[Invited talk] Green’s functions and Well-posedness of Compressible Navier-Stokes equation

There will be an invited talk.

Title: Green’s functions and Well-posedness of Compressible Navier-Stokes equation
Speaker: Prof. Shih-Hsien Yu (National University of Singapore)
Time: 2018 November 13th, 4:00pm~5:00pm
Place: 27-220
Abstract:
A class of decomposition of Green’s functions for the compressilbe Navier-Stokes linearized around a constant state is introduced. The singular structures of the Green’s functions are developed as essential devices to use the nonlinearity directly to covert the 2nd order quasi-linear PDE into a system of zero-th order integral equation with regular integral kernels. The system of integrable equations allows a wider class of functions such as BV solutions. We have shown global existence and well-posedness of the compressible Navier-Stokes equation for isentropic gas with the gas constant γ∈(0,e) in the Lagrangian coordinate for the class of the BV functions and point wise L∞ around a constant state; and the
underline pointwise structure of the solutions is constructed. It is also shown that for the class of BV solution the solution is at most piecewise C2-solution even though the initial data is piecewise C^infty.

[Invited talk] Self-Organized Hydrodynamic models for nematic alignment and the application to myxobacteria

There will be an invited talk.

Title: Self-Organized Hydrodynamic models for nematic alignment and the application to myxobacteria
Speaker: Prof. Hui Yu (Tsinghua University)
Time: 2018 November 1st, 11:00am~11:50am
Place: 27-220
Abstract:
A continuum model for a population of self-propelled particles interacting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but nonconservative system of equations for the nematic mean direction of the flow and the densities of particles flowing parallel or antiparallel to this mean direction. An application to myxobacteria is presented.

[Invited talk] Uncertainty quantification for partial differential equations and their optimal control problems

There will be an invited talk.

Title: Uncertainty quantification for partial differential equations and their optimal control problems
Speaker: Prof. Hyungcheon Lee (Ajou University)
Time: 2018 April 26th, 2:00pm~3:00pm
Place: 27-220
Abstract:
We consider the determination of statistical information about outputs of interest that depend on the solution of a partial differential equation and optimal control problems having random inputs, e.g. coefficients, boundary data, source term, etc. Monte Carlo methods are the most used approach used for this purpose. We discuss other approaches that, in some settings, incur far less computational costs. These include quasi-Monte Carlo, polynomial chaos, stochastic collocation, compressed sensing, reduced-order modelling.

[Invited talk]Fixed Point Method for Backward Stochastic Differential Equations

There will be an invited talk.

Title: Fixed Point Method for Backward Stochastic Differential Equations
Speaker: Prof. Ki-Hoon Nam (Monash Univ. of Melbourne, Australia)
Time: 2017 September 1st 11:00 – 11:50
Place: 129-301
Abstract:
Backward stochastic differential equation (BSDE) is a generalization of martingale representation theorem and it has been widely used for financial derivative pricing and stochastic optimization. Traditionally, most of well-posedness result of BSDE were based on contraction mapping theorem on the space of stochastic processes. In our work, we were able to transform BSDEs into fixed point problems in the space of Lp random variables. The simplicity of our framework enables us to apply various kind of fixed point theorems which have not been tried in previous literature. In particular, this enables to remove infinite dimensionality arise from time and we were able to use white noise analysis to use topological fixed point theorems. As a result, we were able to generalize previous well-posedness results: e.g. time-delayed type, mean-field type, multidimensional super-linear type. The talk is aimed for those who are not familiar with BSDE and it is based on BSE’s, BSDE’s and fixed point theorem (Annals of Probability, 2017, joint work with Patrick Cheridito).