Kinetic and fluid equations for collective behavior

- Every two weeks on Friday: a 40 mn talk
- French time: 11:00-12:00, Korean time: 19:00-20:00
- click on the Zoom link
- ID of the meeting: 819 8821 8704
- Passwd of the meeting: 1234

- 2021/03/12: Prof. Jihoon Lee (Dept. of Math, Chung-Ang Univ.)

Decay estimates of solutions to the fluid equations with rotation or stratification

Abstract: In this talk, we consider the incompressible fluid equations with rotation or stratification.First, we consider three dimensional incompressible Navier-Stokes equations with fractional dissipation and Coriolis force. We find Coriolis force gives extra temporal decay of the solutions under some conditions on the initial data.

Next, we consider the three-dimensional damped Boussinesq equations with strong stratification. We find the global-in-time existence of solutions under some conditions of the initial data and the temporal decay of solutions.

- 2021/02/26: François Golse (Ecole Polytechnique)

PDF Half-space problem for the Boltzmann equation with phase transition at the boundary

Abstract: Y. Sone, K. Aoki and their group have studied numerically the existence of a solution to the steady half-space problem for a rarefied gas whose behavior is described by the Boltzmann equation (with slab symmetry). The gas is assumed to fill a half-space on top of a liquid which is its condensed phase, and the velocity distribution function of molecules entering the half-space from the condensed phase is the centered Maxwelllian parametrized by the temperature at the gas-liquid interface, and the saturating vapor pressure at this temperature. The state at infinity (i.e. far from the interface) is (another) Maxwellian. In a remarkable paper T.-P. Liu and S.-H. Yu [Arch. Rational Mech. Anal. 209 (2013), 869-997] have proposed a complete method for handling this kind of problem. The purpose of this talk is to present an alternative, self-contained proof of one of results in the work of Liu and Yu, specifically the existence and uniqueness of solutions that are decaying as the distance to the interface goes to infinity, uniformly in the Mach number of the Maxwellian at infinity. The proof uses a variant of the generalized eigenvalue problem studied by Nicolaenko in his work on the shock profile for the Boltzmann equation, and the Ukai-Yang-Yu penalization method for half-space problems in kinetic theory (suitably modified). [Work with N. Bernhoff.]

- 2021/02/08: Seok-Bae Yun (Sunkyunkwan University (SKKU))

PDF Ellipsoidal BGK model of the Boltzmann equation with the correct Prandtl number

Ellipsoidal BGK model (ES-BGK) is a generalized version of the Boltzmann-BGK model where the local Maxwellian in the relaxation operator of the BGK model is extended to an ellipsoidal Gaussian with a Prandtl parameter ν, so that the correct Prandtl number can be computed in the Navier-Stokes limit. In this talk, we review some of the recent results on ES-BGK model such as the existence (stationary or non-stationary) theory and the entropy-entropy production estimates. A dichotomy is observed between −1/2 < v < 1 and ν=-1/2. In the former case, an equivalence relation between the local temperature and the temperature tensor enables one to apply theories developed for the original BGK model in a modified form. In the critical case (ν=-1/2), where the correct Prandtl number is achieved, such equivalence break down, and the structure of the flow has to be incorporated to estimate the temperature tensor from below. This is from joint works with Stephane Brull and Doheon Kim.