UABDivulga - Barcelona recerca i innovació

A fluid, such as water or air, is a set of particles bound together by forces acting between each pair and in a homogeneous manner (of the same nature). However, these forces are not strong enough to keep the relative positions fixed, and this facilitates movement and deformation.

The study of fluids, although it is a classical topic, has gained relevance lately due to the development of new materials, with sometimes surprising properties, in the study of plasmas in a macroscopic approach, and also because it poses theoretical and mathematical problems of great difficulty. Our work is framed in a very specific aspect of one of these problems.

If one wants to describe the state and/or evolution of a fluid, one encounters three important parameters: the tendency to form vortices (vorticity), the tendency to compress or expand (divergence), and the dissipation of energy due to internal friction or strong interactions between the particles (viscosity).

The study we have carried out refers to the case where this dissipation does not occur (perfect fluids). In this sense, in a recent previous article¹ we studied the planar case (2-dimensional) where the fluid cannot be compressed (zero divergence) and thus preserves its volume, although not its shape, and can only form vortices. The evolution of this liquid is governed by the so-called "Euler equations". We also analyzed the opposite case where there are no vortices (zero rotation), and the main parameter is compression (divergence). In this case, the evolution is governed by the so-called "aggregation equation".

In the study of both situations, the fluid starts from an ideal initial condition that is constant in a limited region of the 2-dimensional space and zero elsewhere (a "patch"). We found that with an appropriate formulation (using complex numbers), both problems (Euler and aggregation) are extreme cases of the same mathematical phenomenon and the flow, that is, the current caused by the evolution of the fluid, is analytic with respect to time. This means that the knowledge of all the characteristics of the flow at any given moment in time allows one to know its evolution at any moment, both in the past and in the future, unless a catastrophe occurs that would change essential parameters and, therefore, the flow regime.

In both cases, the published proof is completely original and in the second case, that of aggregation, the result is new. The first one (the case of liquids - incompressible fluids) was almost simultaneous with another study by researchers from the USA: they published it a few months earlier, but using a different approach than ours. In fact, our method differs because it allows us to see that the case of fluids and aggregation are two sides of the same phenomenon in the context of "patches".  

Our contribution includes the performance of original and quite sophisticated techniques in both articles. On the other hand, the aggregation equation has been studied in spaces of any dimension. In article² we see that for the aggregation equation, one has the same phenomenon of analyticity of the flow, regardless of the (finite) dimension of the space. This allows modeling seemingly very different phenomena, such as the growth of a population of bacteria on a cultivation plate, various aspects of materials science, and the evolution of the density of certain singularities in superconductors, among others. It is worth noticing that this is a major reason and a characteristic of the use of Mathematics in the description of Nature. This research is situated within the field of Partial Differential Equations, where one makes use of Harmonic Analysis, Complex Analysis, and elements of Combinatorics.