Prof. Dr. Juan J. L. Velázquez

E-mail: velazquez(at)
Phone: +49 228 73 62378
Room: 2.023
Location: Mathematics Center
Institute: Institute for Applied Mathematics
Research Area: Research Area B
Birthdate: 27.Jun 1964
Mathscinet-Number: 289301

Academic Career


PhD, Mathematics, Complutense University of Madrid, Spain

1991 - 1992

Postdoctoral stay, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN, USA

1992 - 1997

Associate Professor, Applied Mathematics Department, Complutense University of Madrid, Spain

1997 - 2008

Professor, Applied Mathematics Department, Complutense University of Madrid, Spain

2008 - 2011

Research Professor, Institute of Mathematical Sciences (ICMAT), Spanish National Research Council (CSIC), Madrid, Spain

Since 2011

Professor (W3), University of Bonn

Research Profile

My field of expertise is the analysis of Partial Differential Equations. In particular, I have been concerned with the study of singularities arising in Nonlinear Differential Equations and the study of the asymptotic behaviour of their solutions in a neighbourhood of the singular points.
My main current research interests is the study of several equations arising in Kinetic Theory. Specific equations in which I have worked recently are in the study of the bosonic Nordheim-Boltzmann equation, the kinetic equations describing Wave Turbulence, and coagulation equations. Some problems which I have studied about these models are the onset of singularities in finite time, the construction of self-similar behaviour which describe the long time asymptotics of the solutions of some of these equations and to determine if they are unique. I am also interested in the study of coarsening models for systems of particles interacting by means of short range potentials, in the study of screening effects in many particle systems and in the study of the properties of large chemical networks as the ones which arise often in problems of Mathematical Biology.

Concerning my future resarch plans, besides continuing with the study of the properties of the Kinetic Models mentioned above, I am interested in understanding the precise connection between the kinetic equations and the underlying particle systems which the equations are expected to approximate. Some specific problems which I intend to address in the future include the study of the mechanical properties of systems described by the Boltzmann equation and the analysis of systems with long range interactions. I am also interested in understanding possible oscillatory behaviours arising in Smoluchowski equations, as well as in the study of coarsening models interacting by means of short range potentials for which might be possible to prove rigorously that the distribution of particle sizes behaves in a probabilistic manner.

Research Projects and Activities

Responsible of two grants of the Spanish Government about “Partial Differential Equations in Mathematical Physics”
2004 - 2007 and 2007 - 2010

Collaborative Research Center SFB 1060 “The mathematics of emergent effects”

Contribution to Research Areas

Research Area B
One of my research directions is the study of pattern formation, in particular aggregation patterns in biological systems (cf. [1], [2], [3]). In these papers questions related to the formation of singularities for classical models describing chemotaxis in biological systems. Recently, I have studied the patterns that arise in models in which the interactions between organisms take place by means of local signals (cf. [4], [5], [6]).

I am also interested in the study of coagulation models as well as kinetic models with fluxes of particles between different regions of the phase space. Well-posedness results for coagulation equations as well as for the quantum Boltzmann equation can be found in [7], [8], [9], as well as in other forthcoming papers.

Selected Publications

[1] Miguel A. Herrero, Juan J. L. Velázquez
Singularity patterns in a chemotaxis model
Math. Ann. , 306: (3): 583--623
ISSN: 0025-5831
DOI: 10.1007/BF01445268
[2] J. J. L. Velázquez
Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions
SIAM J. Appl. Math. , 64: (4): 1198--1223
ISSN: 0036-1399
DOI: 10.1137/S0036139903433888
[3] Elio Eduardo Espejo Arenas, Angela Stevens, Juan J. L. Velázquez
Simultaneous finite time blow-up in a two-species model for chemotaxis
Analysis (Munich) , 29: (3): 317--338
ISSN: 0174-4747
DOI: 10.1524/anly.2009.1029
[4] A. Stevens, J. J. L. Velázquez
Partial differential equations and non-diffusive structures
Nonlinearity , 21: (12): T283--T289
ISSN: 0951-7715
DOI: 10.1088/0951-7715/21/12/T04
[5] Kyungkeun Kang, Benoit Perthame, Angela Stevens, J. J. L. Velázquez
An integro-differential equation model for alignment and orientational aggregation
J. Differential Equations , 246: (4): 1387--1421
ISSN: 0022-0396
DOI: 10.1016/j.jde.2008.11.006
[6] Kyungkeun Kang, Angela Stevens, Juan J. L. Velázquez
Qualitative behavior of a Keller-Segel model with non-diffusive memory
Comm. Partial Differential Equations , 35: (2): 245--274
ISSN: 0360-5302
DOI: 10.1080/03605300903473400
[7] M. Escobedo, S. Mischler, J. J. L. Velázquez
On the fundamental solution of a linearized Uehling-Uhlenbeck equation
Arch. Ration. Mech. Anal. , 186: (2): 309--349
ISSN: 0003-9527
DOI: 10.1007/s00205-007-0084-2
[8] M. Escobedo, S. Mischler, J. J. L. Velázquez
Singular solutions for the Uehling-Uhlenbeck equation
Proc. Roy. Soc. Edinburgh Sect. A , 138: (1): 67--107
ISSN: 0308-2105
DOI: 10.1017/S0308210506000655
[9] Miguel Escobedo, J. J. L. Velázquez
On the fundamental solution of a linearized homogeneous coagulation equation
Comm. Math. Phys. , 297: (3): 759--816
ISSN: 0010-3616
DOI: 10.1007/s00220-010-1058-z
[10] S. Luckhaus, Y. Sugiyama, J. J. L. Velázquez
Finite time blow-up and condensation for the bosonic Nordheim equation
Arch. Rat. Mech. Anal. , 206: : 31-80
[11] M. Escobedo, J. J. L. Velázquez
Finite time blow-up and condensation for the bosonic Nordheim equation
Invent. Math. , 200: (3): 761--847
ISSN: 0020-9910
DOI: 10.1007/s00222-014-0539-7
[16] M. Helmers, B. Niethammer*, J. J. L. Velázquez*
Mathematical analysis of a coarsening model with local interactions
J. Stat. Phys. , 26: (6): 1227-

Publication List


• SIAM Journal on Mathematical Analysis (2001 - 2005)
• Revista Matemática Iberoamericana (2001 - 2008)



A. v. Humboldt-J. C. Mutis Research Award (Alexander von Humboldt Foundation)

Selected Invited Lectures


Keynote speaker, Equadiff Congress, Berlin


PDE session, International Congress of Mathematicians, Madrid, Spain


Keynote speaker, Equadiff Congress, Vienna, Austria

Selected PhD students

Marco A. Fontelos (1997): “Problemas de frontera libre para fluidos viscosos”,
now Professor (on leave), Autonomous University of Madrid, and Researcher, Spanish National Research Council (CISC), Spain

Gerardo Oleaga (2000): “Dinámica de fracturas”,
now Professor (Profesor Contratado Doctor), Complutense University of Madrid, Spain

María Vela (2011): “Ant foraging and minimal paths in simple graphs”,
now Professor, Universidad Europea de Madrid, Spain

Arthur Kierkels (2016): “On a kinetic equation arising in weak turbulence theory for the nonlinear Schrödinger equation”

Supervised Theses

  • Master theses: 6
  • PhD theses: 5, currently 1
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