This project addresses fundamental mathematical problems at the intersection of classical and quantum fluid dynamics, with a focus on compressible and quantum Euler systems. These models capture a range of complex physical phenomena, from high-speed gas flows to quantum fluids like super-fluids and Bose-Einstein condensates. The research aims to advance understanding of the transition from compressible to incompressible flow in the low Mach number regime and to explore how quantum effects influence solution structure and stability. The project also supports graduate training in mathematical analysis and applied partial differential equations (PDEs). The first component focuses on the incompressible limit of global weak solutions to the compressible Euler equations using convex integration, addressing key challenges such as nonlinear oscillations and weak convergence. The second component studies the quantum Euler equations derived from the Schrödinger equation via Madelung’s transformation, with a focus on whether quantum pressure intrinsically regularizes solutions. Analytical tools include polar decomposition and dispersive PDE theory. This research introduces new techniques in nonlinear PDEs and contributes to a deeper theoretical understanding of hyperbolic and dispersive systems in mathematical physics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts