International Conference on Recent Advances in Fluid Dynamics and Nonlinear Dynamics

June 2–June 5, 2019 University, , Conference on “Recent Advances in Fluid Dynamics and Nonlinear Dynamics”

College of Mathematics, , Chengdu, China

June 2-5, 2019

Objectives The aim of this international conference is to bring together experts who works on diverse frontiers of nonlinear PDEs and their appli- cations to survey recent progress and current challenges, to discuss how new ideas and methods could advance the related fields in com- ing years. The conference will contribute in promoting, enhancing, and stimulating cross-continental research interactions and collabo- rations in advanced mathematics applied to nonlinear sciences.

Organizing Committee

Co-Chairs Tian Ma (Sichuan University) Shouhong Wang (Indiana University, USA) Secretary Quan Wang (Sichuan University) Members Daozhi Han (Missouri University of Science and Technology) Hong Luo (Sichuan Normal University) Limei Li (Sichuan Normal University) Ruikuan Liu (Southwest Petroleum University) Zhigang Pan (Southwest Jiaotong University)

Sponsorship College of Mathematics, Sichuan University

1 Conference Logistics

1. Registration: 1:00pm-11:00pm, June 2, 2019. 2. Time and Venue:

• Invited talks, June 3-4, 2019. • Free discussions, June 5, 2019. • Conference Hall 303, College of Mathematics Building, Sichuan Uni- versity.

3. Accommodation: Kehua Hotel, No.141 Kehua North Road,Wuhou Dis- trict, Chengdu, China. 4. Meals: All meals are provided at Kehua Hotel. 5. Transport Information:

• From Shuangliu International Airport to Kehua Hotel: 50RMB by taxi. • From Chengdu East Train Station to Kehua Hotel: 30RMB by taxi. • From Chengdu North Railway Station to Kehua Hotel: 50RMB by taxi. • From Kehua Hotel to Conference Hall 303: 10 minutes walk.

6. Pick-Up Service: We offer pick-up service, please send to Quan Wang (Email: [email protected]) your flight details before May 28, 2019. 7. Contact: Quan Wang, Phone +8613558629124. 8. Registration Fee: There will be no conference fee for this meeting.

College of Mathematics, Sichuan University May 22, 2019

2

The Route from Hotel to Meeting place List of Invited Speakers 1. Dongho Chae (Chung-Ang University) Email: [email protected] 2. Daozhi Han (Missouri University of Science and Technology) Email: [email protected] 3. Chun-Hsiang Hsia (National Taiwan University) Email: [email protected] 4. Song Jiang (Institute of Applied Physics and Computational Mathemat- ics, ) Email: [email protected] 5. Boualem Khouider (University of Victoria) Email: [email protected] 6. Chanh Kieu (Indiana University Bloomington) Email: [email protected] 7. Limei Li (Sichuan Normal University) Email: [email protected] 8. Honghu Liu (Virginia Tech) Email: [email protected] 9. Ruikuan Liu (Southwest Petroleum University) Email: [email protected] 10. Anna L. Mazzucato (Penn State University) Email: [email protected] 11. Ruyun Ma (Xidian University) Email: [email protected] 12. Tian Ma (Sichuan University) Email: [email protected] 13. Taylan Sengul (Marmara University) Email: [email protected] 14. Chunyou Sun (Lanzhou University ) Email: [email protected] 15. Jianwen Sun (Lanzhou University ) Email: [email protected]

3 16. Quan Wang (Sichuan University) Email: [email protected] 17. Shouhong Wang (Indiana University Bloomington) Email: [email protected] 18. Xiaoming Wang (Southern University of Sciences and Technology) Email: [email protected] 19. Yongfu Wang (Southwestern University of Finance and Economics) Email: [email protected] 20. Yao Xu (Nanjing University) Email: [email protected] 21. Peihao Zhao (Lanzhou University) Email: [email protected]

4 Schedule for Invited Lectures

5 June 3, 2019

9:00am–9:10am Opening Ceremony 9:10am–9:20am Group Photo

Chair: Shouhong Wang (Indiana University, USA)

