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

  • Pierluigi Colli

    Università di Pavia, Italy
  • Gianni Gilardi

    Università di Pavia, Italy
  • Jürgen Sprekels

    Weierstrass Institut für Angewandte Analysis und Stochastik, Berlin, Germany
Well-posedness and regularity for a generalized fractional Cahn–Hilliard system cover
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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 and , 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. In this connection, we remark the fact that our definition of the fractional power of operators uses the approach via spectral theory. Typical cases are given by standard second-order elliptic differential 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 in this paper 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 of plays an important und not entirely obvious role: if is positive, then the operators and may be completely unrelated; if, however, equals zero, 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 . 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.

Cite this article

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels, Well-posedness and regularity for a generalized fractional Cahn–Hilliard system. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 3, pp. 437–478

DOI 10.4171/RLM/855