Takashi Arima (NIT Tomakomai, Japan)

**type:** fundamental courses **area:** MAT-07

6 h** schedule:** November 29 - December 13, 2023 / on Wednesdays / 2:00-4:00 pm / November 29 room Aula Vitali - December 6 and 13 room Seminario VIII piano

This course introduces Rational Extended Thermodynamics (RET), a recently developed theory of non-equilibrium thermodynamics, with a focus on its mathema#tical aspects. The theory leads to the derivation of a hyperbolic system of partial differential equations as its closed field equations that describe fluids, which will be discussed. Specifically, the course will focus on the following three points: (1) Issues with the traditional theory of fluid dynamics: First, we will overview the traditional theory of fluid dynamics, the Navier-Stokes-Fourier (NSF) theory, summarize the problems arising from the fact that it is based on a

parabolic system, and discuss the construction methods of hyperbolic partial differential equation systems beyond the NSF theory.

Additionally, we will re-evaluate the kinetic theory of gases and di#scuss the structure of the system of balance equations describing fluids should have. (2) Construction of RET: We will demonstrate that RET can derive a closed set of field equations based on three requirements: the material frame indifference principle, the entropy principle, and causality. Further, we will discuss the mathematical structure of this hyperbolic system and its limits to parabolic theory as a coarse graining theory of RET. (3) Development of RET: It is known that the structure of the system of balance equations in RET varies depending on the fluid being studied. We will introduce the recently proposed RET for polyatomic gases and discuss the differences from the system for monatomic gases and its singular limit to monatomic gases.

**referent:** Andrea Mentrelli