Song Jiang (Institute of Applied Physics and Com- putational Mathematics, Beijing, China) 9:20am–9:50am Magnetic inhibition effect on the Rayleigh-Taylor in- stability in non-resistive magnetohydrodynamics

Honghu Liu (Virginia Tech, USA) 9:50am–10:20am A variational approach to closure of nonlinear dynam- ical systems based on small-scale parameterizations

10:20am–10:40am Tea Break

Chair: Dongho Chae (Chung-Ang University, Korea)

Chanh Kieu (Indiana University, USA) 10:40am–11:10am On the application of the dynamical transition to the tropical cyclone formation problem

Quan Wang (Sichuan University, China) Dynamic transitions in axisymmetric nonlinear sys- 11:10am–11:40am tems and application to a two-layer quasi-geostrophic model in an annular channel

12:00pm–1:00pm Lunch Time

6 June 3, 2019

Chair: Song Jiang (Institute of Applied Physics and Computational Math- ematics, Beijing, China)

Anna Mazzucato (Penn State University, USA) 2:30pm–3:00pm Boundary layers and the vanishing viscosity limit for incompressible flows

Chunyou Sun (Lanzhou University, Guangzhou Uni- 3:00pm–3:30pm versity, China) Inertial manifolds for the 3D modified-Leray-model

Limei Li (Sichuan Normal University, China) 3:30pm–4:00pm L2-asymptotic stability of mild solutions to Magneto- hydrodynamic Benard-Convection system

4:00pm–4:20pm Tea Break

Chair: Dun Zhao (Lanzhou University)

Ruyun Ma (, China) 4:20pm–4:50pm Global structure of radial positive solutions for a pre- scribed mean curvature problem

Taylan Sengul (Marmara University, Turkey) 4:50pm–5:20pm Stability and Transitions of Quasi-Geostrophic Equa- tions

Yao Xu (, China) 5:20pm–5:50pm Homogenization of Elliptic Systems with Stratified Structure

6:20pm–7:20pm Dinner Time

7 June 4, 2019

Chair: Ruyun Ma (Xidian University, China)

Dongho Chae (Chung-Ang University, Korea) 9:00am–9:30am On the Type I blow-up for the incompressible Euler equations

Tian Ma1 and Shouhong Wang2 (Sichuan Univer- sity1, Indiana University2) 9:30am–10:00am Field Theoretical Interpretation of Quantum Mechan- ical Wave Functions and the Mechanism of High Tc Superconductivity

Peihao Zhao (Lanzhou University, China) 10:00am–10:30am Some regularity results in Musielak-Orlicz-Sobolev spaces

10:30am–10:50am Tea Break

Chair: Jianxun Hu (Sun Yat-Sen University, China)

Boualem Khouider (University of Victoria, Canada) 10:50am–11:20am Towards a Stochastic Relaxation for the Quasi- Equilibrium Theory of Cumulus Parameterization: Multicloud Instability, Multiple Equilibria and Chaotic Dynamics

Yongfu Wang (Southwestern University of Finance and Economics, China) 11:20am–11:50am Steady collision of two jets issuing from two axially symmetric channels

12:20pm–1:30pm Lunch Time

8 June 4, 2019

Chair: Peihao Zhao (Lanzhou University, China)

Daozhi Han (Missouri University of Science and Technology, USA) 2:30pm–3:00pm On the design of super-convergent, unconditionally stable and solvable numerical schemes for the Cahn- Hilliard fluid models

Jianwen Sun (Lanzhou University, China) 3:00pm–3:30pm Some Nonlocal Dispersal Equations and Applications

Ruikuan Liu (Southwest Petroleum University, 3:30pm–4:00pm China) Magneto-hydrodynamical Model for Plasma

4:00pm–4:20pm Tea Break

Chair: Tianqing An (, China)

Chun-Hsiung Hsia (National Taiwan University) On the mathematical analysis of the synchronization 4:20pm–4:50pm theory with time-delayed effect

Xiaoming Wang (Southern University of Sciences 4:50pm–5:20pm and Technology, China) Coupling and decoupling of free flow and porous media flow

6:00pm–7:00pm Dinner Time

9 June 5, 2019

Free Discussions

10 Abstracts of Invited Talks

On the Type I blow-up for the incompressible Euler equations

Dongho Chae

Chung-Ang University, Korea

Email: [email protected]

