Regularity of solutions of the Cahn-Hilliard equation with concentration dependent mobility (Q5945996)

The authors establish a global existence result for 2-d Cahn-Hilliard equation with non-constant mobility, NEWLINE\[NEWLINE {\partial u \over \partial t}+ \nabla \cdot (m(u) (k \nabla \Delta u - \nabla \Psi'(u))] =0, \quad k > 0, NEWLINE\]NEWLINE where the mobility \(m(u)\) satisfies the constraints \(m(s) \geq M_1\) and \(|m'(s)|< M_2\) for some positive constants \(M_1, M_2\). NEWLINENEWLINENEWLINEThe main theorem of the paper establishes the existence of a unique classical solution for any initial data \(u_0\) of small enough energy. NEWLINENEWLINENEWLINEThe proof of the theorem proceeds by obtaining Hölder estimates for the solutions (Proposition 3) and then proving that this implies that the solutions are classical (Propositions 1 and 2); this involves the use of Cacciopoli-type inequalities and Campanato spaces. NEWLINENEWLINENEWLINENote that in the theory of Cahn-Hilliard equations with non-constant mobility assertion \(m(s) \geq M_1\) is too restrictive.