The goal of this minicourse is to offer an exploration of the hydrodynamic limits of the Boltzmann equation, illuminating the profound connection between microscopic kinetic theory and macroscopic fluid dynamics. It is ideally suited for researchers and students interested in the rigorous mathematical bridges between kinetic theory and fluid dynamics, providing both foundational knowledge and insights into advanced research directions.

Part I: Foundations of the Boltzmann Equation.

We will begin by establishing a foundational understanding of the Boltzmann equation itself, covering its origin, interpretation, and some essential analytical tools required for its study. Key concepts such as entropy, the H-theorem, molecular chaos, and averaging lemmas will be discussed. Furthermore, we will examine various classes of solutions, including renormalized, dissipative, and weak solutions.

Part II: Deriving Macroscopic Equations from the Boltzmann Equation.

We will then show the systematic derivation of macroscopic equations from their microscopic kinetic origins. This includes a detailed exploration of limits that lead to the compressible Euler system and its linearization, the acoustic waves system. We will also cover incompressible limits, highlighting how familiar fluid dynamics equations, such as the Navier-Stokes equations, emerge from the Boltzmann equation.

Part III: Rigorous Analysis of Viscous Incompressible Limits.

A significant portion of the course will be dedicated to the viscous incompressible regime, culminating in the rigorous derivation of the incompressible Navier-Stokes system. Beyond presenting the formal derivation, a primary objective will be to equip participants with some of the essential mathematical tools necessary for the rigorous analysis of this crucial hydrodynamic limit.

Part IV: Examples of Complex Hydrodynamic Limits (Time Permitting).

Finally, if time permits, we will delve into more recent and advanced topics, showcasing examples of complex hydrodynamic limits. This includes the hydrodynamic limits of Vlasov-Maxwell-Boltzmann equations and the challenging low-temperature regimes, providing a glimpse into the current state of research in the field.