In this talk we discuss the Type I blow up and the related problems in the 3D Euler equa- tions. We say a solution v to the Euler equations satisfies Type I condition at possible blow − ∥∇ ∥ ∞ ∞ up time T∗ if lim supt↗T∗ (T∗ t) v(t) L < + . The scenario of Type I blow up is a natural generalization of the self-similar(or discretely self-similar) blow up. We present some recent progresses of our study regarding this. We first localize previous result that “small Type I blow up” is absent. After that we show that the atomic concentration of energy is excluded under the Type I condition. This result, in particular, solves the problem of re- moving discretely self-similar blow up in the energy conserving scale, since one point energy concentration is necessarily accompanied with such blow up. We also localize the Beale-Kato- Majda type blow up criterion. Using similar local blow up criterion for the 2D Boussinesq equations, we can show that Type I and some of Type II blow up in a region off the axis can be excluded in the axisymmetric Euler equations. These are joint works with J. Wolf.

On the design of super-convergent, unconditionally stable and solvable numerical schemes for the Cahn-Hilliard fluid models

Daozhi Han

Missouri University of Science and Technology

Email: [email protected]

The Cahn-Hilliard fluid systems, such as the Cahn-Hilliard-Navier-Stokes equations andthe Cahn-Hilliard-Darcy equations, are popular diffuse interface models for multiphase moving boundary problems. These systems are difficult to solve due to the stiffness associated with the diffusive interface, the high-order nonlinear nature of the PDE systems, andthemo- tion of the relatively sharp front in advection-dominated regime in many applications. In this talk, we discuss the design of super-convergent, unconditionally stable and uniquely solvable numerical schemes for solving Cahn-Hilliard fluid models. The main ideas include the convex-splitting and the pressure-correction in the temporal discretization, and the hy- bridizable discontinuous Galerkin methods in the spatial discretization. Several numerical simulations will be reported as well.

On the mathematical analysis of the synchronization theory with time-delayed effect

Chun-Hsiung Hsia

National Taiwan University

11 Email: [email protected]

This joint work with Chang-Yeol Jung, Bongsuk Kwon and Yoshihiro Ueda. We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interac- tions and phase lag effect. Both the phase and frequency syn- chronization are inview. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complete and partial phase synchronization for both models with the uniform time-delay and phase lag. Our results show that the Kuramoto models incorporated with small variation of time-delays and/or phase lag effect still exhibit the synchronization. These support that the original Kuramoto model (i.e., no time-delays/phase lags) is qualitatively robust in the perturbation of time- delay and phase lag effects. We also present several numerical experiments sup- porting our main results.

References

[1] C.-H. Hsia, C.-Y. Jung, B. Kwon and Y. Ueda, Synchronization of Kuramoto oscillators with time-delayed interactions and phase lag effect, preprint. [2] C.-H. Hsia, C.-Y. Jung and B. Kwon, On the synchronization theory of Kuramoto oscillators under the effect of inertia, J. Differential Equations, to appear. [3] C.-H. Hsia, C.-Y. Jung and B. Kwon, On the global convergence of frequency synchro- nization for Kuramoto and Winfree oscillators, Discrete Contin. Dyn. Syst. B, to appear.

Magnetic inhibition effect on the Rayleigh-Taylor instability in non-resistive magnetohydrodynamics

Song Jiang

Institute of Applied Physics and Computational Mathematics, Beijing, China

Email: [email protected]

The Rayleigh-Taylor (RT) instability is well known as gravity-driven instability in fluid dy- namics when a heavy fluid is on top of a light one. It appears in a wide range of applications in science and technology, such as in inertia confinement fusion, Tokamak, supernova explosions. In this talk, mathematical analysis of the magnetic RT instability in both incompressible and compressible fluids will be presented, in particular, effects of (impressed) magnetic fieldsupon the growth of the RT instability will be discussed and analyzed quantitatively. We shall show that a sufficiently strong (impressed) magnetic field can inhibit the RT instability; otherwise, instability will still occur in the sense that solutions do not continuously depend on initial data. Moreover, we shall give an explanation of physical mechanism for the magnetic inhibi- tion phenomenon based on mathematical analysis.

