There will be an invited talk.

Title: Approximation of the bitemperature Euler system in 2D
Speaker: Stephane Brull (Professor, Univ. of Bordeaux)
Time: 2022 July 12th, 10:30am~11:30am
Place: 27-325
Abstract:
This talk is devoted to the numerical approximation of the bidimensional bitemperature Euler system. This model is a nonconservative hyperbolic system describing an out of equilibrium plasma in a quasi-neutral regime. This system is non conservative because it involves products between velocity and pressure gradients that cannot be transformed into a divergential form. We develop a second order numerical scheme by using a discrete BGK relaxation model. The second order extension is based on a subdivision of each cartesian cell into four triangles to perform affine reconstructions of the solution. Such ideas have been developed in the litterature for systems of conservation laws. We show here how they can be used in our nonconservative setting. The numerical method is implemented and tested in the last part of the paper.

There will be an invited talk.

Title: Embedding Carnot groups into bounded dimensional Euclidean spaces with optimal distortion
Speaker: Sang Woo Ryoo (Princeton Univ.)
Time: 2020 August 5th & 7th, 09:30am~11:3am
Place: 27-116
Abstract:
In the first part of the talk, I will first introduce the concept of a Carnot group, which generalizes Euclidean spaces, and a differentiation theorem by Pansu which proves that noncommutative Carnot groups cannot be bi-Lipschitzly embedded into Euclidean spaces. I will then present Assouad’s theorem, which shows that the snowflakes of Carnot groups (and more generally doubling metric spaces) can be bi-Lipschitzly embedded into Euclidean spaces, although the target dimension may be very high. After that, I will state a result of Naor and Neiman which significantly reduces the target dimension at the cost of worsening the distortion, and a result of Tao that achieves both optimal distortion and optimal target dimension in the case of the Heisenberg group. My work, which has the above title, is a generalization of Tao’s result to the setting of Carnot groups. If time permits, I will introduce the Lovasz local lemma and some Littlewood-Paley theory on Carnot groups, which are tools used in the proof.

In the second part of the talk, I will explain the key conceptual difficulties in constructing the embedding, by analogy with the Nash embedding theorem. One difficulty is a loss of derivatives problems when solving for a certain linear equation (in the Nash embedding theorem one needs to solve for a quadratic form). This is overcome by a variant of the Nash-Moser iteration scheme pioneered by Tao, and it carries without difficulty to the setting of Carnot groups. The other difficulty is a construction of orthonormal fields which is used in the iterative construction of the embedding. In the Nash embedding theorem this is dealt with using basic fibre bundle topology, and in Tao’s proof this is done using a quantitative homotopy argument, but both seem to fail in the setting of Carnot groups. The main contribution of my paper is that this can be overcome even in the most general setting of doubling metric spaces by using the Lovasz local lemma and concentration of measure phenomenon on the Euclidean sphere.

There will be an invited talk.

Title: Regularity and Long-Time Behavior for Hydrodynamic Flocking Models
Speaker: Prof. Trevor Leslie (Univ. of Wisconsin-Madison)
Time: 2020 March 10th, 04:00pm~05:00pm
Place: 27-220
Abstract:
We consider the Euler Alignment model, a hydrodynamic analog of the discrete Cucker-Smale flocking ODE system. The salient feature of the Cucker-Smale model is the interaction of agents through a so-called “communication protocol” that tends to align the velocities of the agents; this alignment mechanism also drives the Euler Alignment PDEs. We discuss several of the different communication protocols treated in the literature and the corresponding wellposedness theory for the Euler Alignment model in each case. In the case of smooth, bounded protocols in one space dimension, it is possible to completely characterize the initial data leading to the existence of regular solutions, in terms of a certain pointwise balance between the initial velocity and the “total influence” from the density profile. This balance is captured by the sign of a certain quantity that we denote by e. Under the additional assumption of periodicity, a quantity closely related to e determines the long-time distortion of the density profile away from its average value.

There will be an invited talk.

Title: Derivation principle of BGK models
Speaker: Prof. Stephane Brull (Univ. of Bordeaux)
Time: 2020 January 14th, 10:30am~11:20pm
Place: 27-116
Abstract:
In this talk we will present a derivation principle of BGK
models using the resolution of an entropy minimization problem.

