Convergence of a one dimensional Cahn-Hilliard equation with degenerate mobility

Abstract

We consider a one dimensional periodic forward-backward parabolic equation, regularized by a non-linear fourth order term of order ${\epsilon }^2\ll 1$. This equation is known in the literature as Cahn-Hilliard equation with degenerate mobility. Under the hypothesis of the initial data being well prepared, we prove that as $\epsilon \to 0$, the solution converges to the solution of a well-posed degenerate parabolic equation. The proof exploits the gradient flow nature of the equation in ${\mathcal{W}}^2$ and utilizes the framework of convergence of gradient flows developed by Sandier-Serfaty. As an incidental, we study fine energetic properties of solutions to the Thin-film equation ${\partial }_t\nu =(\nu {\nu }_{xxx}{)}_x$.