Well-posedness and regularity for a generalized fractional Cahn-Hilliard system

Abstract

In this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous Cahn--Hilliard system in which nonlinearities of double-well type occur. Standard cases like regular or logarithmic potentials, as well as non-differentiable potentials involving indicator functions, are admitted. The operators appearing in the system equations are fractional versions of general linear operators $A$ and $B$, where the latter are densely defined, unbounded, self-adjoint and monotone in a Hilbert space of functions defined in a smooth domain and have compact resolvents. We remark that our definition of the fractional power of operators uses the approach via spectral theory. Typical cases are given by standard second-order elliptic operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourth-order systems or systems involving the Stokes operator are covered by the theory. We derive general well-posedness and regularity results that extend corresponding results which are known for either the non-fractional Laplacian with zero Neumann boundary condition or the fractional Laplacian with zero Dirichlet condition. It turns out that the first eigenvalue $\lambda_1$ of $A$ plays an important und not entirely obvious role: if $\lambda_1$ is positive, then the operators $\,A\,$ and $\,B\,$ may be completely unrelated; if, however, $\lambda_1=0$, then it must be simple and the corresponding one-dimensional eigenspace has to consist of the constant functions and to be a subset of the domain of definition of a certain fractional power of $B$. We are able to show general existence, uniqueness, and regularity results for both these cases, as well as for both the viscous and the nonviscous system.