I will discuss some recent results about Lp and Schauder estimates for a class of non-local elliptic equations. Compared to previous known results, the novelty of our results is that the kernels of the operators are not necessarily to be homogeneous, regular, or symmetric.

We study the two-phase Stefan problem that models heat propagation and phase transitions in a material with two distinct phases, such as
water and ice. For this problem, we introduce a notion of viscosity
solutions that allows for an appearance of the so-called mushy region. We prove a comparison principle and use this result to establish well-posedness of the viscosity solutions. As a corollary, we show that the viscosity solutions and the weak solutions defined in the sense of distributions coincide.

In recent work, we have investigated various aspects of the asymptotic behavior of solutions to systems that are known to describe the behavior of incompressible flows of binary fluids, that is, fluids composed by either two phases of the same chemical species or phases of different composition. We intend to give an overview on the following issues: existence and main properties
of (trajectory or global) attractors, exponential attractors, convergence to single equilibria, etc.

In this talk, I consider the existence of local-in-time strong solutions to a well established coupled system of partial differential equations arising in Fluid-Structure interactions. The system consisting of an incompressible Navier-Stokes equation and an elasticity equation with velocity and stress matching boundary conditions at the interface in between the two domains where each of the two equations is defined. I discuss new existence results for a range of regularity in the initial data and the differences in the exsitence results when domains with non-flat boundaries are considered.