A free boundary problem involving a cusp : breakthrough of salt water

In this paper we study a two-phase free boundary problem describing the stationary flow of fresh and salt water in a porous medium, when both fluids are drawn into a well. For given discharges at the well ($Q_f$ for fresh water and $Q_s$ for salt water) we formulate the problem in terms of the stream function in an axial symmetric flow domain in {Bbb R^n(n = 2,3). We prove existence of a continuous free boundary which ends up in the well, located on the central axis. Moreover we show that the free boundary has a tangent at the well and approaches it in a $C^1$ sense. Using the method of separation of variables we also give a result about the asymptotic behaviour of the free boundary at the well. For given total discharge ($Q := Q_f + Q_s$) we consider the vanishing $Q_s$ limit. We show that a free boundary arises with a cusp at the central axis, having a positive distance from the well. This work is a continuation of [AD2,3]

A free boundary problem involving a cusp

We consider a stationary free boundary problem describing the stationary flow of fresh and salt water in a porous medium. The salt water is supposed to be stagnant, while the fresh water on top of it is drawn into wells. In a previous work it has been shown, that for pumping rates Q < Q_{cr a solution with smooth interface exists. In this part we study the case Q=Q_{cr in two dimensions. We show that the interface has isolated singularities. At each singularity the free boundary develops a cusp or becomes vertical. By means of local analysis techniques we obtain the asymptotic behaviour of the free boundary at these singularities

Multiscale Partition of Unity

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved independent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the partition of unity property. This precomputation involves the solution of independent problems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for 'cheap' parameter choices.Comment: Proceedings for Seventh International Workshop on Meshfree Methods for Partial Differential Equations, 18 pages, 3 figure

Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Comment: 19 page

A theory of L1L^1-dissipative solvers for scalar conservation laws with discontinuous flux

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes

On spectral minimal partitions: the case of the sphere

We consider spectral minimal partitions. Continuing work of the the present authors about problems for planar domains, [23], we focus on the sphere and obtain a sharp result for 3-partitions which is related to questions from harmonic analysis, in particular to a conjecture of Bishop

The Business Model: Recent Developments and Future Research

This article provides a broad and multifaceted review of the received literature on business models in which the authors examine the business model concept through multiple subject-matter lenses. The review reveals that scholars do not agree on what a business model is and that the literature is developing largely in silos, according to the phenomena of interest of the respective researchers. However, the authors also found emerging common themes among scholars of business models. Specifically, (1) the business model is emerging as a new unit of analysis; (2) business models emphasize a system-level, holistic approach to explaining how firms “do business”; (3) firm activities play an important role in the various conceptualizations of business models that have been proposed; and (4) business models seek to explain how value is created, not just how it is captured. These emerging themes could serve as catalysts for a more unified study of business models

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure