We study a one-dimensional free-boundary problem describing the penetration of carbonation fronts (free reaction-triggered interfaces) in concrete. A couple of decades ago, it was observed experimentally that the penetration depth versus time curve (say s(t) vs. t) behaves like s(t) = C √ t for sufficiently large times t > 0 (with C a positive constant). Consequently, many fitting arguments solely based on this experimental law were used to predict the large-time behavior of carbonation fronts in real structures, a theoretical justification of the √ t-law being lacking until now.The aim of this paper is to fill this gap by justifying rigorously the experimentally guessed asymptotic behavior. We have previously proven the upper bound s(t) ≤ C √ t for some constant C ; now we show the optimality of the rate by proving the right nontrivial lower estimate, i.e. there exists C > 0 such that s(t) ≥ C √ t. Additionally, we obtain weak solutions to the free-boundary problem for the case when the measure of the initial domain vanishes. In this way, our mathematical model is now allowing for the appearance of a moving carbonation front -a scenario that until was hard to handle from the analysis point of view.
We analyze a coupled system of evolution equations that describes the effect of thermal gradients on the motion and deposition of N populations of colloidal species diffusing and interacting together through Smoluchowski production terms. This class of systems is particularly useful in studying drug delivery, contaminant transport in complex media, as well as heat shocks thorough permeable media. The particularity lies in the modeling of the nonlinear and nonlocal coupling between diffusion and thermal conduction. We investigate the semidiscrete as well as the fully discrete a priori error analysis of the finite elements approximation of the weak solution to a thermo-diffusion reaction system posed in a macroscopic domain. The mathematical techniques include energy-like estimates and compactness arguments.
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