The project consisted of eight groups of collaboration with researchers based in Spain, France, Romania, the United States, and Norway. It resulted in two published papers, three preprints, one final stage work, one work in preparation, and one early stage collaboration. All of them, partially or fully, were concerned with the basic theory of PDEs, their fine properties, or numerical simulations of them.
The published papers are quite different in nature. The first is concerned with constructing a continuous self-similar solution from bounded initial data, while the other provides a zero-order approximation of a fully nonlinear nonlocal operator. The latter is exploited in the final stage work to build a theory of existence and also to construct numerical approximations. In addition, it is also proved that there is a linear simplification of the problem in the sense that solutions are asymptotically bounded by a quantity coming from the linear part of the operator.
One of the preprints more or less completes the theory for robust numerical schemes for PDEs with nonlocal and/or local diffusion. It improves convergence results for the schemes, which were developed in a series of papers starting in 2017 by the project researcher and two collaborators. The second preprint provides a quite complete guide to estimates giving bounded solutions of PDEs with nonlocal and/or local diffusion, starting from finite-mass initial data. Although it is rather long, a constructive theory is developed with all necessary details, and constitutes one of the main outcomes of the project. In particular, equivalences and implications between smoothing effects, Green and heat kernel estimates, and functional inequalities are established. Finally, the remaining preprint studies a PDE where there is a competition between convection and diffusion. A key ingredient in the final result was to predict, depending on the data of the problem, which of the two competing terms eventually was strong enough to govern the behaviour of the corresponding solution.
Basic theory of PDEs like existence and uniqueness are cornerstones of the last two works mentioned above. One of them studies equations with both convection and nonlocal diffusion on bounded domains, while the other focuses on equations with only convection.
During the project period, the results were disseminated at various event - both digitally and physically. Of particular interest is "Congreso Bienal de la Real Sociedad Matemática Española (The biyearly Congress of the Royal Spanish Mathematical Society)" and "2nd Norwegian meeting on PDEs" which were arranged in 2022. The events gathered the mathematical communities in Spain and Norway, respectively. The final weeks of the project were reserved for arranging a workshop in Madrid at Universidad Autónoma de Madrid (UAM) and Instituto de Ciencias Matemáticas (ICMAT). Around 35 participants from all over Europe and the United States gathered to exchange knowledge in the topics of the project. This helped forging new contacts, but also maintaining those already established.
