Periodic Reporting for period 1 - techFRONT (Novel techniques for quantitative behaviour of convection-diffusion equations)
Reporting period: 2020-09-01 to 2022-08-31
Physical laws are mathematically encoded into Partial Differential Equations (PDEs). They tell us how certain quantities – like heat, water, or cars – depend on position and time. Precise information on the fundamental processes of the natural world is based to a large extent on PDEs; in turn, these processes will hint at solutions to mathematical problems. The EU-funded techFRONT project studied fine properties of irregular solutions of certain PDEs. The goal was to quantify how much the structure of the equation mattered in addressing various related questions.
Project research found that solutions of mass preserving PDEs become bounded as a consequence either of the diffusion or of the nonlinearity - the latter is a surprising outcome. For one particular equation, a self-similar continuous solution starting from bounded initial data was constructed, providing a sample of the general behaviour of nonnegative solutions. The long-time behaviour of PDEs was also investigated through 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.
Project research found that solutions of mass preserving PDEs become bounded as a consequence either of the diffusion or of the nonlinearity - the latter is a surprising outcome. For one particular equation, a self-similar continuous solution starting from bounded initial data was constructed, providing a sample of the general behaviour of nonnegative solutions. The long-time behaviour of PDEs was also investigated through 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.
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.
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.
The project further developed new tools for PDEs with nonlocal and/or local diffusion. Some of these tools are based on duality and Green function estimates, which combine results from Probability with techniques from Analysis. The innovative aspects constitute a valid alternative to the more classical methods of DeGiorgi-Nash and Moser; the latter is not always applicable when dealing with nonlocal operators. In parallel, the project also investigated long-time behaviour (or asymptotic behaviour) of PDEs both with convection and diffusion, and only diffusion. In any case, scaling techniques were applied in the new setting of nonlocal operators.