Abstract:
Some systems of collective behavior for multiple species can be
described in the continuum limit by cross-diffusion systems,
derived e.g. from lattice models. Examples are coming from
population dynamics, cell biology, and gas dynamics. A common
feature of these strongly coupled parabolic differential equations
is that the diffusion matrix is often neither symmetric nor
positive definite, which makes the mathematical analysis very
challenging.
In this talk, we explain that for certain cross-diffusion systems,
these difficulties can be overcome by exploiting the so-called
entropy structure. This means that there exists a transformation
of variables (called entropy variables) such that the transformed
diffusion matrix becomes positive definite, and there exists a
Lyapunov functional (called entropy) which enables suitable
a priori estimates. Although the maximum principle generally does
not hold for systems, we show that the entropy concept helps to
prove lower and upper bounds for the solutions to systems with
volume-filling effects (boundedness-by-entropy principle).
We detail this theory for several examples coming from tumor-growth
modeling, population dynamics, and multicomponent gas dynamics.
The existence of global weak solutions and their long-time behavior
is investigated and some numerical examples are presented. |