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.