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.

Classification : 35Q99, 35B25, 82C70
Keywords: reaction-diffusion systems, regularity of solutions
Mot clés : systèmes de réaction-diffusion, régularité des solutions

[1] J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations 27 (1978), 224-265. | MR | Zbl

[2] M. Bisi & L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys. 124 (2006), 881-912. | MR | Zbl

[3] H. Brezis, Analyse fonctionnelle. Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, 1983. | MR | Zbl

[4] L. Caffarelli, R. V. Kohn & L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831. | MR | Zbl

[5] L. Caffarelli & A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, to appear in Annals of Math.. | MR | Zbl

[6] J.-F. Collet, Some modelling issues in the theory of fragmentation-coagulation systems, Commun. Math. Sci. 2 (2004), 35-54. | MR | Zbl

[7] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3 (1957), 25-43. | MR | Zbl

[8] L. Desvillettes & K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl. 319 (2006), 157-176. | MR | Zbl

[9] L. Desvillettes & K. Fellner, Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds, Rev. Mat. Iberoam. 24 (2008), 407-431. | EuDML | MR | Zbl

[10] L. Desvillettes, K. Fellner, M. Pierre & J. Vovelle, About global existence for quadratic systems of reaction-diffusion, J. Advanced Nonlinear Studies 7 (2007), 491-511. | MR

[11] P. Erdi & J. Tóth, Mathematical models of chemical reactions, Nonlinear Science: Theory and Applications, Manchester Univ. Press, 1989. | MR | Zbl

[12] H. Federer, Geometric measure theory, Die Grund. Math. Wiss., Band 153, Springer New York Inc., New York, 1969. | MR | Zbl

[13] W. Feng, Coupled system of reaction-diffusion equations and applications in carrier facilitated diffusion, Nonlinear Anal. 17 (1991), 285-311. | MR | Zbl

[14] P. C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomath. 28, Springer, 1979. | MR | Zbl

[15] Y. Giga & R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297-319. | MR | Zbl

[16] O. A. Ladyzenskaia, V. A. Solonnikov & N. N. Uralceva, Linear and quasi-linear equations of parabolic type, Transl. Math. Monographs 23, AMS, 1968. | Zbl

[17] F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure and Appl. Math. 51 (1998), 241-257. | MR | Zbl

[18] A. Mellet & A. Vasseur, $L^p$ estimates for quantities advected by a compressible flow, J. Math. Anal. Appl. 355 (2009), 548-563. | MR | Zbl

[19] J. Morgan, Global existence for semilinear parabolic systems, SIAM J. Math. Anal. 20 (1989), 1128-1144. | MR | Zbl

[20] J. Morgan, Global existence for semilinear parabolic systems via Lyapunov type methods, in Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), Lecture Notes in Math. 1394, Springer, 1989, 117-121. | MR | Zbl

[21] J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM J. Math. Anal. 21 (1990), 1172-1189. | MR | Zbl

[22] J. Morgan, On a question of blow-up for semilinear parabolic systems, Differential Integral Equations 3 (1990), 973-978. | MR | Zbl

[23] J. Morgan & S. Waggonner, Global existence for a class of quasilinear reaction-diffusion systems, Commun. Appl. Anal. 8 (2004), 153-166. | MR | Zbl

[24] J. D. Murray, Mathematical biology, Interdisciplinary Applied Math. 17 & 18, Springer, 2003. | MR | Zbl

[25] M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ. 3 (2003), 153-168. | MR | Zbl

[26] M. Pierre & D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev. 42 (2000), 93-106 (electronic). | MR | Zbl

[27] M. Pierre & R. Texier-Picard, Global existence for degenerate quadratic reaction-diffusion systems, to appear in Ann. IHP Anal. non linéaire. | Numdam | MR | Zbl

[28] F. Rothe, Global solutions of reaction-diffusion systems, Lecture Notes in Math. 102, Springer, 1984. | MR | Zbl

[29] V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math. 66 (1976), 535-552. | MR | Zbl

[30] V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55 (1977), 55-97. | MR | Zbl

[31] L. W. Somathilake & J. M. J. J. Peiris, Global solutions of a strongly coupled reaction-diffusion system with different diffusion coefficients, J. Appl. Math. 1 (2005), 23-36. | MR | Zbl

[32] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton Univ. Press, 1970. | Zbl

[33] A. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 753-785. | Zbl

[34] C. Villani, Hypocoercive diffusion operators, in Proceedings of the International Congress of Mathematicians, 2006. | Zbl

[35] F. B. Weissler, An $L^{\infty }$ blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math. 38 (1985), 291-295. | Zbl