May  2012, 6(2): 163-181. doi: 10.3934/ipi.2012.6.163

Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters

1. 

LMAC, EA 2222, Université de Technologie de Compiègne, BP 20529, 60205 COMPIEGNE Cedex, France

Received  December 2010 Revised  March 2012 Published  May 2012

We are concerned with the inverse problem of detecting sources in a coupled diffusion-reaction system. This problem arises from the Biochemical Oxygen Demand-Dissolved Oxygen model($^1$) governing the interaction between organic pollutants and the oxygen available in stream waters. The sources we consider are point-wise and simulate stationary or moving pollution sources. The ultimate objective is to obtain their discharge location and recover their output rate from accessible measurements of DO when BOD measurements are difficult and time consuming to obtain. It is, as a matter of fact, the most realistic configuration. The subject to address here is the identifiability of these sources, in other words to determine if the observations uniquely determine the sources. The key tool is the study of coupled parabolic systems derived after restricting the global model to regions at the exterior of the observations. The absence of any prescribed condition on the BOD density is compensated by data recorded on the DO which provide over-determined Cauchy boundary conditions. Now, the first step toward the identifiability of the sources is precisely to recover the BOD at the observation points (of DO). This may be achieved by handling and solving the coupled systems. Unsurprisingly, they turn out to be ill-posed. That issue is investigated first. Then, we state a uniqueness result owing to a suitable saddle-point variational framework and to Pazy's uniqueness Theorem. This uniqueness complemented by former identifiability results proved in [2011, Inverse problems] for scalar reaction-diffusion equations yields the desired identifiability for the global model.
Citation: Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems & Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163
References:
[1]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,'', Second edition, 140 (2003).   Google Scholar

[2]

M. Andrle, F. Ben Belgacem and A. El Badia, Identification of moving pointwise sources in an advection-dispersion-reaction equation,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/2/025007.  Google Scholar

[3]

F. Ben Belgacem, Uniqueness for an ill-posed parabolic system,, Comptes Rendus Mathématique Acad. Sci. Paris, 349 (2011), 1161.  doi: 10.1016/j.crma.2011.10.006.  Google Scholar

[4]

A. Bermúdez, Mathematical techniques for some environmental problems related to water pollution control,, in, 27 (1993), 12.   Google Scholar

[5]

C. Bernardi, C. Canuto and Y. Maday, Generalized inf-sup condition for Chebyshev spectral approximation of the Stokes problem,, SIAM J. Numer. Anal., 25 (1988), 1237.  doi: 10.1137/0725070.  Google Scholar

[6]

F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods,'', Springer Series in Computational Mathematics, 15 (1991).   Google Scholar

[7]

L. C. Brown and T. O. Barnwell, "The Enhanced Stream Water Quality Models QUAL2E and QUAL2E-UNCAS: Documentation and User Manual,'', Environmental Protection Agency, (1987).   Google Scholar

[8]

S. C. Chapra, "Applied Numerical Methods with MATLAB,'', McGraw-Hill, (2004).   Google Scholar

[9]

J. Cousteix and J. Mauss, "Analyse Asymptotique et Couche Limite,'', Mathématiques et Applications, 56 (2006).   Google Scholar

[10]

R. Dautray and J.-L. Lions, "Mathematical Analyis and Numerical Methods for Science and Technology,'', Vol. 5, (1992).   Google Scholar

[11]

W. Eckhaus, "Asymptotic Analysis of Singular Perturbations,'', Studies in Mathematics and its Applications, 9 (1979).   Google Scholar

[12]

A. El Badia, T. Ha-Duong and A. Hamdi, Identification of a point source in a linear advection-dispersion-reaction equation: Application to a pollution source problem,, Inverse Problems, 21 (2005), 1121.  doi: 10.1088/0266-5611/21/3/020.  Google Scholar

[13]

A. El Badia and A. Hamdi, Inverse source problem in an advection-dispersion-reaction system: Application to water pollution,, Inverse Problems, 23 (2007), 2103.  doi: 10.1088/0266-5611/23/5/017.  Google Scholar

[14]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Second edition, 224 (1983).   Google Scholar

[15]

G. Gripenberg, S.-O. London and O. Steffans, "Volterra Integral and Functional Equations,'', Encyclopedia of Mathematics and its Applications, 34 (1990).   Google Scholar

[16]

A. Hamdi, Identification of a time-varying point source in a system of two coupled linear diffusion-advection-reaction equations: Application to surface water pollution,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/11/115009.  Google Scholar

[17]

A. Hamdi, The recovery of a time-dependent point source in a linear transport equation: Application to surface water pollution,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/7/075006.  Google Scholar

[18]

G. Jolánkai, "Basic River Water Quality Models: Computer Aided Learning (CAL) Programme on Water Quality Modelling (WQMCAL),'', International Hydrological Programme, (1213).   Google Scholar

