  A novel principle is presented which allows for the proof of bounded weak
solutions to a class of physically relevant, strongly coupled parabolic systems
exhibiting a formal gradient-flow structure. The main feature of these systems
is that the diffusion matrix may be generally neither symmetric nor positive
semi-definite. The key idea of the principle is to employ a transformation of
variables, determined by the entropy density, which is defined by the
gradient-flow formulation. The transformation yields at the same time a
positive semi-definite diffusion matrix, suitable gradient estimates as well as
lower and/or upper bounds of the solutions. These bounds are a consequence of
the transformation of variables and are obtained without the use of a maximum
principle. Several classes of cross-diffusion systems are identified which can
be solved by this technique. The systems are formally derived from
continuous-time random walks on a lattice modeling, for instance, the motion of
ions, cells, or fluid particles.
