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.

This presentation explores recent developments in the theory of duality and gap functions for equilibrium problems ([1,2,3,4]). We investigate the construction of gap functions derived from conjugate duality principles in convex optimization, providing a unified framework for analyzing equilibrium problems ([1]). The presentation concludes with a comprehensive review of recent literature and promising research directions, emphasizing potential applications in variational inequalities and Nash equilibrium problems. This work contributes to the broader understanding of duality theory in equilibrium problems and provides valuable tools for both theoretical analysis and practical implementations ([5]).

References

1. Altangerel, L., Bot, R. I. and Wanka, G. On Gap Functions for Equilibrium Problems via Fenchel Duality, Pacific Journal of Optimization 2 (3), 667-678, 2006.
2. Altangerel, L. On Gap Functions for Quasi-Equilibrium Problems Via Duality, Journal of Mathematical Sciences, 2024.
3. Konnov, I. V., Schaible, S. Duality for Equilibrium Problems under Generalized Monotonicity, Journal of Optimization Theory and Applications 104, 395-408.
4. Mastroeni, G. Gap functions for equilibrium problems, J. Glob. Optim. 27(4):411-426, 2003.
5. Pappalardo, M., Mastroeni, G. and Passacantando, M. Merit Functions: a Bridge Between Optimization and Equilibria, 4OR-Q J Oper Res 12, 1-33, 2014.