SIAM Journal on Applied Mathematics

The DC response of an electrochemical thin film, such as the separator in a microbattery, is analyzed by solving the Poisson--Nernst--Planck equations, subject to boundary conditions appropriate for an electrolytic/galvanic cell. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer, which mediates Faradaic reactions with nonlinear Butler--Volmer kinetics. Analytical results are obtained by matched asymptotic expansions in the limit of thin double layers and compared with full numerical solutions. The analysis shows that (i) decreasing the system size relative to the Debye screening length decreases the voltage of the cell and allows currents higher than the classical diffusion-limited current; (ii) finite reaction rates lead to the important possibility of a reaction-limited current; (iii) the Stern-layer capacitance is critical for allowing the cell to achieve currents above the reaction-limited current; and (iv) all polarographic (current-voltage) curves tend to the same limit as reaction kinetics become fast. Dimensional analysis, however, shows that ``fast' reactions tend to become ``slow' with decreasing system size, so the nonlinear effects of surface polarization may dominate the DC response of thin films.

We study a model electrochemical thin film at DC currents exceeding the classical diffusion-limited value. The mathematical problem involves the steady Poisson--Nernst--Planck equations for a binary electrolyte with nonlinear boundary conditions for reaction kinetics and Stern-layer capacitance, as well as an integral constraint on the number of anions. At the limiting current, we find a nested boundary-layer structure at the cathode, which is required by the reaction boundary condition. Above the limiting current, a depletion of anions generally characterizes the cathode side of the cell. In this regime, we derive leading-order asymptotic approximations for the (i) classical bulk space-charge layer and (ii) another nested highly charged boundary layer at the cathode. The former involves an exact solution to the Nernst--Planck equations for a single, unscreened ionic species, which may apply more generally to Faradaic conduction through very thin insulating films. By matching expansions, we derive current-voltage relations well into the space-charge regime. Throughout our analysis, we emphasize the strong influence of the Stern-layer capacitance on cell behavior.

In many experimental observation systems where the goal is to record a three-dimensional observation of an object, or a set of objects, a lower-dimensional projection of the intended subject is obtained. In some situations only the statistical properties of such objects are desired: the three-dimensional probability density function. This article demonstrates that under special symmetries this function can be obtained from either a one- or two-dimensional probability density function which has been obtained from the observed, projected data. Standard tomographic theorems can be used to guarantee the uniqueness of this function, and a natural basis set can be used in computing the three-dimensional function from the one- or two-dimensional projection. The theory of this inversion is explored using theoretical and computational methods with examples of data taken from scientific experiments.

The capacity for multiple equilibria in an isothermal homogeneous continuous flow stirred tank reactor is determined by the reaction network. Examples show that there is a very delicate relationship between reaction network structure and the possibility of multiple equilibria. We suggest a new method for discriminating between networks that have the capacity for multiple equilibria and those that do not. Our method can be implemented using standard computer algebra software and gives answers for many reaction networks for which previous methods give no information.

Thanks to the use of the Cagniard--De Hoop method, we derive an analytic solution in the time domain for the half-space problem associated with the wave equation with Engquist--Majda higher order boundary conditions. This permits us to derive new convergence results when the order of the boundary condition tends to $+\infty$, as well as error estimates. The theory is illustrated by numerical results.

This article addresses transitions in the topological structure of a family of divergence-free vector fields $u(\cdot, t)$ with Dirichlet boundary conditions. We show that structural bifurcation---i.e., change in topological-equivalence class---occurs at $t_0$ if $u(\cdot, t_0)$ has an isolated degenerate $\partial$-singular point ${\bar x} \in \partial M$ such that $\partial^2 u({\bar x} , t_0)/\partial n \partial t \not=0$. The main results are proved by classifying orbit structures of u near such a point ${\bar x} \in \partial M$ of $u(\cdot, t_0)$. The condition of ${\bar x}$ being a $\partial$-singular point is equivalent to the one originally postulated by Prandtl for boundary-layer separation. Our analysis and classification do contribute, in fact, to a rigorous characterization of boundary-layer separation in 2-D incompressible fluid flows.

