混凝土碳化的自由边界问题：$\sqrt{t}$ -传播定律的严格证明
A free-boundary problem for concrete carbonation: Rigorous justification of the $\sqrt{t}$-law of propagation
摘要：
我们研究了一个一维的自由边界问题，描述渗透的碳化前线（自由反应触发界面）在混凝土中。几十年前，通过实验观察到，对于足够大的时间$t > 0$（$C$为正常数），穿透深度与时间的曲线（例如$s(t)$与$t$）表现得像$s(t)=C\sqrt{t}$。因此，许多拟合参数完全基于这个实验定律被用来预测在实际结构中的碳酸化前沿的大时间行为，直到现在还缺乏$\sqrt{t}$定律的理论依据。%这是我们的论文贡献的地方：本文的目的是通过严格证明实验猜测的渐近行为来填补这一空白。我们以前已经证明了上界$s(t)\leq C'\sqrt{t}$为一些常数$C'$ ;现在我们显示的最优性率证明正确的非平凡下估计，即存在$C">0$，使$s(t)\geq C"\sqrt{t}$。此外，我们得到弱解的自由边界问题的情况下，当测量的初始区域为零。通过这种方式，我们的数学模型现在允许出现一个移动的碳酸化前沿--从分析的角度来看，这种情况直到现在都很难处理。
英文摘要：
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 $s(t)=C\sqrt{t}$ 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 $\sqrt{t}$-law being lacking until now. %This is the place where our paper contributes: 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 $s(t)\leq C'\sqrt{t}$ 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 $s(t)\geq C"\sqrt{t}$. 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.
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