Project description
Understanding natural processes through partial differential 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 will study fine properties of irregular solutions of certain PDEs. Project research will seek to answer if initially irregular solutions become regular after some time, and if the PDEs are well-posed for growing (large) initial data. It will also investigate how solutions behave in the most quantitative way, by using explicit barriers or by understanding the long-time behaviour of the PDEs.
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.
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Funding Scheme
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinator
28049 Madrid
Spain