A time-delayed epidemic model is proposed to describe the dynamics of disease spread among patches. An age structure is incorporated in order to simulate the phenomenon that some diseases only occur in the adult population. Sufficient conditions are established for global extinction and uniform persistence of the disease. One example shows that the disease could undergo persistence-extinction-persistence switches as the migration rate of juveniles increases, although juveniles are immune and cannot transmit the disease. The second example indicates that this switching phenomenon also happens when juveniles migrate with susceptible adults.

We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flow's numerical approximation by finite difference schemes. We discuss, in the continuous case, the theoretical properties of this system and prove the maximum principle in the strong and the weak formulations. In the discrete case, all the second order explicit schemes that are currently used violate, in general, the maximum principle. For these schemes we give a theoretical stability proof, accompanied by several numerical examples.

We consider the general problem on stationary flow of the electrorheological fluid with the constitutive equation developed in [R. H. W. Hoppe and W. G. Litvinov, Comm. Pure. Appl. Anal., 3 (2004), pp. 809--848] in the cylindrical coordinate system. The problem is studied under mixed boundary conditions wherein velocities are specified on one part of the boundary and surface forces are given on the other part. The existence of a solution to this problem and the convergence of Galerkin approximations are established. Then, we consider the occasion where the flow is axially symmetric and study a problem on an electrorheological clutch. This problem is solved numerically, and the results of calculations of the electric field and velocities are presented.

We consider the diffusion limit of a suitably rescaled model transport equation in a slab with multiplying boundary conditions, as the scaling parameter $\varepsilon$ tends to zero. We show that, for sufficiently smooth data, the solution converges in the L² -norm for each t > 0 to the solution of a diffusion equation with Robin boundary conditions corresponding to an incoming flux. The derivation of the diffusive limit is based on an asymptotic expansion, which is rigorously justified.

It is well known that the one-dimensional supercooled Stefan problem possesses solutions that blow up in finite time. The asymptotics of such solutions have been analyzed by Herrero and Velazquez [European J. Appl. Math., 7 (1996), pp. 119--150]. Here we consider the effect of kinetic undercooling as a regularizing mechanism to prevent the formation of such singularities and study the continuation of the solution through the "near blow-up" regime. The asymptotics of solutions and interfaces are described for small values of the kinetic undercooling parameter. It is shown that, in this limit, the interface jumps over an interval determined by the latent heat and by the initial data. Specifically, in dimensionless variables, if the temperature profile at blow-up is denoted by u(x,t_c^-)$, where $t_c$ is the finite blow-up time, then the interface jumps over the interval in which $u(x,t_c^-) < -\lambda$, where $\lambda$ is the latent heat.

This paper contains qualitative and quantitative comparisons between a weakly nonlinear analysis and direct numerical simulations of a free-boundary problem. The former involves modulating the most linearly unstable mode, taking a small perturbation of the neutrally stable value $\nu_c$ of a parameter $\nu$ related to the activation energy. Analogously, we perform the direct numerical computations near the marginally unstable value, namely, $\nu = \nu_c - \epsilon^2$, where $\epsilon$ is rather small.

We delineate the role of a different parameter $\sigma$ (related to the Arrhenius kinetics) in the combustion dynamics when $\nu = \nu_c - \epsilon^2$. In particular, the numerics show that varying $\sigma$ produces a period-doubling scenario when $\epsilon$ lies approximately between 0.08 and 0.12. We describe the $\sigma$ intervals within which complex dynamics occur for various values of $\epsilon$ and for $\nu$ fixed at $\nu_c - \epsilon^2$. When $\epsilon$ drops to approximately 0.06, the asymptotic and numerical solutions agree well for all physical values of $\sigma$.

