Abstract

Fifty minutes Speaker

Recent developments on the surface quasi-geostrophic and related equations

 

Jiahong Wu

 

Oklahoma State University

 

Abstract

Fundamental issues such as the global regularity problem concerning the surface quasi-geostrophic (SQG) and related equations have attracted a lot of attention recently. Significant progress has been made in the last few years. This talk summarizes some current results on the critical and supercritical SQG equations and presents some very recent developments on the generalized SQG equations. In particular, global regularity for the logarithmically supercirtical SQG equation and for the Log-log Euler equation as well as local regularity for the generalized SQG equations with very singular velocity fields will be reported.

 


 

Existence and Stability of Stationary Solutions to Lotka-Volterra Competition with Cross-Diffusion

 

Shoji Yotsutani

 

Ryukoku University, Japan

 

Abstract

  

This is a joint work with Yuan Lou (Ohio State Univ.) and Wei.-Ming Ni (Univ. of Minnesota
and East China Normal Univ).
We have been interested in stationary solutions of a cross-diffusion equation
         
Proposed by Shigesada-Kawasaki-Teramoto in 1979.
   We concentrate on the case ρ11=ρ21=ρ22=0. Lou-Ni(JDE 1996, 1999)
obtained limiting equations as ρ12→∞. One of the limiting equations is
                         

This plays crucial roles to obtain segregated solutions. Lou-Ni-Y (DCDS, 2004) almost
revealed the structure of it in one-dimensional case. We have obtained some
existence and multiplicity results in multi-dimensional case for the limiting equation
and the original equation with sufficiently large ρ12.
In this talk, we investigate the existence, non-existence, and stability of stationary
solutions of the multi-dimensional case by combining mathematical results with
numerical computations.

 


 

The stability of the Hele-Shaw flow with and without surface tension and injection

 

Ching-Hsiao Cheng

 

National Central University

 

Abstract

In this talk, I will briefly introduce the physical model which the Hele-Shaw equations are modeling in fluid dynamics, and talk about some stability results concerning injection of fluids into the fluid domain. For the case without surface tension, the energy estimate alone is enough to conclude the stability results. For the case with surface tension, on the other hand, the techniques of Fourier series has to be used to derive certain decay estimates, and the usual energy estimates combined with the decay estimates give rise to stability results. Our results improve some past conclusions and the methodology used to establish our results can be applied to 3-dimensional problems.

 


 

Generalized Diffusion with Applications in Transport of Charged Particles

 

Chun Liu

 

Penn State University


Abstract

      


 

Boundary Relations for Hyperbolic-Dissipative Systems

 

Tai-Ping Liu

 

Academia Sinica

 

Abstract

Tranditionally, the study of the boundary behavior for partial differential equations is done for the initial-boundary or boundary value problems with specific given boundary conditions. WIth Shih-Hsien Yu, we initiate a new point of view of studying the boundary relation between the Dirichlet and Neumann boundary values. The boundary relation is more basic than the Green's function for a given boundary condition. It is as basic as the fundamental solutions.

Our approach starts with the Fourier transform in the spatial dierctions tangential to the boundary, followed with the Lapalce transform in time and the normal spatial direction. The partial differential equations turn to algebraic equations. The well-posedness consideration yields the Master Relationship, the Dirichlet-Neumann relation in the transformed variables. The main effort is to invert the tranforms to obtain the boundary relation in the physical domain. We aim at the explicit expression of the boundary relation; pointwise description is obtained for several physical systems, such as the dissipative wave equations and the compressible Navier-Sotkes equaitons.

For our analysis, we introduce a new combination of algebraic manipulations, complex contour integrations using the newly introduced notion of Laplace-Fourier paths, the short-long wave decomposition, and the boundary energy estimates.

We feel that this new perspective should prove useful for the study of the boundary behavior in a wide variety of situation such as the inverse problem and virtual computational boundary.


