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:

doi.org/10.1137/21M1454675 

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.

doi.org/10.5802/crmath.374 

11. (with B. Gebhard and J. Hirsch)

On a degenerate elliptic problem arising in the least action principle for Rayleigh-Taylor subsolutions, preprint 2022.

arxiv.org/abs/2208.14495 

Talks

Organizing Responsibilities