Hölder Regularity for Certain Strongly Coupled Parabolic Systems

Abstract

We present a shorter proof to show Ho lder continuity of bounded solutions to a general class of quasilinear parabolic equations. The proof will be extended to obtain regularity results for bounded solutions to certain strongly coupled (or cross-diffusion) quasilinear parabolic systems. Academic PressIn this paper we study the Ho lder continuity of bounded solutions to a class of certain strongly coupled quasilinear parabolic systems of the formis a vector valued function defined in 0 T and div, D denotes the divergence and spatial derivative opeartors. A i , f i are accordingly vector value functions. To the author's knowledge there are only few works on Ho lder regularity of solutions to systems of the type (0.1). In contrast to the case of scalar equations or reaction diffusion systems with the coupling occurs only in the reaction terms, counterexamples (see [26]) indicate that one cannot expect bounded solutions to general strongly coupled systems to be regular everywhere. Also, concerning the problem of global existence of solutions, a priori L bounds are not enough to conclude that the solutions exist on the infinite time interval. The works of Amann [1,3,2] show that, in important cases, it suffices to find a priori L bounds to guarantee global existence provided that we can also prove uniform Ho lder continuity in space and time ([2, Theorem 4.1]).Partial regularity results were obtained by Giaquinta and Struwe in [14] for a fairly general class of systems. Everywhere regularity results for bounded solutions were proven only in few situations assuming additional structure conditions on the system (0.1). Among these are diagonal systems (see Article ID jdeq.1998.3510, available online at http:ÂÂwww.idealibrary.com on 313

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