 

Smoothing Effect for the Homogeneous Boltzmann Equation

 

Tong Yang

 

City Uinversity of Hong Kong

 

Abstract

Department of mathematics, City university of Hong Kong, Hong Kong, P.R. China. In this talk, we will present our recent works about the smoothing effect on weak solutions to the homogeneous Boltzmann equation without angular cutoff. Firstly, in a joint work with Alexandre-Morimoto-Ukai-Xu, we show that every L1 weak solution with finite moments of all order acquires C1 regularity in any positive time. And then in a joint work with Morimoto, we show that Villani conjecture holds for the Maxwellian molecule type cross section. That is, any weak solution with measure initial datum except a single Dirac mass acquires C1 regularity in any positive time. Here, the coercivity estimate plays an important role. In particular, to prove Villani conjecture, a new time degenerate coercivity estimate is given. The reseach was supported in part by the General Research Fund of Hong Kong, CityU # 104511.

      

 

Thirty minutes Speaker

How to analyze boundary singularity in some elliptic equations of Hardy-Sobolev type?

Xiang Fang

 

USA

 

Abstract

In this talk we are interested in how the geometry of boundary singularities can affect the existence of positive solutions. In particular, we study the Dirichlet problem for an elliptic equation of Hardy-Sobolev type with multiple singularities on the boundary of a domain. In the past one usually poses a curvature condition at the boundary singularity in order to obtain positive solutions. In this paper we propose a more refined way to analyze boundary singularities. Then we are able to obtain both solutions obtainable via the curvature method and some new solutions which are unreachable by the curvature method.


 

Non-uniqueness of finite energy solutions to the Navier-Stokes equation on a hyperbolic manifold.

Chi-Hin Chan

 

Department of Applied Mathematics

 

Abstract

In this talk, I will present a recent piece of jointwork with my colleague Magdalena Czubak in which we gave a non-uniqueness result for finite energy, finitely dissipative smooth solutions araising from a specific initial datum to the Navier-Stokes equation on a hyperbolic manifold of negative sectional curvature. I will start the presentation with a brief explaination of the structure of the Navier-Stokes equation as formulated on a general Riemannian manifold. Then, I will bring to the auidances' attention the differential geometric works about the existence of nontrival bounded harmonic functions on hyperbolic manifolds due to D. Sullivan, Michael Anderson, R.
Schoen in the 1980's, since these differential geometric works constituted the foundation upon which this joint work with Magdalena Czubak was based. Also, if time permits, I will also briefly explain how to obtain the finite energy property of the family of non-unique solutions to the Navier-Stokes equation which we construct on a hyperbolic manifold.


 

On the Structure of Multi-Layers Cellular Neural Networks :the Complexity Between Two Layers

 

Jung-Chao Ban

 

National Dong Hwa University

 

Abstract

Let Y be the solution space of an n-layer cellular neural network, and let Y(i) and Y(j) be the hidden spaces, where 1 i n. (Y(n) is called the output space.) The classification and the existence of factor maps between two hidden spaces, that reaches the same topological entropies, are investigated in [Ban et al., J. Differential Equa- tions 252, 4563-4597, 2012]. This work considers the existence of factor maps between those hidden spaces carrying distinct topological entropies. For either case, the Hausdorff dimension dim Y(j) and dim Y(i) can be calculated. Furthermore, the dimension of Y(i) and Y(j) are related upon the factor map between them.


 

BOLTZMANN EQUATION FOR AN INFINITE RANGE POTENTIAL

 

Jin-Cheng Jiang

 

National Tsing Hua University

 

Abstract

We consider the finite range potential with cutoff parameter dm and the resulting Boltzmann equation. Then consider the limit of the resulting Boltzmann equation as dm tends to infinite. We proved that this is indeed the conventional Boltzmann equation with infinite range potential. This is a joint work with T.-P. Liu and Y. Sone.


 

Singular Behavior of Boltzmann Equation on a Planar Boundary

 

I-Kun Chen

 

National Chiao Tung University

 

Abstract

Botlzmann equation describes a gas, from a point of view of statistical mechanics, as a limit of a large amount of particles, constantly colliding with each other. This model behaves significantly differently from fluid mechanics around the boundary and is able to explain more phenomena. In this talk, we are going to study two kinds of singular behavior on a planar boundary. First, the slop of fluid velocity blows up logarithmically on the boundary. Second, the distribution function of particle velocity on the boundary has a jump discontinuity and secondary singularity of logarithmic type. Both results are on thermal transpiration problem for the linearized Boltzmann equation between two plates.

