Research
My current research interests pertain to modelling fluid mechanical phenomena with partial differential equations.
One branch of interest is the evolution of the movement of rigid bodies in fluids, modelled by various equations (Euler, Stokes, Navier-Stokes, etc.), as well as the possibility of controlling the movement of such bodies, by the means of a control acting on the fluid. In particular in the case of the Navier-Stokes equations, this leads to difficulties due to the formation of boundary layers on the bodies. We use various mathematical methods (control theory, analysis of PDEs and ODEs, as well as geometry or complex analysis) in order to achieve our goals.
Another main research direction is that of using convex integration to model turbulence in fluids. In particular in the case of fluid mixing models (Kelvin-Helmholtz instability, Muskat problem, Rayleigh-Taylor instability), the typical results obtained from convex integration (existence of infinitely many weak solutions) are also reflected in the physical instability between the fluids. As such, one aim is to establish new models based on mean flow and differential inclusions, rather than weak solutions of differential equations, in order to mathematically deduce physically relevant information about the behaviour of such highly turbulent and unstable systems.
However, I am also interested in other models from mathematical physics where similar analysis can be carried out.
Publications
Remark: I would like to mention that here J. Kolumbán is my father, who is a Professor at the Babeş-Bolyai University of Cluj-Napoca, working in the field of Optimization. I always use the name József J. Kolumbán or J. J. Kolumbán when publishing articles.
1. (with Sz. András)
On the Ulam–Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Analysis: Theory, Methods & Applications Volume 82, April 2013, Pages 1–11.
http://www.sciencedirect.com/science/article/pii/S0362546X12004609
2. (with J. Kolumbán)
Meaned Spaces and a General Duality Principle, Chapter, Topics in Mathematical Analysis and Applications, Volume 94 of the series Springer Optimization and Its Applications pp 501-522, 11 July 2014.
http://link.springer.com/chapter/10.1007/978-3-319-06554-0_21
3. (with Sz. András)
Existence and localization of solutions for operatorial systems defined on Cartesian product of Fréchet spaces using a new vector version of Krasnoselskii’s cone compression–expansion theorem, Applied Mathematics and Computation Volume 265, 15 August 2015, Pages 40–50.
http://www.sciencedirect.com/science/article/pii/S0096300315005950
4. (with O. Glass and F. Sueur)
External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid, Analysis & PDE 13-3 (2020), 651--684. DOI: 10.2140/apde.2020.13.651
5.
Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid, Journal of Differential Equations, Volume 269, Issue 1, 15 June 2020, Pages 764-831.
https://www.sciencedirect.com/science/article/pii/S0022039619306679
6. (with B. Gebhard and L. Székelyhidi Jr.)
A new approach to the Rayleigh-Taylor instability, Archive for Rational Mechanincs and Analysis, 241, 1243–1280 (2021). DOI: https://doi.org/10.1007/s00205-021-01672-1
7. (with O. Glass and F. Sueur)
Remote trajectory tracking of rigid bodies immersed in a 2D perfect incompressible fluid, Pure and Applied Analysis 3-4 (2021), 613--652. DOI: 10.2140/paa.2021.3.613
8. (with B. Gebhard)
Relaxation of the Boussinesq system and applications to the Rayleigh-Taylor instability, Nonlinear Differential Equations and Applications 29, 7 (2022). DOI: 10.1007/s00030-021-00739-y
9. (with B. Gebhard)
On bounded two-dimensional globally dissipative Euler flows, SIAM Journal on Mathematical Analysis Vol. 54, Iss. 3 (2022). DOI:
10.
Remote trajectory tracking of a rigid body in an incompressible fluid at low Reynolds number, Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1135-1144.
11. (with B. Gebhard and J. Hirsch)
On a degenerate elliptic problem arising in the least action principle for Rayleigh-Taylor subsolutions, Ann. Inst. H. Poincaré Anal. Non Linéaire 41 (2024), no. 6, pp. 1527–1594, DOI: 10.4171/AIHPC/129
Talks
"Control of the motion of a rigid body immersed in a perfect two-dimensional fluid", Colloque Franco-Roumain de Mathématiques Appliquées, August 2016, Iasi, Romania.
"Control of the motion of a rigid body immersed in a perfect two-dimensional fluid", Séminaire des Jeunes Chercheurs du CEREMADE, September 2016, Paris, France.
