This project will provide a rigorous analytical investigation of the behavior near the domain boundary of solutions of partial differential equations (PDEs) modeling quantum and viscous fluids phenomena.  While viscous boundary effects are extensively studied in the mathematical literature, very little is known from an analytical perspective about the behavior of (inviscid) quantum fluids despite their high relevance from the physical viewpoint. 

In this project, we will tackle relevant questions and establish new and exciting results, with the long-term goal of providing valuable insights into the mathematical comprehension of intriguing phenomena observed in experiments and numerical simulations.  A main target of our investigation will be the so-called QHD system, describing a compressible and barotropic, inviscid fluid where quantum effects are relevant at macroscopic scales. This is what happens, for instance, in Bose-Einstein Condensation (BEC) and Superfluidity. Despite extensive investigations in both experimental and theoretical physics, the analytical comprehension of the QHD system is still at an embryonal stage. Partial results (in different regularity frameworks) concerning the existence of solutions are available, but the study of these solutions' qualitative and quantitative properties is far from satisfactory. In particular, very little is known from a rigorous mathematical perspective on fundamental questions concerning, for instance, stability vs instability properties of the solutions near the domain boundary. This project aims to address these questions by establishing a rigorous boundary layers analysis. In the long term, the aim is to provide a rigorous mathematical description of puzzling phenomena that recently received remarkable attention in the physical community. 

We will also study solutions of initial-boundary value problems for inviscid systems that are limits of the incomplete parabolic (physical viscosity) approximation in the boundary characteristic case. While tremendous progress has been achieved in the last 25 years in the noncharacteristic case (when all the characteristic velocities of the inviscid system are bounded away from zero), the characteristic case is much less understood owing to severe technical difficulties. We will establish new stability results that apply to the one-dimensional Navier-Stokes and Magneto-Hydrodynamics (MHD) systems.