Singular Hartree equation in fractional perturbed Sobolev spaces

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.     

Chemically induced phase separated morphologies in epoxy resin‐hyperbranched polymer blends

The morphology build‐up in toughened epoxy blends using reactive hyperbranched dendritic polymers (HBP) and amine‐cured epoxy resins has been investigated in the present work. By changing the processing conditions or the surface chemistry of the modifier, very different morphologies can be obtained, ranging from homogeneous blends to coarse two‐phase systems. The morphology characterised by electron and optical microscopy has been interpreted combining constitutive equations for phase separation and thermodynamic modelling of phase interactions. The latter model, based on the Flory‐Huggins lattice theory, was modified in order to take into account the addition reactivity of the HBP modifiers. This approach proved successful and can be used as a tool for final morphology prediction in any reactive blend formulation.