  Cross-diffusion systems are systems of nonlinear parabolic partial
differential equations that are used to describe dynamical processes in several
application, including chemical concentrations and cell biology. We present a
space-time approach to the proof of existence of bounded weak solutions of
cross-diffusion systems, making use of the system entropy to examine long-term
behavior and to show that the solution is nonnegative, even when a maximum
principle is not available. This approach naturally gives rise to a novel
space-time Galerkin method for the numerical approximation of cross-diffusion
systems that conserves their entropy structure. We prove existence and
convergence of the discrete solutions, and present numerical results for the
porous medium, the Fisher-KPP, and the Maxwell-Stefan problem.