[19]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'', Dunod, (1968).   Google Scholar

[20]

G. I. Marchuk, "Mathematical Models in Environmental Problems,'', Encyclopedia of Mathematics and its Applications, 34 (1986).   Google Scholar

[21]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,'', An extended version of the Japanese edition, 10 (1980).   Google Scholar

[22]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'', Applied Mathematical Sciences, 44 (1983).   Google Scholar

[23]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations,, Journal of Differential Equations, 66 (1987), 118.  doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[24]

C. N. Sawyer, P. L. McCarty and G. F. Parkin, "Chemistry for Environmental Engineering and Science,'', Unpublished proceedings of the Delaware Conference on Ill Posed Inverse Problems, (1980).   Google Scholar

[25]

G. Wahba, "Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems With Noisy Data,'', McGraw-Hill, (2003).   Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,'', Second edition, 140 (2003).   Google Scholar

[2]

M. Andrle, F. Ben Belgacem and A. El Badia, Identification of moving pointwise sources in an advection-dispersion-reaction equation,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/2/025007.  Google Scholar

[3]

F. Ben Belgacem, Uniqueness for an ill-posed parabolic system,, Comptes Rendus Mathématique Acad. Sci. Paris, 349 (2011), 1161.  doi: 10.1016/j.crma.2011.10.006.  Google Scholar

[4]

A. Bermúdez, Mathematical techniques for some environmental problems related to water pollution control,, in, 27 (1993), 12.   Google Scholar

[5]

C. Bernardi, C. Canuto and Y. Maday, Generalized inf-sup condition for Chebyshev spectral approximation of the Stokes problem,, SIAM J. Numer. Anal., 25 (1988), 1237.  doi: 10.1137/0725070.  Google Scholar

[6]

F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods,'', Springer Series in Computational Mathematics, 15 (1991).   Google Scholar

[7]

L. C. Brown and T. O. Barnwell, "The Enhanced Stream Water Quality Models QUAL2E and QUAL2E-UNCAS: Documentation and User Manual,'', Environmental Protection Agency, (1987).   Google Scholar

[8]

S. C. Chapra, "Applied Numerical Methods with MATLAB,'', McGraw-Hill, (2004).   Google Scholar

[9]

J. Cousteix and J. Mauss, "Analyse Asymptotique et Couche Limite,'', Mathématiques et Applications, 56 (2006).   Google Scholar

[10]

R. Dautray and J.-L. Lions, "Mathematical Analyis and Numerical Methods for Science and Technology,'', Vol. 5, (1992).   Google Scholar

[11]

W. Eckhaus, "Asymptotic Analysis of Singular Perturbations,'', Studies in Mathematics and its Applications, 9 (1979).   Google Scholar

[12]

A. El Badia, T. Ha-Duong and A. Hamdi, Identification of a point source in a linear advection-dispersion-reaction equation: Application to a pollution source problem,, Inverse Problems, 21 (2005), 1121.  doi: 10.1088/0266-5611/21/3/020.  Google Scholar

[13]

A. El Badia and A. Hamdi, Inverse source problem in an advection-dispersion-reaction system: Application to water pollution,, Inverse Problems, 23 (2007), 2103.  doi: 10.1088/0266-5611/23/5/017.  Google Scholar

[14]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Second edition, 224 (1983).   Google Scholar

[15]

G. Gripenberg, S.-O. London and O. Steffans, "Volterra Integral and Functional Equations,'', Encyclopedia of Mathematics and its Applications, 34 (1990).   Google Scholar

[16]

A. Hamdi, Identification of a time-varying point source in a system of two coupled linear diffusion-advection-reaction equations: Application to surface water pollution,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/11/115009.  Google Scholar

[17]

A. Hamdi, The recovery of a time-dependent point source in a linear transport equation: Application to surface water pollution,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/7/075006.  Google Scholar

[18]

G. Jolánkai, "Basic River Water Quality Models: Computer Aided Learning (CAL) Programme on Water Quality Modelling (WQMCAL),'', International Hydrological Programme, (1213).   Google Scholar

[19]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'', Dunod, (1968).   Google Scholar

[20]

G. I. Marchuk, "Mathematical Models in Environmental Problems,'', Encyclopedia of Mathematics and its Applications, 34 (1986).   Google Scholar

[21]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,'', An extended version of the Japanese edition, 10 (1980).   Google Scholar

[22]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'', Applied Mathematical Sciences, 44 (1983).   Google Scholar

[23]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations,, Journal of Differential Equations, 66 (1987), 118.  doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[24]

C. N. Sawyer, P. L. McCarty and G. F. Parkin, "Chemistry for Environmental Engineering and Science,'', Unpublished proceedings of the Delaware Conference on Ill Posed Inverse Problems, (1980).   Google Scholar

[25]

G. Wahba, "Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems With Noisy Data,'', McGraw-Hill, (2003).   Google Scholar

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