On the application of the dynamical transition to the tropical cyclone formation problem

Chanh Kieu

Indiana University

12 Email: [email protected]

This study examines the application of the dynamical transition framework recently devel- oped by Ma and Wang to understand the formation of tropical cyclones (TC) from the large- scale perspective. It is shown that the large-scale formation of TCs can be understood as a result of the Principle of Exchange of Stabilities in the barotropic model for the Intertropical Convergence Zone (ITCZ). Analyses of the transition dynamics at the critical point reveal that the maximum number of TC disturbances that the Earth’s tropical atmosphere can support at any instant of time has an upper bound, which is 12 for the current atmospheric condition. Additional numerical estimation of the transition structure on the central man- ifold of the ITCZ model confirms this important finding, which offers an explanation fora fundamental question of why the Earth’s atmosphere can support a limited number of TCs globally each year.

Towards a Stochastic Relaxation for the Quasi-Equilibrium Theory of Cumulus Parameterization: Multicloud Instability, Multiple Equilibria and Chaotic Dynamics

Boualem Khouider

University of Victoria

Email: [email protected]

The representation of clouds and organized tropical convection remains one of the biggest sources of uncertainties in climate and long-term weather prediction models. Some of the most common cumulus parameterization schemes, namely mass-flux schemes, rely on the quasi-equilibrium (QE) closure, which assumes that convection consumes the large scale in- stability and restores large scale equilibrium instantaneously. However the QE hypothesis has been challenged both conceptually and in practice. In existing work, the QE assumption was relaxed and instead prognostic equations for the cloud work function (CWF) and the cu- mulus kinetic energy (CKE) were derived and used. It was shown that when the CWF kernel merely acts to decrease the CWF, the prognostic closure system leads to damped oscillations on a time scale of a few hours, giving parameterized cumulus clouds enough memory to in- teract with each other, with the environment, and with stratiform anvils in particular. Here we take a few steps forward and show that when cloud-cloud interactions are reintroduced into the CWF-CKE equations, the coupled system becomes unstable. More importantly, a refinement for the CWF-CKE prognostic closure including their coupling to the meanfield equations, from a stochastic multicloud model (SMCM), for the cloud area fraction (CAF) is proposed. Qualitative analysis and numerical simulations show that in the case of a single cloud type, the CKE-CWF-CAF equations exhibit interesting dynamics including multiple equilibria, limit cycles, and chaotic behaviour both when the large-scale forcing is held fixed and when it oscillates with various frequencies, representative of cumulus convection unre- solved variability.

L2-asymptotic stability of mild solutions to Magnetohydrodynamic Benard-Convection system

Limei Li

Sichuan Normal University

13 Email: [email protected]

In this talk, we consider global-in-time small solutions of the initial value problem to Mag- netohydrodynamic Benard-Convection system in R3. For such solutions, we discuss their asymptotic stability under arbitrarily large initial L2-pertubations.

A variational approach to closure of nonlinear dynamical systems based on small-scale parameterizations

Honghu Liu

Virginia Tech

Email: [email protected]

The modeling of physical phenomena oftentimes leads to partial differential equations (PDEs) that are usually nonlinear and can also be subject to various uncertainties. Solutions of such equations typically involve multiple spatial and temporal scales, which can be numerically expensive to fully resolve. On the other hand, for many applications, it is often some large- scale features of the solutions that are of interest. The closure problem of a given PDE system seeks essentially for a smaller system that governs to a certain degree the evolution of such large-scale features, in which the effects of small-scale variables are modeled through various parameterization schemes based on the large-scale variables. In this talk, we discuss a new approach to deal with the parameterization problem of small spatial scales by large ones for nonlinear stochastic PDEs [1, 2, 3, 2]. The approach is variational in nature, and relies on manifolds that aim to provide in a mean square sense approximate small-scale parameterizations. We will highlight a simple semi-analytic approach to determine such manifolds based on backward-forward auxiliary systems. We will then illustrate the approach on some academic test cases. This is joint work with Mickaël D. Chekroun (UCLA), James C. McWilliams (UCLA) and Shouhong Wang (IU).

