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We compare three thermodynamically consistent numerical fluxes known in the literature, appearing in a Voronoi finite volume discretization of the van Roosbroeck system with general charge carrier statistics. Our discussion includes an extension of the Scharfetter–Gummel scheme to non-Boltzmann (e.g. Fermi–Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

We use the notion of energy solutions of the stochastic Burgers equation to give a short proof of the Hairer-Quastel universality result for a class of stationary weakly asymmetric stochastic PDEs., "Stochastic Analysis on Large Scale Interacting Systems". October 26～29, 2015. edited by Ryoki Fukushima, Tadahisa Funaki, Yukio Nagahata, Makoto Nakashima, Hirofumi Osada and Yoshiki Otobe. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as Voronoi finite-volume discretization schemes for the drift-diffusion equations describing charge transport in bulk semiconductor devices. Three common challenges, that can corrupt the precision of numerical solutions, will be discussed: boundary layers at Ohmic contacts, discontinuties in the doping profile, and corner singularities in L-shaped domains. The influence on the order of convergence is assessed for each computational challenge and the different discretization schemes. Additionally, we provide an analysis of the inner boundary layer asymptotics near Ohmic contacts to support our observations.

In this work we present a new mixture theory of a liquid solvent containing completely dissociated ions to study the space charge layer of electrolytes in contact with some inert metals. We incorporate solvation shell effects (i) in our derivation of the mixing entropy and (ii) in the pressure model. Chemical potentials of ions and solvent molecules in the incompressible limit are then derived from a free energy function. For the thermodynamic equilibrium the coupled equation system of mass and momentum balance, the incompressibility constraint and the Poisson equation are summarized. With that we study the space charge layer of the electrolytic solution for an applied half cell potential and compare our results to historic and recent interpretations of the double layer in liquid electrolytes. The novelties of the model are: (i) the potential and pressure dependence of the free charge density in equilibrium (ii) the calculation of entropic contributions due to solvated ions, and (iii) the natural prediction of a solvated anion saturation layer which we finally reinterpret as Stern layer. Keywords: Double layer, Poisson–Boltzmann, Solvation, Mixture theory, Gouy–Chapman–Stern model

Using an idea going back to Madelung, we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov-Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta-shock solution of the continuity equation for a discontinuous velocity field. We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics). (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Using the concept of strong necessary conditions (CSNC), we derive a complete decomposition of the minimal Skyrme model into a sum of three coupled BPS submodels with the same topological bound. The bounds are saturated if corresponding Bogomolny equations, different for each submodel, are obeyed.

International audience; In this paper, we consider an open tube of diameter $\varepsilon>0$, on the side of which a small hole of size $\varepsilon^2$ is pierced. The resonances of this tube correspond to the eigenvalues of the Laplacian operator with homogeneous Neumann condition on the inner surface of the tube and Dirichlet one the open parts of the tube. We show that this spectrum converges when $\varepsilon$ goes to $0$ to the spectrum of an explicit one-dimensional operator. At a first order of approximation, the limit spectrum describes the note produced by a flute, for which one of its holes is open.

In this paper we prove the existence, uniqueness and stability of solutions of an inverse problem to a time-dependent Ginzburg--Landau model for superconductivity with the final overdetermination.

We study the Smoluchowski-Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation, several profiles of blowup solution have been noticed. Here, we study blowup in infinite time in accordance with the residual vanishing and Hamiltonian control on collapse dynamics.

In this work, along with the companion work Oh (2012), we propose a novel approach to the problem of gauge choice for the \emph{Yang-Mills equations} on the Minkowski space $\mathbb{R}^{1+3}$. A crucial ingredient is the associated \emph{Yang-Mills heat flow}. As this approach does not possess the drawbacks of the previous approaches (as in Klainerman-Machedon (1995) and Tao (2003)), it is expected to be more robust and easily adaptable to other settings. Building on the results proved in the companion article Oh (2012), we prove, as one of the first applications of our approach, finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$. This is a classical result first proved by S. Klainerman and M. Machedon (1995) using local Coulomb gauges. As opposed to their method, the present approach avoids the use of Uhlenbeck's lemma (1982), and hence does \emph{not} involve localization in space-time., Comment: 39 pages, 2 tables. v2: Version accepted for publication

We prove the existence and uniqueness theorems for solutions of an initial-boundary value problem to the system of equations, which describes dynamics of viscoelastic continuous medium with a variable boundary and a memory along the trajectories of particles in classes of summable functions.

