# American Institute of Mathematical Sciences

December  2012, 7(4): 989-1018. doi: 10.3934/nhm.2012.7.989

## Analysis of a simplified model of the urine concentration mechanism

 1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, France 2 Univ Paris 06, Univ Paris 05, INSERM UMRS 872, and CNRS ERL 7226, Laboratoire de génomique, physiologie et physiopathologie rénales, Centre de Recherche des Cordeliers, 75006, Paris, France 3 UMR 7598 Laboratoire J.-L. Lions, UPMC Univ Paris 06, Paris, F-75005

Received  September 2011 Revised  October 2012 Published  December 2012

We study a nonlinear stationary system of transport equations with specific boundary conditions describing the transport of solutes dissolved in a fluid circulating in a countercurrent tubular architecture, which constitutes a simplified model of a kidney nephron. We prove that for every Lipschitz and monotonic nonlinearity (which stems from active transport across the ascending limb), the dynamic system, a PDE which we study through contraction properties, relaxes toward the unique stationary state. A study of the linearized stationary operator enables us, using eigenelements, to further show that under certain conditions regarding the nonlinearity, the relaxation is exponential. We also describe a finite volume scheme which allows us to efficiently approach the numerical solution to the stationary system. Finally, we apply this numerical method to illustrate how the countercurrent arrangement of tubes enhances the axial concentration gradient, thereby favoring the production of highly concentrated urine.
Citation: Magali Tournus, Aurélie Edwards, Nicolas Seguin, Benoît Perthame. Analysis of a simplified model of the urine concentration mechanism. Networks & Heterogeneous Media, 2012, 7 (4) : 989-1018. doi: 10.3934/nhm.2012.7.989
##### References:

show all references

##### References:
 [1] Youcef Mammeri, Damien Sellier. A surface model of nonlinear, non-steady-state phloem transport. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1055-1069. doi: 10.3934/mbe.2017055 [2] Zhiming Chen, Weibing Deng, Huang Ye. A new upscaling method for the solute transport equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 941-960. doi: 10.3934/dcds.2005.13.941 [3] Tehuan Chen, Chao Xu, Zhigang Ren. Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1251-1269. doi: 10.3934/jimo.2018052 [4] Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735 [5] Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018 [6] Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283 [7] Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic & Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955 [8] Jorge Clarke, Christian Olivera, Ciprian Tudor. The transport equation and zero quadratic variation processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2991-3002. doi: 10.3934/dcdsb.2016083 [9] Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789 [10] Atul Kumar, R. R. Yadav. Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity. Conference Publications, 2013, 2013 (special) : 457-466. doi: 10.3934/proc.2013.2013.457 [11] Karthik Elamvazhuthi, Piyush Grover. Optimal transport over nonlinear systems via infinitesimal generators on graphs. Journal of Computational Dynamics, 2018, 5 (1&2) : 1-32. doi: 10.3934/jcd.2018001 [12] Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159 [13] Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114 [14] Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641 [15] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 [16] Lei Wu. Diffusive limit with geometric correction of unsteady neutron transport equation. Kinetic & Related Models, 2017, 10 (4) : 1163-1203. doi: 10.3934/krm.2017045 [17] Roman Romanov. Estimates of solutions of linear neutron transport equation at large time and spectral singularities. Kinetic & Related Models, 2012, 5 (1) : 113-128. doi: 10.3934/krm.2012.5.113 [18] Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic & Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395 [19] Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic & Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79 [20] Gianluca Crippa, Laura V. Spinolo. An overview on some results concerning the transport equation and its applications to conservation laws. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1283-1293. doi: 10.3934/cpaa.2010.9.1283

2018 Impact Factor: 0.871