A new model for the study of one-dimensional walls in magnetic multilayers is presented. We obtain the optimal scaling of this energy functional for low anisotropy double layers with magnetic layers of equal thickness. We prove that the optimal scaling may be attained by opposing Néel walls. We obtain the core length of the Néel wall and a detailed description of its structure. We illustrate our findings numerically.

Concentration gradients of morphogens are known to be instrumental in cell signaling and tissue patterning. Of interest here is how the presence of a competitor of BMP ligands affects cell signaling. The effects of Sog on the binding of Dpp with cell receptors are analyzed for dorsal-ventral morphogen gradient formation in vertebrate and Drosophila embryos. This prototype system includes diffusing ligands, degradation of morphogens, and cleavage of Dpp-Sog complexes by Tolloid to free up Dpp. Simple and biologically meaningful necessary and sufficient conditions for the existence of a steady state gradient configuration are established, and existence theorems are proved. For high Sog production rates (relative to the Dpp production rate), it is found that the steady state configuration exhibits a more intense Dpp-receptor concentration near the dorsal midline. Numerical simulations of the evolution of the system show that, beyond some threshold Sog production rate, the transient Dpp-receptor concentration at the dorsal midline would become more intense than that of the steady state, before subsiding and approaching a nonuniform steady state of lower magnitude. The magnitude of the transient concentration has been found to increase by several fold with increasing Sog production rate. The highly intense Dpp activity at and around the dorsal midline is consistent with available experimental observations and other analytical studies.

An asymptotic analysis of a quasi-geostrophic model is presented to investigate the vertically propagating internal Rossby wave structure in the presence of a sharp density interface. For the case of a nonconstant density gradient, the pattern of the vertical structure is nontrivial and depends on the parameter of varying stratification, denoted by $\delta$. As this parameter approaches a critical value from above (i.e., $\delta \approx\delta_{cri}$ ), the internal wave amplitudes increase continually until wave breaking occurs. When $\delta < \delta_{cri}$, two cases are considered. For the case without a turning-point, the requirement of appropriate interfacial conditions provides a general matching solution, available only for a certain range of $\delta$. In the presence of a turning-point, which describes the solution transition from monotonic to oscillatory behavior or vice versa, three asymptotic forms of the solutions are derived, even for small values of $\delta$.

An SIRS epidemic model, with a generalized nonlinear incidence as a function of the number of infected individuals, is developed and analyzed. Extending previous work, it is assumed that the natural immunity acquired by infection is not permanent but wanes with time. The nonlinearity of the functional form of the incidence of infection, which is subject only to a few general conditions, is biologically justified. The stability analysis of the associated equilibria is carried out, and the threshold quantity ($\R$) that governs the disease dynamics is derived. It is shown that $\R$, called the basic reproductive number, is independent of the functional form of the incidence. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms are derived for the different types of bifurcation that the model undergoes, including Hopf, saddle-node, and Bogdanov--Takens. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as forward or backward, subcritical or supercritical. The existence of a saddle-node bifurcation, at the turning point of backward bifurcation, is established by applying Sotomayor's theorem. The Bogdanov--Takens normal form is used to formulate the local bifurcation curve for a family of homoclinic orbits arising when a Hopf and a saddle-node bifurcation merge. These theoretical results are detailed and numerically illustrated for two different kinds of incidence, corresponding to unbounded and saturated contact rates. The coexistence of two limit cycles, due to the occurrence of a backward subcritical Hopf bifurcation, is also demonstrated. These results lead to the determination of ranges for the periodicity behavior of the model based on two critical parameters: the basic reproductive number and the rate of loss of natural immunity.

The total variation--based image denoising model of Rudin, Osher, and Fatemi [Phys. D, 60, (1992), pp. 259--268] has been generalized and modified in many ways in the literature; one of these modifications is to use the L¹ -norm as the fidelity term. We study the interesting consequences of this modification, especially from the point of view of geometric properties of its solutions. It turns out to have interesting new implications for data-driven scale selection and multiscale image decomposition.