| Abstract: |
| Aggregation-diffusion equations and systems have grow rapidly in their population. Models featuring nonlocal interactions through spatial convolution have been applied to several areas, including the physical, chemical, and biological sciences. A typical strategy to establish well-posedness is to use regularity properties of the kernels themselves; however, for many model problems such regularity is not available. One such example is the top-hat kernel which is discontinuous.
In this talk, I will present recent progress in establishing a robust well-posedness theory for a class of nonlocal aggregation-diffusion models with minimal regularity requirements on the interaction kernel in any spatial dimension on either the whole space or the torus. Starting with the scalar equation, we first establish the existence of a global weak solution in a small mass regime for merely bounded kernels. Under some additional hypotheses, we show the existence of a global weak solution for any initial mass. In typical cases of interest, these solutions are unique and classical. I will then discuss the generalisation to the n-species system for the regimes of small mass and arbitrary mass. We will conclude with some consequences of these theorems for several models typically found in ecological applications.
This is joint work with Dr. Skrzeczkowski and Prof. Jose Carrillo. |
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