

1990  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 


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 NordheimBoltzmann 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 selfsimilar 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.


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”


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. Wellposedness 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. 


[ 1] Miguel A. Herrero, Juan J. L. Velázquez
Singularity patterns in a chemotaxis model Math. Ann. , 306: (3): 583623 1996 ISSN: 00255831 DOI: 10.1007/BF01445268[ 2] J. J. L. Velázquez
Point dynamics in a singular limit of the KellerSegel model. I. Motion of the concentration regions SIAM J. Appl. Math. , 64: (4): 11981223 2004 ISSN: 00361399 DOI: 10.1137/S0036139903433888[ 3] Elio Eduardo Espejo Arenas, Angela Stevens, Juan J. L. Velázquez
Simultaneous finite time blowup in a twospecies model for chemotaxis Analysis (Munich) , 29: (3): 317338 2009 ISSN: 01744747 DOI: 10.1524/anly.2009.1029[ 4] A. Stevens, J. J. L. Velázquez
Partial differential equations and nondiffusive structures Nonlinearity , 21: (12): T283T289 2008 ISSN: 09517715 DOI: 10.1088/09517715/21/12/T04[ 5] Kyungkeun Kang, Benoit Perthame, Angela Stevens, J. J. L. Velázquez
An integrodifferential equation model for alignment and orientational aggregation J. Differential Equations , 246: (4): 13871421 2009 ISSN: 00220396 DOI: 10.1016/j.jde.2008.11.006[ 6] Kyungkeun Kang, Angela Stevens, Juan J. L. Velázquez
Qualitative behavior of a KellerSegel model with nondiffusive memory Comm. Partial Differential Equations , 35: (2): 245274 2010 ISSN: 03605302 DOI: 10.1080/03605300903473400[ 7] M. Escobedo, S. Mischler, J. J. L. Velázquez
On the fundamental solution of a linearized UehlingUhlenbeck equation Arch. Ration. Mech. Anal. , 186: (2): 309349 2007 ISSN: 00039527 DOI: 10.1007/s0020500700842[ 8] M. Escobedo, S. Mischler, J. J. L. Velázquez
Singular solutions for the UehlingUhlenbeck equation Proc. Roy. Soc. Edinburgh Sect. A , 138: (1): 67107 2008 ISSN: 03082105 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): 759816 2010 ISSN: 00103616 DOI: 10.1007/s002200101058z[ 10] S. Luckhaus, Y. Sugiyama, J. J. L. Velázquez
Finite time blowup and condensation for the bosonic Nordheim equation Arch. Rat. Mech. Anal. , 206: : 3180 2012[ 11] M. Escobedo, J. J. L. Velázquez
Finite time blowup and condensation for the bosonic Nordheim equation Invent. Math. , 200: (3): 761847 2015 ISSN: 00209910 DOI: 10.1007/s0022201405397[ 16] M. Helmers, B. Niethammer*, J. J. L. Velázquez*
Mathematical analysis of a coarsening model with local interactions J. Stat. Phys. , 26: (6): 1227 2015



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


2005  A. v. HumboldtJ. C. Mutis Research Award (Alexander von Humboldt Foundation) 


1999  Keynote speaker, Equadiff Congress, Berlin  2006  PDE session, International Congress of Mathematicians, Madrid, Spain  2007  Keynote speaker, Equadiff Congress, Vienna, Austria 


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”


 Master theses: 6
 PhD theses: 5, currently 1


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