Reduction of a model for sodium exchanges in kidney nephron

<p style='text-indent:20px;'>This work deals with a mathematical analysis of sodium's transport in a tubular architecture of a kidney nephron. The nephron is modelled by two counter-current tubules. Ionic exchange occurs at the interface between the tubules and the epithelium and between the epithelium and the surrounding environment (interstitium). From a mathematical point of view, this model consists of a 5<inline-formula><tex-math id="M1">\begin{document}$ \times $\end{document}</tex-math></inline-formula>5 semi-linear hyperbolic system. In literature similar models neglect the epithelial layers. In this paper, we show rigorously that such models may be obtained by assuming that the permeabilities between lumen and epithelium are large. We show that when these permeabilities grow, solutions of the 5<inline-formula><tex-math id="M2">\begin{document}$ \times $\end{document}</tex-math></inline-formula>5 system converge to those of a reduced 3<inline-formula><tex-math id="M3">\begin{document}$ \times $\end{document}</tex-math></inline-formula>3 system without epithelial layers. The problem is defined on a bounded spacial domain with initial and boundary data. In order to show convergence, we use <inline-formula><tex-math id="M4">\begin{document}$ {{{\rm{BV}}}} $\end{document}</tex-math></inline-formula> compactness, which leads to introduce initial layers and to handle carefully the presence of lateral boundaries. We then discretize both 5<inline-formula><tex-math id="M5">\begin{document}$ \times $\end{document}</tex-math></inline-formula>5 and 3<inline-formula><tex-math id="M6">\begin{document}$ \times $\end{document}</tex-math></inline-formula>3 systems, and show numerically the same asymptotic result, for a fixed meshsize.</p>

Riesz basis of exponential family for a hyperbolic system

This paper studies a linear hyperbolic system with boundary conditions that was first studied under some weaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a {C_{0}} -semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system {\{e^{\lambda_{n}t}\}_{n\geq 1}} constitutes a Riesz basis in {L^{2}[0,T]} . Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

A Discrete Analog of the Lyapunov Function for Hyperbolic Systems

We study the difference splitting scheme for the numerical calculation of stable solutions of a two-dimensional linear hyperbolic system with dissipative boundary conditions in the case of constant coefficients with lower terms. A discrete analog of the Lyapunov function is constructed and an a priori estimate is obtained for it. The obtained a priori estimate allows us to assert the exponential stability of the numerical solution.