A moving-boundary problem for concrete carbonation: Global existence and uniqueness of weak solutions

This paper deals with a one-dimensional coupled system of semi-linear parabolic equations with a kinetic condition on the moving boundary. The latter furnishes the driving force for the moving boundary. The main result is a global existence and uniqueness theorem of positive weak solutions. The system under consideration is modelled on the so-called carbonation of concrete - a prototypical chemical-corrosion process in a porous solid - concrete - which incorporates slow diffusive transport, interfacial exchange between wet and dry parts of the pores and, in particular, a fast reaction in thin layers, here idealized as a moving-boundary surface in the solid. We include simulation results showing that the model captures the qualitative behaviour of the carbonation process.     

Well-posedness of a moving-boundary problem with two moving reaction strips

We deal with a one-dimensional coupled system of semi-linear reaction-diffusion equations in two a priori unknown moving phases driven by a non-local kinetic condition. The PDEs system models the penetration of gaseous carbon dioxide in unsaturated porous materials (like concrete). The main issue is that the strong competition between carbon dioxide diffusion and the fast reaction of carbon dioxide with calcium hydroxide–which are the main active reactants–leads to a sudden drop in the alkalinity of concrete near the steel reinforcement. This process–called concrete carbonation–facilitates chemical corrosion and drastically influences the lifetime of the material. We present details of a class of moving-boundary models with kinetic condition at the moving boundary and address the local existence, uniqueness and stability of positive weak solutions. We also point out our concept of global solvability. The application of such moving-boundary systems to the prediction of carbonation penetration into ordinary concrete samples is illustrated numerically.     

Large time behavior of a solution of carbon dioxide transport model in concrete carbonation process

In [7] we show the global existence and uniqueness of a solution of carbon dioxide transport model in concrete carbonation process. This model is governed by a parabolic-type equation which has a non-local term depending on the unknown function itself. In this paper, we show the large time behavior of that solution.     

On the Existence of Weak Solutions for a 1-D Free-Boundary Concrete Carbonation Problem

In this paper we deal with a one-dimensional free-boundary problem arising in the modeling of concrete carbonation process. More precisely, we investigate global existence, uniqueness and large-time behavior of weak solutions for the problem under consideration. We also obtain the existence of a weak solution when the measure of the initial domain vanishes.     

Multiscale model for moisture transport with adsorption phenomenon in concrete materials

Abstract

We consider a new two-scale problem which is given as a mathematical model for moisture transport arising in a concrete carbonation process. In research for moisture transport, it is a crucial step how to describe the relationship between the relative humidity and the degree of saturation, mathematically. Here, we have proposed the two-scale model consisting of the diffusion equation for the relative humidity in a macro domain and the free boundary problems describing the relationship in infinitely micro domains. Accordingly, the structures of the micro domains are unknown in our model. This is a significant feature of our new model to emphasize. The aim of this paper is to establish local existence in time and uniqueness of a solution to the model.     

Asymptotics of nonlinear processes of a controlled approach

Stability of a terminal system of some class that appears in the projection of a control approach is studied, a mathematical model of which is described by a nonlinear system of ordinary differential equations for which no conditions of the existence theorem for solutions constructed by the Lapunov function are satisfied. We obtain criteria of stability, effectiveness of which is shown in concrete examples. nht]mis|Translated from Dinamicheskie Sistemy, No. 7, pp. 62–71, 1988.     

Cycling in proofs and feasibility

There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh's lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles.

     

Numerical approximation of a concrete carbonation model: Study of the ‐law of propagation

In this paper, we are interested in the long time behavior of approximate solutions to a free boundary model which appears in the modeling of concrete carbonation [1]. In particular, we study the long time regime of the moving interface. The numerical solutions are obtained by an implicit in time and finite volume in space scheme. We show the existence of solutions to the scheme and, following [2‐3], we prove that the approximate free boundary increases in time following a ‐law. Finally, we supplement the study through numerical experiments.     

A method for wave decomposition in computations for nonlinear dynamical three-dimensional towing systems

We propose a new method for computing multimode wave interactions in towing systems; our method is based on wave splitting of the equations of a model in such a way that each computational step requires computation for only one mode. The efficiency of the method is demonstrated by analysis of a concrete towing system with a turning towboat. Translated fromDinamicheskic Sistermy. Vol. 12, pp. 55–64, 1993.     

Finite-parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearities

We study the problem of global exponential stabilization of original Burgers' equations and the Burgers' equation with nonlocal nonlinearities by controllers depending on finitely many parameters. We investigate both equations by employing controllers based on finitely many Fourier modes and the latter equation by employing finitely many volume elements. To ensure global exponential stabilization, we have provided sufficient conditions on the control parameters for each problem. We also show that solutions of the controlled equations are steering a concrete solution of the non-controlled system as t → ∞ with an exponential decay rate.     

