PDE Seminar (2019/4/12): Fractional Cahn-Hilliard system and Łojasiewicz-Simon gradient inequality

2019-4-12 16:30 - 2019-4-12 18:00
Faculty of Science Building #3, Room 309
Goro Akagi (Tohoku University)
This talk concerns convergence of weak solutions to the Cauchy-Dirichlet problem for a variant of the Cahn-Hilliard system along with the fractional Laplacian (defined via Fourier transform) and homogeneous (solid) Dirichlet boundary conditions posed on bounded domains as time goes to infinity. Fractional (and usual) Cahn-Hilliard systems are variationally characterized as a gradient flow in a Sobolev space of negative order for a (coercive but non-convex) free energy functional. Therefore solutions are expected to converge to an equilibrium of the system, which is a critical point of the free energy functional. On the other hand, even minimizers of the free energy functional may be neither unique nor isolated, and hence, the convergence of each solution (along the whole sequence of time) is a somewhat delicate issue. Moreover, the Cahn-Hilliard system does not enjoy the order-preserving property (i.e., comparison and maximum principles) of solutions. Therefore, the situation is more involved than second-order parabolic equations such as the Allen-Cahn equation. In this talk, we shall focus on the so-called Łojasiewicz-Simon gradient inequality, which is an infinite-dimensional version of the Łojasiewicz inequality proved for analytic functions. More precisely, we shall develop a Łojasiewicz-Simon type gradient inequality for the fractional (Dirichlet) Laplacian and some class of non-analytic (but C^1) nonlinearities, and moreover, we shall apply it to prove the convergence. To this end, we shall set up a variational formulation of the system in a proper way, and we also need to overcome some difficulties arising from the deficiency of regularity properties for fractional Poisson equations equipped with solid Dirichlet conditions. This talk is based on a joint work with G. Schimperna and A. Segatti (Univ. of Pavia, Italy).