- Date”§ 15:00-18:00 on Apr 25, 2023
- Venue: Kawai Hall, Tohoku University

- Xuwen Chen (University of Rochester)
- Giulio Schimperna (University of Pavia)

**15:00-15:05** Opening

**15:05-15:50** Xuwen Chen (University of Rochester)

From Quantum Particles to Compressible Inviscid Fluid

**15:50-16:20** Discussion and break

**16:20-17:05** Giulio Schimperna (University of Pavia)

On the "simplest" thermodynamically consistent phase-field model: existence of solutions and weak-strong uniqueness

**17:05-17:35** Discussion

**17:35-17:40** Closing

**17:40-18:00** Free discussion

From Quantum Particles to Compressible Inviscid Fluid

Xuwen Chen (University of Rochester)

**Abstract:**
We derive the classical compressible Euler equation as the limit of 3D
quantum N-particle dynamics as N tends to infinity and Planck's
constant tends to zero. We establish strong and quantitative
convergence up to the 1st blow up time of the limiting Euler equation.
During the course of the proof, we prove, as theoretically predicted,
that the macroscopic pressure emerges from the space-time averages of
microscopic interactions, which are in fact, Strichartz-type bounds
and we have hence found a physical meaning for the Strichartz type
bounds.

On the "simplest" thermodynamically consistent phase-field model: existence of solutions and weak-strong uniqueness

Giulio Schimperna (University of Pavia)

**Abstract:**
In this talk we will present a very simple model describing
non-isothermal phase transitions in a thermodynamically consistent
setting. As far as we know, this model was first introduced in the
90's
by following a general derivation scheme proposed by Michel Fr«±mond;
then, in the subsequent decades, the very same equations were obtained
by other authors by using different strategies. From the point of view
of analytical results, existence of weak solutions was proved more
than 20 years ago in one space dimension; on the other hand, in two-
and three- dimensions, the problem remained open until nowadays. Here
we will show that existence of weak solutions can be shown in two- and
three- dimensions by using a somehow "ad hoc" strategy, which however
works only in the case of configuration potentials of controlled
growth (so excluding the case of "singular" potential, e.g., of
logarithmic type). We will also discuss local in time existence of
strong solutions serving as a starting point for a weak-strong
uniqueness theorem. The results presented in the talk have been
obtained in collaboration with Robert Lasarzik (WIAS - Berlin) and
Elisabetta Rocca (Pavia).

This workshop is supported by JSPS KAKENHI 21KK0044.

- Goro Akagi (Mathematical Institute, Tohoku University)
- goro.akagi [AT] tohoku.ac.jp