where α > 1 2 and p > 1. Firstly I review some related existence results on local

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1 Titles and Abstracts 1. Dongfen Bian, Beijing Institute of Technology, China Title: Some results about Boussinesq-MHD equations Abstract: This talk is concerned with initial and initial boundary value problems for two-dimensional Boussinesq equations for MHD convection. For Cauchy problem, the global well-posedness of this system is obtained without any smallness assumption for variation of initial data. For initial boundary value problem, we show that the system has a unique classical solution for H^3 initial data, and the non-slip boundary condition for velocity field and the perfectly conducting wall condition for magnetic field. In addition, we show that the kinetic energy is uniformly bounded in time. 2. Haiqiang Chen, Xiamen University, China Title: A new estimator for integrated volatility with microstructure noise and jumps Abstract: This paper develops an improved estimator for integrated volatility of an Ito semi-martingale in the presence of jumps and market microstructure noise. The estimator is based on the joint use of pre-averaging multi-power variation estimation and threshold technique, severing to remove the impact from microstructure noise and jumps respectively. Asymptotic properties of the proposed estimator, such as consistency and associated central limit theorems are provided. Monte Carlo simulations show that the estimator is robust to both Le vy jumps and microstructure noise and provides less biased estimate, compared to the existing estimators in the literature. As an application, we study the problem of volatility forecasting in the presence of market microstructure noise and jumps. 3. Hua Chen, Wuhan University, China Title: Multiple solutions for semi-linear corner degenerate elliptic equations with singular potential term Abstract: In this talk, we establish the correspinding corner type Sobolev inequality, Poincare inequality and Hardy inequality, and then we study the Dirichlet problem of semi-linear corner-degenerate elliptic equations with singular potentials. We obtain the result for existence of infinitely many solutions and the result for existence of infinitely many sign changing solutions in the subcritical case. 4. Jianqing Chen, Fujian Normal University, China Title: On a class of Schro dinger equation with second order derivative on the nonlinear term Abstract: In this talk, I will discuss some results on the following Schro dinger equation iφ t + φ + 2α( φ 2α ) φ 2α 2 φ + φ p 1 φ = 0, xεr N where α > 1 2 and p > 1. Firstly I review some related existence results on local

2 solutions of Cauchy problem. Secondly, with the help of local solution, we discuss the orbital stability and instability of standing waves. Thirdly we derive an existence of blow-up solution due to the effect of second order derivative on the nonlinear term. 5. Wengu Chen, Institute of Applied Physics and Computational Mathematics, China Title: Bourgain space X s,b with applications to the KdV type equations Abstract: This talk will review the local and global well-posedness of the Cauchy problem for the KdV type equations by using harmonic analysis technique in the framework of Bourgain space including multilinear estimate, multilinear multiplier and I-method. 6. Shijin Ding, South China Normal University, China Title: Global Existence and Time-decay Estimates of Solutions to the Compressible Navier-Stokes-Smoluchowski Equationss Abstract: This talk is concerned with the Cauchy problem of the compressible Navier-Stokes-Smoluchowski equations in R 3. Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces, the existence theory of global solutions in H 3 to the stationary profile is established. Moreover, when the initial perturbation is bounded in L p -norm with 1 p < 6, we obtain the optimal convergence rates of 5 the solution in L q -norm with 2 p 6, and its first order derivative in L 2 -norm. 7. Lili Du, Sichuan University, China Title: Two-dimensional Impinging Jets in Hydrodynamic Rotational Flows Abstract: In this talk, we will discuss the well-posedness of the impinging jets in steady incompressible, rotational, plane flows. More precisely, given a mass flux and a vorticity of the incoming flows in the inlet of the nozzle, there exists a unique smooth impinging plane jet. Moreover, there exists a smooth free streamline, which goes to infinity and initiates at the endpoint of the nozzle smoothly. In addition, asymptotic behavior in upstream and downstream, uniform direction and other properties of the impinging jet are also obtained. 8. Hongjun Gao, Nanjing Normal University, China Title: Steady Periodic Rotational Gravity Waves with Negative Surface Tension Abstract: Consideration in this paper is the two-dimensional steady periodic rotational gravity waves with negative surface tension. Local curves of small amplitude solutions of the resulting problem are obtained by using the Crandall-Rabinowitz local bifurcation theory. By means of the global bifurcation theory combined with the Schauder theory of elliptic equations with the Venttsel boundary conditions, the curves of small amplitude solutions is extended to the

