  The paper entitled "Well posedness of general cross-diffusion systems", by C.
Choquet, C. Rosier, L. Rosier, J. Diff. Eq. 2021, is devoted to the
mathematical analysis of the Cauchy problem for general cross-diffusion systems
without any assumption about its entropic structure. The absence of this type
of hypothesis is strongly felt for two questions: the uniqueness of the
solution, despite the nonlinear coupling of the highest derivatives terms, and
the maximum principle. The article is therefore largely devoted to these two
points. The answers are provided at the cost of certain assumptions or
technicalities, mainly: (i) the ratios between the diffusion and
cross-diffusion coefficients has to be drastically controlled for sufficiently
enhancing the regularity of the solution, namely its gradient belongs to the
space $ L ^ 4 ((0,T)\times \Omega )$; the regularity is obtained by adapting
the classical Meyer's to the nonlinear parabolic setting under consideration ;
(ii) the source terms have to ensure the confinement of the solution. The
present "companion" paper aims at showing where more classical analysis tools
fail to solve these questions and gives some additional clarifications.