References

[1] M. D. Chekroun and H. Liu. Finite-horizon parameterizing manifolds, and applications to suboptimal control of nonlinear parabolic PDEs. Acta Applicandae Mathematicae, 135(1):81–144, 2015. [2] M. D. Chekroun, H. Liu, and S. Wang. Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. Springer Briefs in Mathematics, Springer, New York, 2015. [3] M. D. Chekroun, H. Liu, and S. Wang. Stochastic Parameterizing Manifolds and Non- Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II. Springer Briefs in Mathematics, Springer, New York, 2015. [4] M.D. Chekroun, H. Liu, and J.C. McWilliams. The emergence of fast oscillations in a reduced primitive equation model and its implications for closure theories. Computers & Fluids, 151:3–22, 2017.

Magneto-hydrodynamical Model for Plasma

14 Ruikuan Liu

Southwest Petroleum University

Email: [email protected]

Basing on the Newton’s second law and the Maxwell equations for the electromagnetic field, we establish a new N-dimensional(N ≥ 2) incompressible magneto-hydrodynamics (MHD) equations for the motion of plasma under the standard Coulomb gauge. By using the Galerkin method, we prove the existence of a global weak solution for this new N-dimensional model. Moreover, based on Agmon, Douglis and Nirenberg’s estimates for the stationary Stokes equation and the Solonnikov’s theorem of Lp-Lq-estimates for the evolution Stokes equation, we show that this 2-dimensional MHD equation exists a global strong solution. In addition, the uniqueness of global strong solution is obtained.

Boundary layers and the vanishing viscosity limit for incompressible flows

Anna Mazzucato

Penn State University, University Park, USA

Email: [email protected]

I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Eu- ler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or mono- tonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a result on concentration of vorticity at the boundary for symmetric flows and the justifica- tion of Prandtl approximation for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains, quantifying the effect of curvature on the pressure correction.

References

[1] G.-M. Gie, J. P. Kelliher, A. L. Mazzucato. Boundary layers for the Navier- Stokes equations linearized around a stationary Euler flow. J. Math. Fluid Mech. 20(4), 1405– 1426,2018. [2] G.-M. Gie, J. P. Kelliker, M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussen- zveig Lopes. The vanishing viscosity limit for some symmetric flows.. An- nales de l’Institut Henri Poincare, online Dec 5, 2018.

Global structure of radial positive solutions for a prescribed mean curvature problem

Ruyun Ma

Xidian University

15 Email: [email protected]

We are concerned with the global structure of radial positive solutions of boundary value problem ( ) div ϕN (∇v) + λf(|x|, v) = 0 in B(R), v = 0 on ∂B(R), √ y N N where ϕN (y) = , y ∈ R , λ is a positive parameter, B(R) = {x ∈ R : |x| < R}, and 1−|y|2 | · | denote the Euclidean norm in RN . All results, depending on the behavior of nonlinear term f near 0, are obtained by using global bifurcation techniques.

Inertial manifolds for the 3D modified-Leray-α model

Chunyou Sun

Lanzhou University, Guangzhou University

Email: [email protected]

Inertial manifold (IM) is a finite-dimensional Lipschitz invariant manifold that contains the global attractor and attracts all the orbits at an exponential rate. From the viewpoint of physical or turbulent, an IM is an interaction law relating small and large eddies in turbulent flow. In this talk, I will talk about our recent results on the existence ofan N-dimensional IM for the critical modified-Leray-α model in T3. This is a joint work with Dr. Xinhua Li.

Some Nonlocal Dispersal Equations and Applications

Jianwen Sun

Lanzhou University

Email: [email protected]

In this talk, we shall give some basic results of recent work of nonlocal dispersal equations. The main results are nonlocal eigenvalue problems with parameter and the asymptotic be- havior of positive solutions. This is a joint work with Pros W.-T. Li and Z.-C. Wang

Stability and Transitions of Quasi-Geostrophic Equations

Taylan Sengul

Marmara University, Turkey

Email: [email protected]

The dynamical processes involved in the description of geophysical flows in the atmosphere and the ocean are extremely complex. There are a multitude of physical variables, several processes that determine the evolution of these variables and a wide range of spatial and temporal scales. The simplest set of equations that meaningfully describe the large scale flows in the atmosphere or mesoscale flows in the oceans are the so called quasi-geostrophic equations. I will present some recent results on the stability and transitions of the exact solutions in various models of barotropic quasi-geostrophic equations.