Center manifolds of infinite dimensions I: Main results and applications

In this paper, the first of a bipartite work, we consider an abstract, nonautonomous system of evolution equations of hyperbolic type, related to semilinear wave equations. Theorem 1 states that under certain assumptions the system admits a global center manifold, or equivalently a global decoupling function which is continuously differentiable with respect to its arguments, among which timet occurs. The difficult proof is presented in part II, i.e. the continuation of the present paper. For purposes of applications a local version of Theorem 1 is proved, i.e. the local center manifold Theorem 2. We obtain a series of applications both to abstract, nonautonomous wave equations and to concrete nonautonomous, semilinear wave equations subject to Neumann and Dirichlet boundary conditions.     

Min–Max Game Theory and Non-standard Differential Riccati Equations Under the Singular Estimate for e At B and e At G in the Absence of Analyticity

We consider an abstract dynamical system which is characterized by “singular estimates”, as it arises from many concrete hyperbolic/parabolic coupled PDE models, subject to boundary/point control and (deterministic) disturbance. If B is the control operator, G the disturbance operator and e ^At the free dynamic semigroup (which is not assumed to be analytic), then e ^At B and e ^At G are assumed to possess singular estimates. For such a system, we study a min–max game theory problem with quadratic cost functional over a finite horizon. The problem is fully solved in feedback form via a Riccati operator which satisfies a non-standard differential Riccati equation. Several PDE-illustrations are presented.     

On the stressed state of concrete at low temperatures

We give the results of experimental study of the influence of internal stresses that arise in concrete under cyclic freezing on its frost-resistance. On this basis we exhibit the causes of destruction of concrete tested for frost-resistance in a working temperature range.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 81–83.     

Modelling of macroscopical transient frost processes and microscopical driving forces

The artificial saturation phenomenon due to freeze-thaw cycles is described by a multi-phase and multiscale model (1; 2; 3) formulated within the Theory of Porous Media, (4). It represents partially saturated concrete as a mixture of 5 interacting constituents φ^a, namely the solid skeleton φ^s, the bulk water φ^l, the pore volume occupied by vapour φ^v, the ice φⁱ and the gel water phase φ^p. Most relevant for the model is the distinction between two length scales and their characteristic time scales. The boundary is marked where macroscopic bulk conditions change to surface physics and chemistry. Surface physics and chemistry acting on the nano-scale affect fundamental properties of concrete and consequently the durability of concrete against freeze-thaw. At the macroscopic scale the model describes transient conditions (i.e. water-uptake, heat transport, volume dilatation of 9 %, phase change of first order considering hysteresis) which are characterized by a relatively long time period to reach equilibrium in contrast to the processes modelled on the microstructure. At the microscopic scale the model represents the nanoscopic CSH-gel system consisting of solid CSH and water as a linked system of both components basing on the concept of the “Solid-Liquid Gel System” (5). In the constribution the numerical results of the model are presented with focus on the evaluation of the process zone during the penetration of the melting front into the matrix. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pairs,sets and sequences in first-order theories

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.      

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynkin Diagram Sequences and Stabilization Phenomena

We continue the study of stabilization phenomena for Dynkin diagram sequences initiated in the earlier work of Kleber and the present author. We consider a more general class of sequences than that of this earlier work, and isolate a condition on the weights that gives stabilization of tensor product and branching multiplicities. We show that all the results of the previous article can be naturally generalized to this setting. We also prove some properties of the partially ordered set of dominant weights of indefinite Kac–Moody algebras, and use this to give a more concrete definition of a stable representation ring. Finally, we consider the classical sequences B _n , C _n , D _n that fall outside the purview of the earlier work, and work out some easy-to-describe conditions on the weights which imply stabilization.     

Consistency on cubic lattices for determinants of arbitrary orders

We introduce a class of k-potential submanifolds in pseudo-Euclidean spaces and prove that for an arbitrary positive integer k and an arbitrary nonnegative integer p, each N-dimensional Frobenius manifold can always be locally realized as an N-dimensional k-potential submanifold in ((k + 1)N + p)-dimensional pseudo-Euclidean spaces of certain signatures. For k = 1 this construction was proposed by the present author in a previous paper (2006). The realization of concrete Frobenius manifolds is reduced to solving a consistent linear system of second-order partial differential equations.     

On the essential spectra of coupled operators matrix and application

We define an abstract setting to treat essential spectra of unbounded coupled operator matrix. We prove a well‐posedness result and develop a spectral theory which also allows us to prove an amelioration to many earlier works. We point out that a concrete example from integro‐differential equation fit into this abstract framework involving a general class of regular operator in L₁ spaces.