Abstract
This work’s major intention is the investigation of the well-posedness of certain cross-diffusion equations in the class of bounded functions. More precisely, we show existence, uniqueness and stability of bounded weak solutions under a smallness assumption on the intial data. As an application, we provide a new well-posedness theory for a diffusion-dominant cross-diffusion system that originates from a hopping model with size exclusions. Our approach is based on a fixed point argument in a function space that is induced by suitable Carleson-type measures.
Highlights
Systems of partial differential equations with cross-diffusion have developed into a large field of research in the last decades
Cross-diffusion, the phenomenon in which the gradient in the concentration of a species causes a flux of another species, appears in various applications as the modelling of population dynamics, e.g. [7,8,9,21,33] or electrochemistry, e.g. [5]
Another important biological field that is mathematically described by systems with cross-diffusion is cell-sorting or chemotaxis-like problems, e.g. [29,30]
Summary
Systems of partial differential equations with cross-diffusion have developed into a large field of research in the last decades. The reason why we choose to work in the setting of bounded functions is motivated by the following specific example, which apparently belongs to the class of cross-diffusion systems modelled in (1), (2). Ud to the cross-diffusion system (5), (6) This solution is unique in the class of functions satisfying (11). The gradient estimate (3) or (11) is natural in this perturbative setting (10), as it is the standard gradient estimate for the homogeneous heat equation with L∞ data—observe that the control over the gradient deteriorates as t → 0 with a rate proportional to the diffusion length In this sense, we consider the conditions for well-posedness imposed in the present paper as optimal.
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