary problems,
Xavier Ros-Oton, Marvin Weidner,
preprint arXiv (2024).
Schauder and Cordes-Nirenberg estimates for nonlocal elliptic equations with singular kernels,
Xavier Fernandez-Real, Xavier Ros-Oton,
Proc. Lond. Math. Soc. 129 (2024), e12629, 47pp.
Optimal regularity and fine asymptotics for the porous medium equation in bounded domains,
Tianling Jin, Xavier Ros-Oton, Jingang Xiong,
J. Reine Angew. Math. 809 (2024), 269-300.