International Workshop on PDEs and Computations in Fluid Dynamics

McGill University, Montreal, Canada

May 23, 2023

Purpose:

This one-day workshop, associated with the CRM Applied Mathematics Laboratory, is aimed to bring together the leading experts as well as promising young researchers to present their recent results in partial differential equations with applications and numerical computations in fluid dynamics. Key topics focus on the most challenging open problems in the area such as global regularity, uniqueness of solutions, singular limits, boundary layers behavior, and free boundary problems, etc. It also provides a premier interdisciplinary forum for senior and junior researchers to exchange their experiences in the study of partial differential equations. The talks will span from analysis through modeling and computation to applications of partial differential equations.

Organizers: 

Ming Mei (McGill University & Champlain College St-Lambert)

Jean-Christophe Nave (McGill University)

Speakers:

Yue-Jun Peng   (University of Clermont Auvergne, France)

Bruno Rubino (University of L’Aquila, Italy)

Haifeng Hu (McGill University & Changchun University)

Seth Taylor (McGill University)

Xiaowen Li      (McGill University & Northeast Normal University)

Shufang Xu (McGill University & Shanghai University)

Lizhen Zhang (McGill University & Shanghai Jiao Tong University)

Location:

Room 1104, Burnside Building, McGill University,

Financial Support

CRM Applied Mathematics Lab

Schedule

10:00-11:00 Yue-Jun Peng (University of Clermont Auvergne, France)

Quasi-neutral limit in Euler-Poisson systems

11:00-12:00 Bruno Rubino (University of L’Aquila, Italy)

Hybrid Models for Semiconductor Devices

Lunch break

1:30 - 2:30 Haifeng Hu (McGill University & Changchun University)

Structural stability for 1D semiconductor hydrodynamic model with

sonic boundary

2:30 – 3:30 Seth Taylor (McGill University)

TBA

3:30 – 4:00 Xiaowen Li (McGill University & Northeast Normal University)

Stability of stationary solutions to the multidimensional chemotaxis-consumption

models with physical boundary conditions

4:00 – 4:30 Shufang Xu (McGill University & Shanghai University)

The Riemann problem for the blood flow model with body force term

4:30 – 5:00 Lizhen Zhang (McGill University & Shanghai Jiao Tong University)

Mild solutions to stochastic incompressible nonhomogeneous Navier-Stokes

equations with fractional Brownian motion

Abstracts

Title: Quasi-neutral limit in Euler-Poisson systems

Speaker: Yue-Jun Peng (University of Clermont Auvergne, France)

Abstract: Euler-Poisson systems are widely used in the mathematical modeling of plasmas in which the Debye-length is a small parameter. The quasi-neutral limit leads to the quasi-neutrality of the plasma, which is realized as the parameter tends to zero. In this talk, I will present mathematical studies and results of the quasi-neutral limit in Euler-Poisson systems in stationary or non-stationary case. This concerns nonlinear partial differential equations of hyperbolic or elliptic type. Various mathematical tools
are needed to justify the limit in mathematical frameworks.

Title: Hybrid Models for Semiconductor Devices

Speaker: Bruno Rubino (University of L’Aquila, Italy)

Abstract: Semiconductor device modelling spans a wide range of areas in solid state physics, applied and computational mathematics. In this talk, we discuss some results about hybrid models linking the quantum hydrodynamics equation with classical hydrodynamics deriving the transmission conditions between the two PDE systems modelling the quantum and the classical dynamics.

Title: Structural stability for 1D semiconductor hydrodynamical model with sonic boundary

Speaker: Haifeng Hu (McGill University & Changchun University)

Abstract: In this talk, I will present our recent research on the structural stability of interior subsonic steady states to the hydrodynamic model for semiconductors with sonic boundary. More precisely, we show that the small perturbation in the subsonic doping profiles leads to the small difference between the corresponding interior subsonic solutions. While it has been proved that this model possesses various physical steady states such as the interior subsonic, interior supersonic, shock transonic and smooth transonic solutions, the singularities at the sonic boundary make it difficult to investigate the structural stability of these solutions. To address this issue, we propose a novel approach, which combines the weighted multiplier technique, local singularity analysis, monotonicity argument and squeezing skill. Our work indicates that the interior subsonic solutions are at least amenable to this approach. Numerical approximations further confirm our theoretical results. These results to some extent provide insights into the structural stability of other types of solutions. This is the joint work with Yuehong Feng and Ming Mei.

Title: TBA

Speaker: Seth Taylor (McGill University)

Abstract:

Title: Stability of stationary solutions to the multidimensional chemotaxis-consumption models

with physical boundary conditions

Speaker: Xiaowen Li (McGill University & Northeast Normal University)

Abstract: In this talk, we present the spectral analysis method to study the dynamical stability of nontrivial steady states to a multidimensional parabolic-parabolic chemotaxis-consumption model in a bounded domain with physical boundary conditions. Following the framework of Shi and Wang, it can be shown that the spectrum set of the linearized operator at the steady state only consists of eigenvalues with finite algebraic multiplicity. Then we show that all eigenvalues have negative real part under some assumptions on the parameters. This result of spectral analysis implies the local asymptotic stability of the nontrivial steady states in $W^{1,p}$ with $p>n$.

Title: The Riemann problem for the blood flow model with body force term

Speaker: Shufang Xu (McGill University & Shanghai University)

Abstract: This talk is concerned with the Riemann problem for the blood flow model with body force term. The Riemann solutions involve the rarefaction waves, the shock waves and the collapsed state. Due to the effect of body force term, the solutions are not self-similar. The existence of the simple wave solutions to the non-reducible blood flow model has been studied by the method of characteristic decomposition. All possible Riemann solutions are determined for any given Riemann initial data.

Title: Mild solutions to stochastic incompressible nonhomogeneous Navier-Stokes equations with fractional Brownian motion

Speaker: Lizhen Zhang (McGill University & Shanghai Jiao Tong University)

Abstract: In this talk, we study the global mild solution of stochastic nonhomogeneous incompressible Navier-Stokes equation perturbed by the addictive fractional Brownian motion in the torus with dimension . Mild solution is presented in a form of time integral, with regularity between weak solution and strong solution. We use operator splitting method to estimate the stochastic integral part and deterministic part separately. Based on the priori estimates, finally we obtain the local existence of mild solution by Banach fixed theorem and we further show the global existence of mild solution. This is a joint work with Prof. Ming Mei and Prof. Yachun Li.