RT Journal Article
T1 Existence and Uniqueness of Solution of a Continuous Flow Bioreactor Model with Two Species.
A1 Crespo Moya, María
A1 Ivorra, Benjamin
A1 Ramos del Olmo, Ángel Manuel
AB In this work, we study the mathematical analysis of a coupled system of two reaction-diffusion-advection equations and Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacterias) and a diluted substrate (e.g., nitrate) in a continuous flow bioreactor. This type of bioreactor can be used, for instance, for water treatment. First, we prove the existence and uniqueness of solution, under the hypothesis of linear reaction by using classical results for linear parabolic boundary value problems. Next, we prove the existence and uniqueness of solution for some nonlinear reactions by applying \textit{Schauder Fixed Point Theorem} and the theorem obtained for the linear case. Results about the nonnegativeness and boundedness of the solution are also proved here.
PB Springer
SN 1578-7303
YR 2016
FD 2016
LK https://hdl.handle.net/20.500.14352/22974
UL https://hdl.handle.net/20.500.14352/22974
LA eng
NO 1. Antonsev, S.N., Díaz, J.I., Shmarev, S.: Progress in Nonlinear Differential Equations and Their Applications.Energy methods for free boundary problems: applications to nonlinear PDEs and fluid mechanics.Birkhäuser, Boston (2002)2. Bailey, J.E., Ollis, D.F.: Biochemical Engineering Fundamentals. McGraw-Hill, New York (1986)3. Ballyk, M., Dung, L., Jones, D.A., Smith, H.L.: Effects of random motility on microbial growth andcompetition in a flow reactor. SIAM J. Appl. Math. 59(2), 573–596 (1998)4. Bello, J.M., Ivorra, B., Ramos, A.M., Rapaport, A.: Bioreactor shape optimisation. Modeling, simulation,and shape optimization of a simple bioreactor for water treatment. In: 7th STIC & Environnement, p. 344 (2011) (transvalor—Presses des Mines)5. Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Universitext. Springer,New York (2010)6. Dramé, A.K.: A semilinear parabolic boundary-value problem in bioreactors theory. Electron. J. Differ. Equ. (EJDE) (electronic only) (2004)7. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, New York (2010)8. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice Hall, London (1964)9. Gelfand, I.M., Shilov, G.E.: Generalized Functions: Properties and Operations. Academic Press, New York (1964)10. Harmand, J., Rapaport, A., Trofino, A.: Optimal design of interconnected bioreactors: new results. AIChEJ. 49(6), 1433–1450 (2003)11. Kung, C.M., Baltzis, B.C.: The growth of pure and simple microbial competitors in a moving distributedmedium. Math. Biosci. 111(2) (1992)12. Ladyzhenskaîa, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasi-Linear Equations of ParabolicType. American Mathematical Society, New York (1968) (translations of mathematical monographs)13. Lauffenburger, D., Aris, R., Keller, K.H.: Effects of random motility on growth of bacterial populations.Microb. Ecol. 7(3), 207–227 (1981)14. Lauffenburger, D., Calcagno, P.B.: Competition between two microbial populations in a nonmixed environment:effect of cell random motility. Biotechnol. Bioeng. 25(9), 2103–2125 (1983)15. Lieberman, G.M.: Intermediate schauder theory for second order parabolic equations. IV. Time irregularityand regularity. Differ. Integral Equ. 5(6), 1219–1236 (1992)16. Lions, J.L.: Sur les problèmes mixtes pour certains systèmes paraboliques dans les ouverts non cylindriques.Ann. l’inst. Fourier 7, 143–182 (1957)17. Lions, J.L.: Contrôle optimal de systèmes governés par des équations aux dérivées partiales (1968)18. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary value Problems and Applications, vol. I (1972)19. Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. FEMS Symposium. Plenum Press, New York (1992)20. Pao, C.V., Ruan, W.H.: Positive solutions of quasilinear parabolic systems with nonlinear boundaryconditions. J. Math. Anal. Appl. 333(1), 472–499 (2007)21. Qiu, Z., Wang, K., Zou, Y.: The asymptotic behavior of flowreactor models with two nutrients. Math.Comput. Model. 40(56), 465–479 (2004)22. Ramos, A.M.: Introducción al Análisis Matemático del Método de Elementos Finitos. Editorial Complutense, Madrid (2012)23. Rapaport, A., Harmand, J., Mazenc, F.: Coexistence in the design of a series of two chemostats. NonlinearAnal. Real World Appl. 9(3), 1052–1067 (2008)24. Shangerganesh, L., Balachandran, K.: Existence and uniqueness of solutions of predator-prey type modelwith mixed boundary conditions. Acta Appl. Math. 116(1), 71–86 (2011)25. Smith, H.L., Waltman, P.: The theory of the chemostat. In: Cambridge Studies in Mathematical Biology,vol. 13. Cambridge University Press, Cambridge (1995)26. Teschl, G.: Graduate Studies in Mathematics. Ordinary differential equations and dynamical systems. American Mathematical Society, New York (2012)27. Water Treatment: Principles and Practices of Water Supply Operations Series. American Water Works Association, New York (2003)28. Wen, C.Y., Fan, L.T.: Models for Flow Systems and Chemical Reactors. Chemical Processing and Engineering.Dekker, New York (1975)
NO Ministry of Science and Innovation
DS Docta Complutense
RD 8 may 2024