Global Regularity And Uniqueness Of Solutions In A Surface Growth Model Using Rigorous A Posteriori Methods

Global Regularity and Uniqueness of Solutions in a Surface Growth Model Using Rigorous A-Posteriori Methods

The use of rigorous numerical methods to approach problems which can not be solved using standard methods (yet) has increased signifiantly in recent years. In this book, riogorous a-posteriori methods are used to study the time evolution of a surface growth model, given by a fourth order semi-linear parabolic partial differential equation, where standard methods fail to verify global uniqueness and smoothness of solutions. Based on an arbitrary numerical approximation, a-posteriori error-analysis is applied in order to prevent a blow up analytically. This is a method that in a similar way also applies to the three dimensional Navier-Stokes equations. The main idea consists of energy-estimates for the error between solution and approximation that yields a scalar differential equation controlling the norm of the error with coefficients depending solely on the numerical data. This allows the solution of the differential equation to be bounded using only numerical data. A key technical tool is a rigorous eigenvalue bound for the nonlinear operator linearized around the numerical approximation. The presented method succeeds to show global uniqueness for relatively large initial conditions, which is demonstrated in many numerical examples.

Mathematical Reviews

Chemical Sensors

Chemical sensors are integral to the automation of myriad industrial processes and everyday monitoring of such activities as public safety, engine performance, medical therapeutics, and many more. This 4 volume reference work covering simulation and modeling will serve as the perfect complement to Momentum Press's 6 volume reference works 'Chemical Sensors: Fundamentals of Sensing Materials' and 'Chemical Sensors: Comprehensive Sensor Technologies', which present detailed information related to materials, technologies, construction and application of various devices for chemical sensing. This 4 volume comprehensive reference work analyzes approaches used for computer simulation and modeling in various fields of chemical sensing and discusses various phenomena important for chemical sensing such as bulk and surface diffusion, adsorption, surface reactions, sintering, conductivity, mass transport, interphase interactions, etc. In this work it will be shown that theoretical modeling and simulation of the processes, being a basic for chemical sensors operation, could provide considerable progress in choosing both optimal materials and optimal configurations of sensing elements for using in chemical sensors. Each simulation and modeling volume in the present series reviews modeling principles and approaches peculiar to specific groups of materials and devices applied for chemical sensing. Volume 2: Conductometric-Type Sensors covers phenomenological modeling and computational design of conductometric chemical sensors based on nanostructured materials such as metal oxides, carbon nanotubes and graphene. This volume contains an overview of the approaches used to quantitatively evaluate characteristics of sensitive structures in which electric charge transport depends on the interaction between the surfaces of the structures and chemical compounds in the surrounding.