"Control of the motion of a rigid body immersed in a perfect two-dimensional fluid", Réunion ANR IFSMACS, November 2016, Toulouse, France.
"External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid", VII Partial differential equations, optimal design and numerics workshop, August 2017, Benasque, Spain.
"External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid", Analysis Seminar, 9th of November 2017, Max Planck Institute for Mathematics, Leipzig, Germany.
"External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid", Analysis Seminar, 13th of November 2017, Basel University, Basel, Switzerland.
"Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid", Séminaire des Jeunes Chercheurs du CEREMADE, February 2018, Paris, France.
"Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid", Groupe de Travail Intéraction fluide-solide, 8th of March 2018, Bordeaux, France.
"Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid", Colloque Franco-Roumain de Mathématiques Appliquées, August 2018, Bordeaux, France.
"Control of the motion of a rigid body immersed in a perfect two-dimensional fluid", Leipzig-Prague Workshop, December 2018, Oberwiesenthal, Germany.
"Fluid-solid interaction and control", HIM Trimester on Evolution of Interfaces, January 2019, Bonn, Germany.
"Rayleigh-Taylor turbulence via convex integration", Leipzig-Prague Workshop, June 2019, Labská Stráň, Czech Republic.
"Rayleigh-Taylor turbulence via convex integration", 7th International Conference on Mathematics and Informatics, September 2019, Târgu Mureş, Romania.
"A new approach to the Rayleigh-Taylor instability", Oberseminar ANALYSIS - PROBABILITY, 15th of May 2020, Leipzig, Germany, video broadcast only.
"Convex Integration and Turbulence: From Nash's flat torus to mushroom clouds and nuclear fusion", A Magyar Tudomány Napja Erdélyben (Day of Hungarian Science in Transylvania), 13th of November 2020, Cluj-Napoca, Romania, video broadcast only.
"Remote motion planning of rigid bodies immersed in a 2D perfect incompressible fluid", 8th International Conference on Mathematics and Informatics, September 2021 Târgu Mureş, Romania, video broadcast only.
"A New Approach to the Rayleigh-Taylor Instability", Convex Integration and Nonlinear Partial Differential Equations Workshop, 10th of November 2021, Edinburgh, Scotland, video broadcast only. Full video available here.
"A New Approach to the Rayleigh-Taylor Instability", Journées Jeunes EDPistes 2022, 24th of March 2022, Lyon, France.
"Remote motion planning for the movement of rigid bodies in a 2D perfect incompressible fluid", Oberseminar Analysis, Felix Klein Lecture Room, 1st of June 2022, Leipzig, Germany.
"On recent results regarding the mathematical modelling of turbulent fluid mixing via convex integration", Optimization and Data Analysis Research Seminar, Budapest University of Technology and Economics, 13th of October 2022, Budapest, Hungary.
"A Generalized Least Action Principle for Rayleigh-Taylor subsolutions", Focused Workshop on Moffat's Magnetic Relaxation Problem, Erdős Center, Rényi Institute, 6th of July 2023, Budapest, Hungary.
"A Generalized Least Action Principle for Rayleigh-Taylor subsolutions", 9th International Conference on Mathematics and Informatics, September 2023, Târgu Mureş, Romania.
"Constructing potential solutions to the Euler equations, with applications to some control problems in fluid-solid interactions", Harmonic and Spectral Analysis 2023 Online Conference, 6th of October 2023.
"A Generalized Least Action Principle for Rayleigh-Taylor subsolutions", Analysis Seminar, Rényi Institute, 2nd of November 2023, Budapest, Hungary.
Organizing Responsibilities
"Fluid Mixing" Workgroup at the HIM Trimester on Evolution of Interfaces, each Thursday from 11 AM between January 24 and April 26 2019, Bonn, Germany.
Convex Integration Workgroup at the Budapest University of Technology and Economics/Rényi Institute (date and location may vary for certain sessions, contact me for details), starting from the 8th of November 2022, Budapest, Hungary.
"PDE Tasting: An invitation to some current problems in mathematical fluid mechanics" Spring School, Max Planck Institute for Mathematics in the Sciences, 27-31 March 2023, Leipzig, Germany. https://www.mis.mpg.de/calendar/conferences/2023/pde-tasting.html
"Harmonic and Spectral Analysis 2023" Conference (co-organizer), 4-6 October 2023, online only. http://mathspectral.hu/hsa-2023/