techFRONT · Novel techniques for quantitative behaviour of convection-diffusion equations
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2020-09-01 → 2022-08-31
- EU contribution
- €172,932
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
Novel techniques for quantitative behaviour of convection-diffusion equations
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.
Data: CORDIS, © European Union
Project objective
Physical laws are mathematically encoded into partial differential equations (PDEs). They tell us how certain quantities---like heat, water, or even cars---depend on position and time. Even without knowing the solutions explicitly, the ultimate goal of this project is to investigate fine properties of irregular solutions of certain classes of PDEs: can we predict the behaviour of the solution by using barriers; how will the solution behave after a long time has passed; can irregular solutions become regular---possibly classical; are the problems well-posed even for growing initial data? In practice, such properties describe the underlying physical model. Indeed, the mathematical insight provides new knowledge about the real-world applications, and information about the application gives hints to solutions of mathematical problems.We aim to use new and innovative techniques to prove fine properties of solutions of generalized porous medium equations (GPME). We intend to build a solution theory for a new class of weak solutions. This includes general well-posedness, regularity theory, and asymptotic behaviour. Our approach will provide an alternative to established methods due to DeGiorgi-Nash and Moser which seems to be unsuitable in this context. When there is convection present in GPME, that is, when we have a convection-diffusion equation (CDE), we plan to explore the possibilities of using the new to theory for GPME to shed new light on the asymptotic behaviour for CDE.
Original text from CORDIS.
Participants
- UNIVERSIDAD AUTONOMA DE MADRID · MadridCoordinatorSpain
Links
Data: CORDIS, © European Union