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In this article we show a $C^{0,\alpha}$-partial regularity result forsolutions of a certain class of cross-diffusion systems with entropy structure.Under slightly more stringent conditions on the system, we are able to obtain a$C^{1,\alpha}$-partial regularity result. Amongst others, our results yield thepartial $C^{1,\alpha}$-regularity of weak solutions of the Maxwell-Stefansystem, as well as the partial $C^{1,\alpha}$-regularity of bounded weaksolutions of the Shigesada-Kawasaki-Teramoto model. The classical partialregularity theory for nonlinear parabolic systems as developed by Giaquinta andStruwe in the 80s proceeds by Campanato iteration which relies on energymethods. Our analysis here centers around the insight that, in the Campanatoiteration strategy, we can replace the use of energy estimates by "entropydissipation inequalities" and the use of the squared $L^2$-distance to measurethe distance between functions by the use of the "relative entropy". In orderfor our strategy to work, it is necessary to regularize the entropy structureof the cross-diffusion system, thereby introducing a new technical tool, whichwe call the "glued entropy".

Partial H\"{o}lder Regularity for Solutions of a Class of Cross-Diffusion Systems with Entropy Structure