Financial support
Here you can see my curriculum vitae.
My research
Among others, I am interested in the following research topics:
Water waves
Water waves have been the object of study of many different researchers along History. This is a familiar problem in the sense that everyone has seen water waves in a beach at some point. However, this is also a very challenging problem. Actually, in the Feynmann Lecture on Physics there is already a warning about the difficulty of the motion of water waves: Now, the next waves of interest, that are easily seen by everyone and which are usually used as an example of waves in elementary courses, are water waves. As we shall soon see, they are the worst possible example, because they are in no respects like sound and light; they have all the complications that waves can have. (The Feynman Lectures on Physics Vol. I Ch. 51: Waves, Richard Feynmann) My research is then focused on the the derivation and study of asymptotic models for water waves in different physical regimes. Due in part to this and other contributions, I was awarded with the 2020 'Antonio Valle' prize from the Spanish Society of Applied Mathematics (SeMA).
The Muskat Problem
Part of my research focuses on the flow of two incompressible fluids in porous media. These fluids are assumed to be immiscible and, as such, they are separated by a sharp interface. This is known in the literature as the Muskat problem (named after M. Muskat in the 1930's). Here, due to the effect of the solid matrix of the porous medium, the usual fluid equations for the conservation of momentum (Euler or Navier-Stokes equations) are replaced with the empirical Darcy's Law. Remarkably, the Muskat problem is mathematically analogous to the Hele-Shaw cell problem (named after H. Hele-Shaw's classical 1898 paper) that studies the movement of a fluid trapped between two parallel vertical plates, which are separated by a very narrow distance. Besides its mathematical interest, the Muskat problem is also relevant in geosciences (aquifers, oil wells or geothermal reservoirs). Moreover, it has also been considered as a model for the velocity of cells in tumour growth. In my thesis (you can download it here) I studied the effect of impervious walls on the qualitative behaviour of the interface separating both fluids. Here you can see some simulations of the interface. I also considered the case where the porous medium has a permeability given by two different values (i.e. the porous medium is inhomogeneous). For that case you can also see some videos. One of my (favourite) results for the Muskat problem is related to wave breaking. In the case of water waves in the ocean, breaking of waves may occur anywhere where the amplitude is sufficient. In particular, water waves may break in mid-ocean. However, it is particularly common on beaches. This occurrence is due to the effect of the ocean floor: as a wave approaches the coastline and comes into contact with the ocean floor, the waves' velocity decreases. Under stationary conditions, a decrease in speed must be compensated by an increase in height in order to maintain a constant energy flux. This process is called shoaling and results in wave breaking. Our result studies the effect of the porous medium's floor in the breaking of the interface. In particular, we, using a computer assisted proof, we were able to prove that there exist initial interfaces such that if the depth is finite, they lead to solutions that turn, while if the depth is infinite, they lead to solutions that do not turn. Or, in other words, the finiteness of the depth enhances the turning behaviour of the system. This could be though as a mathematical proof of the fact that Smooth runs the water where the brook is deep (Shakespeare's Henry VI. Part II, 1592.) Due in part to this contributions, I was awarded with the 2015 'Vicent Caselles' prize from the Royal Spanish Mathematical Society (RSME).
Cancer growth
Free boundary problems for partial differential equations are one of the most arduous areas in Mathematical Analysis. These problems are mathematically challenging and physically interesting. Moreover, their applications are really spread, from geothermal reservoirs to tumour growth, passing through weather forecasting. Free boundary problems typically arise during the interaction between two fluids or between a fluid and an elastic solid, i.e. when the dynamics of both fluids or the fluid and the elastic solid are connected. The study of fluid - fluid and fluid - solid interactions are classical problems in Applied Sciences and Mathematics. In the last decade many different techniques to study free boundary problems arising in fluid dynamics have been developed. Part of my research focuses on such interactions in problems arising in Biology and in particular in tumour growth. Although cancer research has improved the available treatments and our understanding of this disease, there is still a lack of a good mathematical description. As pointed out in the literature, mathematical models could provide unexpected insight into the underlying mechanisms and generate novel hypotheses for experimentation. Part of my research in this topic has been awarded with a 2021 Leonardo fellowship for researchers and cultural creators from the BBVA Foundation.
Cell aggregation
Even if as of 2016 no "standard model" of the origin of life has yet emerged, most currently accepted models state that life arose on Earth between 3800 and 4100 million years ago. These first forms of live where single-celled organisms. For most of the history of life on Earth there were only single-celled organisms. However, now there are many different fungi, algae, plants and animals that are multicellular organism (they are formed by aggregations of cells working together). Thus, even if we know that these single-celled organism eventually formed multicellular organisms (around 1500 millions of years ago), the origins of multicellarity are one of the most interesting topics in biology because we still do not know the way multicellarity arises. A particular situation where cells form a cluster, in a process known as cell aggregation, arises when the motion of the cells is driven by a chemical gradient, i.e. the cells attempt to move towards higher (or lower) concentration of some chemical substance. This process is usually called chemotaxis. Then, multicellular aggregates and eventually tissue-like assemblies are formed when individual cells attach to each other as a consequence of the chemotactic movement and when this aggregation leads to subsequent cellular differentiation. That is, for instance, the case of the slime mold Dictyostelium Discoideum and bacterial populations, such as of Escherichia coli and Salmonella typhimurium. A preliminary step towards a better understanding of chemotaxis and cell aggregation, was given by Keller and Segel with their 1970's classical papers (see also the prior work by Patlak in the 1950's). In these papers, Keller and Segel proposed a PDE system as a model of cell aggregation as a consequence of chemotactic movement. Then, mathematically, the appearance of cell aggregation is translated to the formation of a finite time singularity. My research focuses on whether the solution exists globally in time (i.e. no aggregation occurs). Here you can find some videos for these systems. Among other results, we were able to prove that the presence of a logistic term, the diffusive forces needed to prevent the finite time singularity are actually much weaker than the standard Laplacian.
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Popular Science & Outreach
Ondas no lineales en fluidos incompresibles. La Gaceta de la RSME, vol 24, num 3, pag 507-531, 2021.
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Pint of Science 2021- Rafael Granero: "Matemáticas, fluidos y arte".
"Buscamos un modelo matemático que marque cómo se comportan los tumores".
La ecuación de Burgers como un paso previo al estudio de los fluidos incompresibles La Gaceta de la RSME, vol 15, num 3, pag, 489-512, 2012 (with José Manuel Moreno).
El Problema de Basilea: Historia y algunas demostraciones. La Gaceta de la RSME, vol 12, num 4, pag 721-737, 2009.