Joint work with: Hitoshi Funagane, Shigeru Takata, and Tai-Ping Liu


 

Equilibrating Effects of Boundary and Collision in Rarefied Gases

 

Hung-Wen Kuo

 

Academia Sinica

 

Abstract

We will discuss the time-asymptotic behavior for rarefied gases in the spherical domain with variable boundary temperature under the diffuse reflection boundary condition. First, we consider the dissipative effects of boundary for free molecular flow. Next, we use this to construct the stationary solution of the Boltzmann equation for sufficiently large Knudsen number and small boundary temperature variation. The exponential asymptotic stability of the stationary solution is also proved. This is based on joint work with Tai-Ping Liu and Li-Cheng Tsai.


 

On the spatially homogeneous Boltzmann equation for relativistic particles

 

Seok-Bae Yun

 

Academia Sinica

 

Abstract

The relativistic Boltzmann equation describes the evolution of the statistical distribution of gaseous particles in the Minkowski space-time. In this talk, we consider the Cauchy problem, asymptotic behavior and the propagation of moments for the relativistic Boltzmann equation in the case when the dynamics does not depend on spatial variables.
This is a joint work with Robert Strain.


 

Turbulent Flame Speeds in Unsteady Flows

 

Yu-Yu Liu

 

National Cheng Kung University

 

Abstract

G-equations are well-known front propagation models in combustion theory. By level set formulation, G-equation models the motion of the flame front by a laminar speed and a velocity field.  Some nonlinear effects, like curvature or diffusion, may be also built into the basic model. We aim to study turbulent flame speeds modeled by G-equations and cellular flow, in particular when the flow is highly advective or unsteady.


 

Hsin-Yuan Huang

 

National Sun Yat-Sen University

 

Abstract       


 

On a Free Boundary Problem for a Two-Species Weak Competition System

 

Chang-Hong Wu

 

National Chiao Tung University

 

Abstract

IIn this talk, we will introduce a Lotka–Volterra type weak competition model with a free boundary in a one-dimensional habitat. The main objective is to understand the asymptotic behavior of two competing species spreading via a free boundary. Some sufficient conditions for spreading success and spreading failure are provided, respectively.
Also, when spreading successfully, we provide an estimate for the spreading speed. This is a joint work with Prof. Jong-Shenq Guo.


 

Multiscale Methods for High Contrast Elliptic Equations

 

Chia-Chieh Chu

 

National Tsing Hua University

 

Abstract

In this talk, I will introduce multiscale numerical methods for the elliptic equations arising in interface and two-phase flow problems. The model problems we consider are motivated by the multiscale computations of flow and transport of two-phase flow in strongly heterogeneous porous media. Although the analysis is carried out for simplified model problems, it does provide valuable insight in designing accurate multiscale methods for more realistic applications. The method is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasi- uniform meshes, and without resolving the interfaces. he method has an optimal convergence rate and the hidden constants in these estimates are independent of the `’contrast'' of the PDE's coefficients. I conduct some numerical experiments to confirm the optimal rate of convergence of the proposed method and its independence from the aspect ratio of the coefficients.       


 

Global Dynamics of Zooplankton and Harmful Algae in Flowing Habitats Where the Algal Toxin Contains Little Limiting Nutrient

 

Feng-Bin Wang

 

Chang Gung University

 

Abstract

Prymnesium parvum (a haptophyte alga), also called “golden alga”, occurs worldwide and is capable of forming large fish-killing blooms in coastal and inland water environments. A viable phytoplankton community can emerge during the occurrence of harmful algal blooms in riverine ecosystems. Recently, it has been suggested that management of flow is a possible strategy to control harmful algal blooms and mitigate their effects in some river systems. It is important to understand the persistence of algae in the presence of flow and the spatial variation of algal abundance and toxicity during bloom and flow events.

Motivated by those facts, Grover et al. proposed one-dimensional models with simple habitat geometry and transport processes to study spatial variations of harmful algae and toxin production and decay, in riverine reservoirs. In this talk, we shall focus on a continuum approach using an advection-dispersion-reaction system to resolve transport and biochemical reaction kinetics along the main stem of a riverine reservoir.

This is a joint work with Professors Sze-Bi Hsu and Xiao-Qiang Zhao