Regularity analysis for systems of reaction-diffusion equations

Goudon, Thierry; Vasseur, Alexis

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 1, pp. 117-142.

Ce travail est consacré à l'étude de la régularité des solutions de certains systèmes d'équations de réaction-diffusion. En particulier, nous montrons que les solutions peuvent être bornées et régulières en dimensions un et deux alors qu'en dimensions supérieures nous discutons la dimension de Hausdorff de l'ensemble des points singuliers. L'approche proposée ici s'inspire de la méthode de De Giorgi pour étudier la régularité de problèmes elliptiques avec des coefficients discontinus. La preuve exploite la stucture spécifique des systèmes considérés et n'est pas une simple adaptation de techniques scalaires. L'entropie associée naturellement au système joue un rôle crucial dans cette analyse.

This paper is devoted to the study of the regularity of solutions to some systems of reaction-diffusion equations. In particular, we show the global boundedness and regularity of the solutions in one and two dimensions. In addition, we discuss the Hausdorff dimension of the set of singularities in higher dimensions. Our approach is inspired by De Giorgi's method for elliptic regularity with rough coefficients. The proof uses the specific structure of the system to be considered and is not a mere adaptation of scalar techniques; in particular the natural entropy of the system plays a crucial role in the analysis.

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