The construction is based as on the introduction of relaxation
coefficients and a principle of entropy minimization under
constraints for moments. These free parameters are next ajusted to
transport coefficients when performing a Chapman-Engskog expansion
ip to Navier-Stokes. Firstly, the methodology will be explained and
illustrated for a monoatomic and polyatomic single gas.
Next the case of gas mixtures is considered. In this part, after
clarifying the Chapman-Engskog, a BGK model is derived. This BGK
model is proved to satisfy Fick and Newton laws. In a last part, we
will explain how to extend our model to reacting gas mixtures.

There will be an HYKE Intensive Lectures.

Title: HYKE Intensive Lectures on Symmetric Hyperbolic Systems of Balance Laws and State
Lecturers: Prof. Tommaso Ruggeri (Univ. of Bologna)
                   Prof. Masaru Sugiyama (Nagoya Institute of Technology)
Time: 2019 December 3rd, 9:00am~1:00pm
           2019 December 6th, 9:00am~5:00pm
           2019 December 11th, 9:00am~11:00am
Place: 27-116 (3rd), 27-220 (6th, 11th)
Lecture Topics:
Prof. Ruggeri:
Lecture 1,2: Balance Laws, Hyperbolic Systems, Symmetric Systems. Main field and theorem of symmetrization. Examples of Euler fluids, non-linear elasticity, MHD, Born-Infeld nonlinear electrodynamics, relativistic fluids.
Lecture 3,4: Compatibility between Galilean invariance and Entropy Principle. Qualitative analysis, K -condition and global existence of smooth solutions. Principal subsystems and nesting theories with examples. The mixture of gas and connection with Cucker-Smale models.
Lecture 5,6: Shock wave and growth of entropy across the shock, shock structure and sub-shocks formation. Problematic of Rational Extended Thermodynamics in mono and polyatomic gas.
Prof. Sugiyama:
0. Preliminary Discussion
1. Introduction
  1.1 How to describe macroscopic phenomena?
  1.2 Three levels of description of macroscopic systems
2. Equilibrium Statistical Mechanics
  2.1 Distribution function and the Liouville equation
  2.2 Gibbs ensembles
     2.2.1 Microcanonical ensemble, and its application to ideal gas
     2.2.2 Canonical ensemble, and its applications to ideal gas and a system of harmonic oscillators
     2.2.3 Grandcanonical ensemble, and its application to ideal gas
  2.3 Remarks
     2.3.1 Gibbs entropy and information entropy
     2.3.2 Fluctuation
     2.3.3 Validity range of classical statistical mechanics
3. Some Topics I
  3.1 Phase transition
     3.1.1 Virial expansion and the van der Waals equation of state
     3.1.2 Yang and Lee theory of phase transition
     3.1.3 Ising model
     3.1.4 Critical phenomena: Renormalization group approach
  3.2 Fermi statistics and Bose statistics
4. Non-Equilibrium Statistical Mechanics
  4.1 BBGKY hierarchy of the distribution functions
  4.2 Boltzmann equation and the system of moment equations
5. Some Topics II
  5.1 Extended thermodynamics – A new non-equilibrium thermodynamics –
  5.2 Linear response theory
  5.3 Stochastic processes
6. Concluding Remarks and Outlook

There will be an invited talk.

Title: A condition for blow-up solutions to discrete semilinear wave equations on networks
Speaker: Dr. Min-Jun Choi (Seoul National University)
Time: 2019 November 6th, 10:00am~11:00am
Place: 27-220
Abstract:

There will be an invited talk.

Title: Hyper-elastic Ricci Flow
Speaker: Prof. Marshall Slemrod (University of Wisconsin-Madison)
Time: 2019 November 5th, 11:00am~12:00pm
Place: 27-220
Abstract:
In this talk, I will introduce the concept of hyper-elastic Ricci flow. The equation of hyper-elastic Ricci flow amends the classical Ricci flow by the addition of Cauchy stress tensor which itself is derived from the free energy. The main implication of the theory is a uniformization of material behavior which follows from application of a parabolic minimum principle.