16 References

[1] Kieu, Chanh, Taylan Sengul, Quan Wang, and Dongming Yan. “On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents.” Communi- cations in Nonlinear Science and Numerical Simulation 65 (2018): 196-215. [2] Dijkstra, Henk, Taylan Sengul, Jie Shen, and Shouhong Wang. “Dynamic transitions of quasi-geostrophic channel flow.” SIAM Journal on Applied Mathematics 75, no. 5 (2015): 2361-2378.

Dynamic transitions in axisymmetric nonlinear systems and application to a two-layer quasi-geostrophic model in an annular channel

Quan Wang

Sichuan University

Email:[email protected]

Dynamic transitions of dissipative systems can be classified into three categories: continu- ous type, catastrophic type, and mixed type. In fluid dynamics, it is essential to know the specific transition type involved in many unstable phenomena. In this article, weaimto completely classify the dynamic transitions arising in the axisymmetric systems modeled by the equations with a fourth-second order structure, thus covering a broad class of problems in geophysical fluid dynamics. A transition theorem is established by reducing the governing equations to a complex-valued ODE, derived by employing approximate invariant manifolds. We also provide an algorithm by which one can numerically determine the transition type for any problem whose governing equations are equipped with the required fourth–second order structure. To illustrate how to use the transition theorem and algorithm, we apply our re- sults to examine the transition types associated with the baroclinic instability in a two–layer quasi-geostrophic system in an annular channel and with different bathymetry profiles. This is a joint work with Dr. Marco Hernandez.

Field Theoretical Interpretation of Quantum Mechanical Wave Functions and the Mechanism of High Tc Superconductivity

Tian Ma1 and Shouhong Wang2

Sichuan University1, Indiana University2

Email: [email protected], [email protected]

First, we introduce a new field theoretical interpretation of quantum mechanical wave func- tions, by postulating that the wave function is the common wave function for all particles in the same class determined by the external potential V, the square of the modulus of the wave function represents the distribution density of the particles, and the gradient of phase of the wave function provides the velocity field of the particles. Second, we show that the keyfor condensation of bosonic particles is that their interaction is sufficiently weak to ensure that a large collection of boson particles are in a state governed by the same condensation wave function field under the same bounding potential V. For superconductivity, the formation of superconductivity comes down to conditions for the formation of electron-pairs, and for

17 the electron-pairs to share a common wave function. Thanks to the recently developed PID interaction potential of electrons and the average-energy level formula of temperature, these conditions for superconductivity are explicitly derived. Furthermore, we obtain both micro- scopic and macroscopic formulas for the critical temperature. Third, we derive the field and topological phase transition equations for condensates, and make connections to the quantum phase transition, as a topological phase transition.

Coupling and decoupling of free flow and porous media flow

Xiaoming Wang

Southern University of Sciences and Technology

Email: [email protected]

Many natural and engineering problems involve the cou- pled system of free flow and flows in porous media. Here we discuss recent developments in terms of physical interface boundary condi- tions that couple the free flow system with flows in porous media together with their analysis. We also discuss the physically impor- tant small Darcy number regime and the decoupling of the free flow system from the porous media system.

Steady collision of two jets issuing from two axially symmetric channels

Yongfu Wang

Southwestern University of Finance and Economics

Email: [email protected]

In this talk, we will discuss the mathematical theory on collision of two incompressible jets emerging from two axially symmetric nozzles. This work is motivated by the famous survey in 1989 by A. Friedman. More precisely, we showed that for any given mass fluxes M1 > 0 and M2 < 0 in two nozzles respectively, that there exists an incompressible, inviscid im- pinging outgoing jet with contact discontinuity. Furthermore, we showed that there is no stagnation point in the flow field and its closure, except one point on the symmetric axis. Some asymptotic behavior of the impinging jet in upstream and downstream and geometric properties of the free stream surfaces are also obtained. This is a joint work with Lili Du (SCU).

