| Abstract: |
| In this talk, recent advances in nonlinear diffusion equations as
asymptotic limits of Cahn--Hilliard systems is treated. The weak
formulation of the Stefan problem, the porous media equation,
the fast diffusion equation, the weak formulation of Hele-Shaw equation and many other
nonlinear diffusion equations of the form
$$\frac{\partial u}{\partial t}-\Delta \beta (u) = g$$
are target problems.
In Fukao (2016) and Colli--Fukao (2016),
an idea of asymptotic limits of Cahn--Hilliard system is introduced for
the characterization of the solution.
More precisely, as the level of approximation the term of nonlinear diffusion $\beta (u)$
is treated as a monotone term in the Cahn--Hilliard system.
$$\frac{\partial u}{\partial t}-\Delta \mu = 0, \quad
\mu =-\varepsilon \Delta u + \beta (u) + \pi_\varepsilon (u)-f.$$
This approach has an advantage to
improve the growth condition of $\hat{\beta }$, the primitive of nonlinear term $\beta$.
The problem is considered under the Robin type boundary condition in this talk.
This study is based on the joint work with Taishi Motoda, Kyoto University of Education. |
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