Special Session 144: 

A doubly-nonlinear Cahn-Hilliard system with nonlinear viscosity

Luca Scarpa
University College London
England
Co-Author(s):    Elena Bonetti, Pierluigi Colli, Giuseppe Tomassetti
Abstract:
We prove existence and uniqueness of solutions to a doubly-nonlinear Cahn-Hilliard system with nonlinear viscosity of the form \begin{align*} \partial_t u- \Delta\mu = 0 \qquad&\text{in } \Omega\times(0,T)\,,\ \mu\in\varepsilon\partial_t u + \beta(\partial_t u) - \delta\Delta u + \psi`(u) - g \qquad&\text{in } \Omega\times(0,T)\,,\ \mu=0\,, \quad \partial_{\bf n} u =0 \qquad&\text{in } \partial\Omega\times(0,T)\,,\ u(0)=u_0 \qquad&\text{in } \Omega \end{align*} on a smooth bounded domain $\Omega\subseteq\mathbb{R}^3$, where $\varepsilon,\delta$ are positive parameters, $\beta$ is a maximal monotone graph in $\mathbb{R}\times\mathbb{R}$, $\psi:(a,b)\to\mathbb{R}$ is a so-called double-well potential and $g$ is a given source. Existence of solutions is proved by approximating the problem and passing to the limit through monotonicity and compactness arguments, under some growth assumptions either on $\beta$ or on $\psi$. Moreover, the asymptotic behaviour of the solutions as $\delta\searrow0$ and $\varepsilon\searrow0$ (separately) is shown. This study is based on a joint work with Elena Bonetti (Universit\`a degli Studi di Milano), Pierluigi Colli (Universit\`a degli Studi di Pavia) and Giuseppe Tomassetti (Universit\`a degli Studi ``Roma Tre``).