Homogenization of Elliptic Systems with Stratified Structure

Yao Xu

Nanjing University

Email: [email protected]

18 This talk mainly concerns with the homogenization of second-order elliptic systems with stratified structure on bounded Lipschitz domains, i.e., { −div(A(x, ρ(x)/ε)∇u ) = f in Ω, ε (0.1) uε = g on ∂Ω, where ε > 0, Ω ⊂ Rd, the matrix A(x, y) defined on Ω × Rn (n ≤ d) is bounded, elliptic and 1-periodic in y, ρ ∈ C1(Ω; Rn) is nondegenerate. A brief review on the homogenization theory with periodic structure will be presented first. Under rather general smoothness assumptions on A and ρ, the sharp-order scale-invariant convergence rate in L2d/(d−1)(Ω) for system (0.1) is established. We also discuss about the measurability of coefficient A(x, ρ(x)/ε) and the qualitative homogenization result under a weak condition of vector-valued Banach space type on A.

Some regularity results in Musielak-Orlicz-Sobolev spaces

Peihao Zhao

Lanzhou University

Email: [email protected]

In this talk, I will present some regularity results in our latest research on the regularity results in Musielak-Orlicz-Sobolev (MOS) spaces, including: 1. Local bounded and Cα es- timates for the local minimizers of some energy functional and some fully nonlinear elliptic equations in MOS spaces; 2. LΦ(A(x,·)) estimate for the gradient in W 1,A(x,·) for the local minimizers of some energy functional.

19 The List of Participants

20 Participants from Abroad

Name Position Affiliation Email 1 Dongho Chae Chung-Ang University [email protected] 2 Boualem Professor University of Victoria [email protected] Khouider 3 Chanh Kieu Assistant Pro- Indiana University [email protected] fessor Bloomington 4 Anna L. Mazzu- Professor Penn State University [email protected] cato 5 Taylan Sengul Associate Pro- Marmara University [email protected] fessor 6 Shouhong Wang Professor Indiana University [email protected] 7 Xiaoming Wang Professor Southern University of [email protected] Sciences and Technol- ogy

Participants from Hong Kong, Macao and Taiwan

Name Position Affiliation Email 1 Chun-Hsiang Professor National Taiwan Uni- [email protected] Hsia versity

Participants from Mainland China

Name Position Affiliation Email 1 Tianqing An (安 Professor Hohai University [email protected] 天庆) 2 Xuefeng Cai (蔡 Master Can- Southwest Jiaotong [email protected] 学峰) didate University 3 Xueyong Chen Lecturer Xuchang University [email protected] (陈学勇) 4 Dong Deng (邓 Ph.D Candi- Sichuan University [email protected] 栋) date 5 Guowei Dai (代国 Professor Dalian University of [email protected] 伟) Technology

21 Name Position Affiliation Email 6 Mingshu Fan (樊 Associate Southwest Jiaotong [email protected] 明书) Professor University 7 Yanlong Fan (范 Master Can- Sichuan University [email protected] 炎龙) didate 8 Zhihui Fan (樊智 Ph.D Candi- Sichuan University [email protected] 辉) date 9 Shuai Gao (高帅) Master Can- Southwest Jiaotong [email protected] didate University 10 Bangsheng Han Lecturer Southwest Jiaotong hanbangsheng@home. (韩帮胜) University swjtu.edu.cn 11 Daozhi Han (韩道 Assistant Missouri University of [email protected] 志) Professor Science and Technology 12 Jianxun Hu (胡建 Professor Sun Yat-Sen University [email protected] 勋) 13 Zhibo Hou (侯智 Lecturer [email protected] 博) 14 Lan Jia (贾澜) Ph.D Candi- Sichuan University [email protected] date 15 Qian Jiang (姜倩) Master Can- Sichuan Normal Univer- [email protected] didate sity 16 Song Jiang (江松) Professor Institute of Applied [email protected] Physics and Compu- tational Mathematics, Beijing 17 Gang Li (李刚) Lecturer Jincheng College [email protected] 18 Hong Luo (罗宏) Professor Sichuan Normal Univer- [email protected] sity 19 Honghu Liu (刘鸿 Assistant Virginia Tech [email protected] 鹄) Professor 20 Jiao Luo (罗娇) Master Can- Sichuan Normal Univer- [email protected] didate sity 21 Junyan Li (李 军 Lecturer Jincheng College [email protected] 燕) 22 Liang Li (李良) Master Can- Sichuan University [email protected] didate 23 Limei Li (李俐玫） Associate Sichuan Normal Univer- [email protected] Professor sity 24 Mei Liu (刘梅) Master Can- Sichuan Normal Univer- [email protected] didate sity 25 Ruikuan Liu (刘 Lecturer Southwest Petroleum [email protected] 瑞宽) University 26 Ruyun Ma (马如 Professor Xidian University [email protected] 云) 27 Tian Ma （马天） Professor Sichuan University [email protected]

