| Abstract: |
| The following system is a mathematical model for the first stage of brewing Japanese Sake.
It's configured by differential equations (with initial conditions, boundary conditions), and a constraint condition.
This model is studied by A. Ito and Y. M.
\[
\left\{
\begin{array}{ll}
\theta_t-d_0 \Delta \theta+g_0(\theta,u_1,u_2)=f_0&\mbox{a.e. in } Q\vspace{0.1cm}\
(u_1)_t-d_1 \Delta u_1 +g_1(\theta,u_1,u_2)=f_1&\mbox{a.e. in } Q\vspace{0.1cm}\
(u_2)_t-d_2 \Delta u_2 +g_2(\theta,u_1,u_2)=f_2&\mbox{a.e. in } Q\vspace{0.1cm}\
(u_1,u_2)\in K(\theta)&\mbox{a.e. in } Q\vspace{0.1cm}\
(v)_t-d_3 \Delta v =-c_3vu_1+f_3&\mbox{a.e. in } Q\vspace{0.1cm}\
(w)_t-d_4 \Delta w =c_6vu_1-(c_7u_1+c_8u_2)w+f_4&\mbox{a.e. in } Q\vspace{0.1cm}\
\frac{\partial \theta}{\partial \mathbf{n}} +c_9 \theta =h &\mbox{a.e. on }\Sigma \vspace{0.1cm}\
\frac{\partial u_1}{\partial \mathbf{n}}=\frac{\partial u_2}{\partial \mathbf{n}}=\frac{\partial v}{\partial \mathbf{n}}=\frac{\partial w}{\partial \mathbf{n}}=0
&\mbox{a.e. on } \Sigma\vspace{0.1cm}\
\theta(0)=\theta_0,\ \ \ \ u_1(0)=u_{1,0},\ \ \ \ u_2(0)=u_{2,0}&\vspace{0.1cm}\
v(0)=v_0,\ \ \ \ w(0)=w_0&\mbox{a.e. on } \Omega
\end{array}
\right.\vspace{0.1cm}
\]
The constraint condition is one of the characteristic point of our model.
We can see that equations for $u_1,u_2$ and the constraint condition configure "Quasi-variational inequality".
In my talk, we discuss existence of weak solutions for our model,
and some numerical simulations are presented. |
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