We study the decay of the semigroup generated by the damped waveequation in an unbounded domain. We first prove under the natural {\em geometriccontrol condition} the exponential decay of the semigroup. Then we prove undera weaker condition the logarithmic decay of the solutions (assuming that theinitial data are smoother). As corollaries, we obtain several extensionsof previous results of stabilisation and control., On étudie la décroissance du semi-groupe des ondes amorties dans un domainenon borné. Notre premier résultat est que, sous une hypothèse naturelle decontrôle géométrique, le semi-groupe décroît exponentiellementvite. On démontre ensuite sous une hypothèse plus faible la décroissancelogarithmique des solutions associées à des données initiales plusrégulières. On obtient en corollaire plusieurs généralisations derésultats de stabilisation et de contrôle.

In this paper we investigate the motion of one-dimensional graphs under anisotropic non-convex mean curvature flow regularized via a Willmore term. Aiming at understanding the evolution problem when we let the regularization parameter tend to zero, we first present rigorous analytical results for the stationary case. Subsequently we discuss the time-dependent problem, focussing mainly on numerical simulations. We discretize by finite elements, and provide a semi-implicit scheme and a number of numerical experiments.

We consider an evolution equation arising in the Peierls--Nabarro model for crystal dislocation. we study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. these dislocation points evolve according to the external stress and an interior repulsive potential

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs—through their fundamental solutions—on stochastic integral equations (SIEs) driven by space–time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase in the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in [Discrete Contin. Dyn. Syst. Ser. A 33 (2013) 413–463, Stoch. Dyn. 6 (2006) 521–534], where we introduced two different stochastic versions of the fourth order memoryful PDE associated with the Brownian-time Brownian motion (BTBM): (1) the BTBM SIE and (2) the BTBM SPDE, both driven by space–time white noise. Under wide conditions, we showed the existence of random field locally-Hölder solutions to the BTBM SIE with striking and unprecedented time-space Hölder exponents, in spatial dimensions $d=1,2,3$. In particular, we proved that the spatial regularity of such solutions is nearly locally Lipschitz in $d=1,2$. This gave, for the first time, an example of a space–time white noise driven equation whose solutions are smoother than the corresponding Brownian sheet in either time or space. ¶ In this paper, we introduce the $2\beta^{-1}$-order $\beta$-inverse-stable-Lévy-time Brownian motion ($\beta$-ISLTBM) SIEs, $\beta\in \{1/2^{k};k\in\mathbb{N}\}$, driven by space–time white noise. Based on the dramatic regularizing effect of the BTBM density ($\beta=1/2$), and since the kernels in these $\beta$-ISLTBM SIEs are fundamental solutions to higher order Laplacian PDEs; one may suspect that we get even more dramatic spatial regularity than the BTBM SIE case. We show, however, that the BTBM SIE spatial regularity and its random field third spatial dimension limit are maximal among all $\beta$-ISLTBM SIEs—no matter how high we take the order $1/\beta$ of the Laplacian. This gives a limit as to how far we can push the SIEs spatial regularity when driven by the rough white noise. Furthermore, we show that increasing the order of the Laplacian $\beta^{-1}$ beyond the BTBM bi-Laplacian manifests entirely as increased temporal regularity of our random field solutions that asymptotically approaches that of the Brownian sheet as $\beta\searrow0$. Our solutions are both direct and lattice limit solutions. We treat many stochastic fractional PDEs and their corresponding higher order SPDEs, including BTBM and $\beta$-inverse-stable-Lévy-time Brownian motion SPDEs, in separate articles.

We consider the equation $$v_t=L_s v-W'(v)+\sigma_\epsilon(t,x) \quad {\mbox{ in }} (0,+\infty)\times\R,$$ where $L_s$ is an integro-differential operator of order $2s$, with $s\in(0,1)$, $W$ is a periodic potential, and $\sigma_\epsilon$ is a small external stress. The solution $v$ represents the atomic dislocation in the Peierls--Nabarro model for crystals, and we specifically consider the case $s\in(0,1/2)$, which takes into account a strongly nonlocal elastic term. We study the evolution of such dislocation function for macroscopic space and time scales, namely we introduce the function $$ v_{\epsilon}(t,x):=v\left(\frac{t}{\epsilon^{1+2s}},\frac{x}{\epsilon}\right). $$ We show that, for small $\epsilon$, the function $v_\epsilon$ approaches the sum of step functions. From the physical point of view, this shows that the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. We also show that the motion of these dislocation points is governed by an interior repulsive potential that is superposed to an elastic reaction to the external stress.