22 Name Position Affiliation Email 28 Chen Peng (彭晨) Lecturer Jincheng College [email protected] 29 Jiaojiao Pan (潘 Master Can- Sichuan Normal Univer- [email protected] 娇娇) didate sity 30 Zhe Pu (蒲哲) Master Can- Sichuan Normal Univer- [email protected] didate sity 31 Zhigang Pan (潘 Lecturer Southwest Jiaotong [email protected] 志刚) University 32 Qian Qi (漆前) master Can- Sichuan Normal Univer- [email protected] didate sity 33 Tinwei Ruan (阮 Master Can- Sichuan Normal Univer- [email protected] 庭伟) didate sity 34 Chunyou Sun (孙 Professor Lanzhou University [email protected] 春友) 35 Jianwen Sun (孙 Associate Lanzhou University [email protected] 建文) Professor 36 Siyuan Shang (商 Master Can- Southwest Jiaotong [email protected] 嗣源) didate University 37 Fosheng Wang Lecturer Mianyang Teachers’ [email protected] (王佛生) College 38 Huichao Wang Lecturer Xuchang University [email protected] (王会超) 39 Quan Wang (王 Lecturer Sichuan University [email protected] 泉) 40 Ruili Wu (武 瑞 Ph.D Candi- Sichuan University [email protected] 丽) date 41 Xinxing Wu (吴新 Researcher Southwest Petroleum [email protected] 星) University 42 Yongfu Wang (王 Lecturer Southwestern Univer- [email protected] 永富) sity Of Finance And Economics 43 Yilong Wang (王 Associate Southwest Petroleum [email protected] 义龙) Professor University 44 Chao Xing (邢超) Ph.D Candi- Sichuan University [email protected] date 45 Yao Xu (徐侥) Ph.D Candi- Nanjing University [email protected] date 46 Bo You (尤波) Associate Xi’an Jiaotong Univer- [email protected] Professor sity 47 Dongming Yan Lecturer University of [email protected] (闫东明) Finance and Economics 48 Guoju Ye (叶 国 Professor Hohai University [email protected] 菊) 49 Huan Yuan (袁 Master Can- Civil Aviation Flight [email protected] 桓) didate University of China

23 Name Position Affiliation Email 50 Jiayan Yang (杨 Lecturer Southwest Medical Uni- [email protected] 佳艳) versity 51 Mengling Yu (余 Master Can- Sichuan Normal Univer- [email protected] 梦玲) didate sity 52 Tao Yang (杨韬) Master Can- Sichuan Normal Univer- [email protected] didate sity 53 Dun Zhao (赵敦) Professor Lanzhou University [email protected] 54 Dongpei Zhang Ph.D Candi- Sichuan University [email protected] (张东培) date 55 Peihao Zhao（赵 Professor Lanzhou University [email protected] 培浩） 56 Qiang Zhang (张 Associate Civil Aviation Flight [email protected] 强) Professor University of China 57 Shengui Zhang Associate Northwest Minzu Uni- [email protected] (张申贵) Professor versity 58 Junyilang Zhao Lecturer Sichuan University [email protected] (赵君一朗) 59 Yutong Zhao (赵 Master Can- Southwest Jiaotong [email protected] 钰彤) didate University 60 Xiaolan Zhou (周 Master Can- Southwest Jiaotong [email protected] 小兰) didate University

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