| Abstract: |
| Recently, we have investigated a multi-scale system consisting of a nonlinear diffusion equation and
a free boundary problem. The system was proposed as a mathematical model to concrete carbonation process in a three-dimensional domain.
We note that the free boundary problem describes adsorption phenomena appearing carbonation process.
In this talk we focus on a mathematical treatment for the free boundary problem.
Here, let $u$ and $s$ be the relative humidity and the degree of saturation, respectively.
Then, these variables satisfy
\begin{eqnarray*}
& & \rho_g u_t - \kappa u_{xx} = 0 \quad \mbox{ in } Q_s(T), \
& & u(t,1) = h(t) \quad \mbox{ for } 0 < t < T, \
& & u(0,x) = u_0(x) \quad \mbox{ for } s_0 < x < 1, s(0) = s_0, \
& & s`(t) = a (u(t,s(t) - \varphi(s(t))) \quad \mbox{ for } 0 < t < T, \
& & \kappa u_x(t,s(t)) = (\rho_a - \rho_g u(t,s(t))) \alpha(s(t), u(t,s(t)))
\quad \mbox{ for } < t < T,
\end{eqnarray*}
where $Q_s(T) := \{ (t,x)| s(t) < x < 1, 0 < t < T\}$, $s_0$ and $u_0$ are initial values of $s$ and $u$,
$h$ is a given boundary data, $\rho_g$ (resp. $\rho_a$) is the density of water in liquid (resp. air), $\kappa$ is a diffusion coefficient, $a$ is a positive constant, $\varphi: {\mathbb R}^2 \to {\mathbb R}$
is a non-negative continuous function.
In our previous works we have established uniqueness and existence of a solution, globally in time, under the assumption $0 \leq h \leq 1 - \delta$, where $\delta$ is a small positive constant.
However, when we consider the multi-scale system, it is not easy to prove existence of a solution because of the assumption for $h$. Then, in order to prove the existence without this assumption we provide a weak formulation for the free boundary problem. Our aims of this talk are to formulate a weak solution of the free boundary problem and give a result on existence of an approximate solution. |
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