# American Institute of Mathematical Sciences

March 2019, 18(2): 977-998. doi: 10.3934/cpaa.2019048

## A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results

 1 Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy 2 Istituto di Matematica Applicata e Tecnologie Informatiche "Enrico Magenes", CNR, Via Ferrata 1, 27100 Pavia, Italy 3 Dipartimento di Ingegneria e Architettura, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy 4 Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, Italy 5 Istituto per le Applicazioni del Calcolo "M. Picone", CNR, Via dei Taurini 19, 00185 Roma, Italy

* Corresponding author

Received  February 2018 Revised  June 2018 Published  October 2018

We consider an evolution system describing the phenomenon of marble sulphation of a monument, accounting of the surface rugosity. We first prove a local in time well posedness result. Then, stronger assumptions on the data allow us to establish the existence of a global in time solution. Finally, we perform some numerical simulations that illustrate the main feature of the proposed model.

Citation: Elena Bonetti, Cecilia Cavaterra, Francesco Freddi, Maurizio Grasselli, Roberto Natalini. A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results. Communications on Pure & Applied Analysis, 2019, 18 (2) : 977-998. doi: 10.3934/cpaa.2019048
##### References:
 [1] D. Aregba-Driollet, F. Diele and R. Natalini, A mathematical model for the sulphur dioxide aggression to calcium carbonate stones: numerical approximation and asymptotic analysis, SIAM J. Appl. Math., 64 (2004), 1636-1667. doi: 10.1137/S003613990342829X. [2] C. Baiocchi, Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl., 76 (1967), 233-304. doi: 10.1007/BF02412236. [3] W. Bangerth, R. Hartmann and G. Kanschat, deal.II - A general-purpose object-oriented finite element library, ACM Trans. Math. Softw., 33 (2007), 1-27. doi: 10.1145/1268776.1268779. [4] E. Bonetti and M. Frémond, Analytical results on a model for damaging in domains and interfaces, ESAIM Control Optim. Calc. Var., 17 (2011), 955-974. doi: 10.1051/cocv/2010033. [5] F. Clarelli, A. Fasano and R. Natalini, Mathematics and monument conservation: free boundary models of marble sulfation, SIAM J. Appl. Math., 69 (2008), 149-168. doi: 10.1137/070695125. [6] F. Freddi and G. Royer-Carfagni, Regularized variational theories of fracture: A unified approach, Journal of the Mechanics and Physics of Solids, 58 (2010), 1154-1174. doi: 10.1016/j.jmps.2010.02.010. [7] M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [8] C. Giavarini, M. L. Santarelli, R. Natalini and F. Freddi, A non-linear model of sulphation of porous stones: Numerical simulations and preliminary laboratory assessments, Journal of Cultural Heritage, 9 (2008), 14-22. doi: 10.1016/j.culher.2007.12.001. [9] F. R. Guarguaglini and R. Natalini, Global existence of solutions to a nonlinear model of sulphation phenomena in calcium carbonate stones, Nonlinear Anal., 6 (2005), 477-494. doi: 10.1016/j.nonrwa.2004.09.007. [10] F. R. Guarguaglini and R. Natalini, Nonlinear transmission problems for quasilinear diffusion systems, Netw. Heterog. Media, 2 (2007), 359-381. doi: 10.3934/nhm.2007.2.359. [11] F. R. Guarguaglini and R. Natalini, Fast reaction limit and large time behavior of solutions to a nonlinear model for sulphation phenomena, Comm. Partial Differential Equations, 32 (2007), 163-189. doi: 10.1080/03605300500361438. [12] A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016. [13] R. Natalini, C. Nitsch, G. Pontrelli and S. Sbaraglia, A numerical study of a nonlocal model of damage propagation under chemical aggression, European J. Appl. Math., 14 (2003), 447-464. doi: 10.1017/S0956792503005205. [14] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [15] M. Taylor, Partial Differential Equations I. Basic Theory. Applied Mathematical Sciences, 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8. [16] D. Whitehouse, Surfaces and Their Measurement, Butterworth-Heinemann, Boston, 2012. [17] BS EN ISO 4287:2000, Geometrical product specification (GPS). Surface texture. Profile method. Terms, definitions and surface texture parameters.

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##### References:
 [1] D. Aregba-Driollet, F. Diele and R. Natalini, A mathematical model for the sulphur dioxide aggression to calcium carbonate stones: numerical approximation and asymptotic analysis, SIAM J. Appl. Math., 64 (2004), 1636-1667. doi: 10.1137/S003613990342829X. [2] C. Baiocchi, Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl., 76 (1967), 233-304. doi: 10.1007/BF02412236. [3] W. Bangerth, R. Hartmann and G. Kanschat, deal.II - A general-purpose object-oriented finite element library, ACM Trans. Math. Softw., 33 (2007), 1-27. doi: 10.1145/1268776.1268779. [4] E. Bonetti and M. Frémond, Analytical results on a model for damaging in domains and interfaces, ESAIM Control Optim. Calc. Var., 17 (2011), 955-974. doi: 10.1051/cocv/2010033. [5] F. Clarelli, A. Fasano and R. Natalini, Mathematics and monument conservation: free boundary models of marble sulfation, SIAM J. Appl. Math., 69 (2008), 149-168. doi: 10.1137/070695125. [6] F. Freddi and G. Royer-Carfagni, Regularized variational theories of fracture: A unified approach, Journal of the Mechanics and Physics of Solids, 58 (2010), 1154-1174. doi: 10.1016/j.jmps.2010.02.010. [7] M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [8] C. Giavarini, M. L. Santarelli, R. Natalini and F. Freddi, A non-linear model of sulphation of porous stones: Numerical simulations and preliminary laboratory assessments, Journal of Cultural Heritage, 9 (2008), 14-22. doi: 10.1016/j.culher.2007.12.001. [9] F. R. Guarguaglini and R. Natalini, Global existence of solutions to a nonlinear model of sulphation phenomena in calcium carbonate stones, Nonlinear Anal., 6 (2005), 477-494. doi: 10.1016/j.nonrwa.2004.09.007. [10] F. R. Guarguaglini and R. Natalini, Nonlinear transmission problems for quasilinear diffusion systems, Netw. Heterog. Media, 2 (2007), 359-381. doi: 10.3934/nhm.2007.2.359. [11] F. R. Guarguaglini and R. Natalini, Fast reaction limit and large time behavior of solutions to a nonlinear model for sulphation phenomena, Comm. Partial Differential Equations, 32 (2007), 163-189. doi: 10.1080/03605300500361438. [12] A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016. [13] R. Natalini, C. Nitsch, G. Pontrelli and S. Sbaraglia, A numerical study of a nonlocal model of damage propagation under chemical aggression, European J. Appl. Math., 14 (2003), 447-464. doi: 10.1017/S0956792503005205. [14] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [15] M. Taylor, Partial Differential Equations I. Basic Theory. Applied Mathematical Sciences, 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8. [16] D. Whitehouse, Surfaces and Their Measurement, Butterworth-Heinemann, Boston, 2012. [17] BS EN ISO 4287:2000, Geometrical product specification (GPS). Surface texture. Profile method. Terms, definitions and surface texture parameters.
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