Research in brief:
I work on non-linear evolution partial differential equations. In particular, I am interested in the study of front propagation phenomena in equations of pseudo-parabolic, kinetic, parabolic and hyperbolic type. This amounts to study travelling waves and rarefaction waves, moving interfaces and fronts invading unstable states. Some of my research on the above topics was motivated by physical problems arising in infiltration in porous media. I have looked at some fluid dynamic problems related to this issue from a more applied, although still mathematical, point of view. This includes modelling and rigorous mathematical analysis of fluid accumulation in thin-film viscous flows over a smooth substrate in a limit in which both gravity and surface tension balance. I am also interested in understanding some processes of two-phase flow by probabilistic analysis of simple models of porous media. Most recently, I have been working on developing numerical methods for non-local operators, where the main motivation is the simulation of "very rapid" front solutions of certain non-linear evolution equations with non-local diffusion.
Some collaborators: M. Calle (formerly at Universidad Autónoma de Madrid), J.J.L. Velázquez (University of Bonn), F. Achleitner (Vienna University of Technology) and S. Hittmeir (University of Vienna), J.R. King (University of Nottingham) and F. de la Hoz (UPV/EHU).
PhD Students:
J. Cayama (PhD 2020, supervised with F. de la Hoz)
X. Diez (PhD 2021)
I. Girona (PhD 2024, supervised with F. de la Hoz)
L. Constantin (expected 2026, supervised with J. Giacomoni and G. Warnault, co-tutelle with Univ. Pau).
Research Projects:
`Análisis Matemático y Numérico de Ecuaciones en Derivadas Parciales' PID2021-126813NB-I00, MICINN Sep'22 -Aug'25 (PIs: F. de la Hoz, L. Vega).
Grupo Consolidado: `Fourier Analysis and Partial Differential Equations'. IT1615-22. GV. Jan'22 - Dic'25 (PI: C. Pérez).
Past projects:
'Mathematical and Numerical Analysis of some Partial Differential Equations and its Applications' PGC2018-094522-B-I00, MICINN Jan'19-Oct'22. (PIs: C. Cuesta, L. Vega).
Grupo Consolidado 'Fourier Analysis and Partial Differential Equations' IT1247-19. GV Jan'19-Jan'22 (PI: C. Pérez).
'Análisis Teórico y Numérico de Ecuaciones de Evolución' MTM2014-53145-P. MINECO, Jan'15-Jan'18. (PIs: L. Vega, C. Cuesta).
Grupo consolidado `Mathematical Physics, Mathematical Analysis and Partial Differential Equations, and Numerical Analysis'. IT641-13, GV. Jan'13-Jan'19. (PI: L. Vega).
'Proyecto de Excelencia Severo Ochoa' at ICMAT, SEV-2011-0087, MINECO (a member from Jan'12 to Ag'12, PI: M. de León).
'Problemas matemáticos que surgen en fenómenos de propagación. MTM2011-24109. MICINN, Jan'12-Jan'15 (PI: C. Cuesta).
'Propagation phenomena in nonlinear partial differential equations' RYC-2008-02127. Nov'09-Nov'11 (PI: C. Cuesta).
'The role of regularisations in front propagation phenomena' EP/D06834/1. EPSRC, Nov'06-Nov'09 (PI: C. Cuesta).
'Nonlinear Waves in Kinetic and Macroscopic Models II'. P18367. FWF, April'06-April'09 (PI: C. Schmeiser).
'Nonlinear Waves in Kinetic and Macroscopic Models'. P16174. FWF, April'03-April'06 (PI: C. Schmeiser).
RTN HYKE (Hyperbolic and Kinetic Equations). UE “FP5 Human Potential” a member in April-July'03 .(PIs: several, G. Rein at Vienna).
RTN FRONTS-SINGULARITIES. UE “FP5 Human Potential” a member in several periods in 2002, 2003, 2005 and 2006 .(PIs: several, J. King at Nottingham).
'Dynamic Capillary Pressure in Porous Media'. REF: 620-61-386. NWO. PhD-project. Nov'97 (Supervisors: J. Hulshof and C.J. van Duijn).