This volume presents seven invited lecture series from the DFG Priority Programme Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness, surveying recent developments in mathematical fluid dynamics across a broad range of scales, from kinetic theory to turbulence, mixing, and relativistic flows. The contributions reflect current research directions in the field. Hana Mizerová and Simon Markfelder discuss aspects of the emerging theory of non-standard weak and generalized solutions to the compressible Euler equations. Theodore Drivas addresses turbulence, Gianluca Crippa studies mixing phenomena, Tai‑Ping Liu and Shih‑Hsien Yu develop the theory of nonlinear waves for the Boltzmann equation, and Marcelo Disconzi focuses on dissipative relativistic fluid dynamics.The volume is intended for researchers and advanced graduate students in partial differential equations and applied mathematics.

This volume presents seven invited lecture series from the DFG Priority Programme Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness, surveying recent developments in mathematical fluid dynamics across a broad range of scales, from kinetic theory to turbulence, mixing, and relativistic flows. The contributions reflect current research directions in the field. Hana Mizerová and Simon Markfelder discuss aspects of the emerging theory of non-standard weak and generalized solutions to the compressible Euler equations. Theodore Drivas addresses turbulence, Gianluca Crippa studies mixing phenomena, Tai‑Ping Liu and Shih‑Hsien Yu develop the theory of nonlinear waves for the Boltzmann equation, and Marcelo Disconzi focuses on dissipative relativistic fluid dynamics.The volume is intended for researchers and advanced graduate students in partial differential equations and applied mathematics.