PDE Seminar (2019/4/12): Fractional CahnHilliard system and ŁojasiewiczSimon gradient inequality
 Date

2019412 16:30 
2019412 18:00
 Place
 Faculty of Science Building #3, Room 309

Speaker/Organizer
 Goro Akagi (Tohoku University)

 This talk concerns convergence of weak solutions to the CauchyDirichlet problem for a variant of the CahnHilliard 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) CahnHilliard systems are variationally characterized as a gradient flow in a Sobolev space of negative order for a (coercive but nonconvex) 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 CahnHilliard system does not enjoy the orderpreserving property (i.e., comparison and maximum principles) of solutions. Therefore, the situation is more involved than secondorder parabolic equations such as the AllenCahn equation. In this talk, we shall focus on the socalled ŁojasiewiczSimon gradient inequality, which is an infinitedimensional version of the Łojasiewicz inequality proved for analytic functions. More precisely, we shall develop a ŁojasiewiczSimon type gradient inequality for the fractional (Dirichlet) Laplacian and some class of nonanalytic (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).