| Abstract: |
| Self-assembling is an ubiquitous phenomenon in several applications, ranging from biology to materials science. In this talk, we consider a diffuse interface model describing a ternary system, constituted by a diblock copolymer and a homopolymer acting as solvent, interacting with an electric field. The dynamics of the ternary system is fully coupled with that of the electric field, hence the whole system is modeled by two Cahn--Hilliard--Oono equations for the copolymer blocks, accounting for long-range interactions; a classical Cahn--Hiliard equation for the homopolymer and, finally, the Maxwell equation for the electric displacement field. A multiphase singular potential is employed in order to ensure physical consistency. First, we show existence of global weak solutions in two and three dimensions. Uniqueness of weak solutions is estabilished in the constant mobility case, and a conditional result is given in the general case. Instantaneous regularization and long-time behavior are also investigated, the latter in the case of affine-linear electric permittivity, showing in particular that solutions converge to a single stationary state. |
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