3 global continuum of solutions. Furthermore, it is shown that those waves are necessarily symmetric about the crest under the assumption that their surface profiles are monotonic between troughs and crests and locally strictly monotonic near the troughs. 9. Zhenhua Guo, Northwest University, Xia an, China Title: On Free Boundary Value Problem for Compressible Navier-Stokes Equations with Temperature dependent Heat Conductivity Abstract: We prove existence of global strong solution to free boundary value problem in one-dimensional compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when heat conductivity depends on temperature in power law of Chapman-Enskog. In addition, the free boundary is shown to expand outward at an algebraic rate in time. 10. Yongqian Han, Institute of Applied Physics and Computational Mathematics, China Title: Diffusion Limit of Radiative Transfer Equation Coupled with Energy Equation Abstract: The radiative transfer equation coupled with the energy equation, which describes the spatial transport of radiation in a material medium, is considered. In this talk, we introduce some results on the well-posedness of solutions and diffusion limit of small mean free path of the nonlinear transfer equations. By using comparison principle, we obtain the lower bound and upper bound of the solution, and then we prove the existence and uniqueness of the global solution. We show that the nonlinear transfer equation has a diffusion limit as the mean free path tends to zero. Our proof is based on asymptotic expansions. 11. Daiwen Huang, Institute of Applied Physics and Computational Mathematics, China Title: On 3D Sstochastic Primitive Equations of the Large-Scale Ocean Abstract: In this talk, we give some results on three-dimensional (3D) stochastic primitive equations of the large-scale ocean. Firstly, we recall the global well-posedness and long-time dynamics for the 3D viscous primitive equations describing the large-scale oceanic motion with white noise. Secondly, we introduce some results on the viscous primitive equations describing the large-scale oceanic motions under fast oscillating random perturbation, such as, the solution to the initial boundary value problem (IBVP) of the 3D stochastic primitive equations converging in distribution to that of IBVP of the limit equation, 3D primitive equations with white noise. 12. Zhaohui Huo, University of Chinese Academy of Sciences Title: Well-posedness for the Zakharov-Kuznetsov Equation Abstract: The Cauchy problem of the Zakharov-Kuznetsov equation

4 u t + x,1 u = 0, (x, t) R n R, x = (x,1, x,2,, x,n ); is considered. It is shown that it is globally well-posed in H 1 in three dimension, and is locally well-posed in H 1/2 in two dimension. It answers two open problems: Is it globally well-posed in energy space H 1 (R 3 )for 3D Z-K equtation? Is it locally well-posed in endpoint space H 1/2 (R 2 ) for 2D Z-K equtation? 13. Huaiyu Jian, Tsinghua University, China Title: A Few Results of Monge-Ampere Equation Abstract: This talk is based on the co-works with Xu-Jia Wang at Australian National University. The existence and uniqueness of the Monge-Ampere from Gauss curvature ow and Lp-Minkowski problem will be studied. Then we prove the optimal regularity and establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary, including fully nonlinear elliptic equations and equations of Monge-Ampere type. 14. Jie Jiang, Chinese Academy of Sciences, China Title: Well-posedness and Long-time Behavior of a Non-autonomous Cahn-Hilliard-Darcy System with Mass Source Modeling Tumor Growth Abstract: In this talk, we study an initial boundary value problem of the Cahn-Hilliard-Darcy system with a non-autonomous mass source term S that models tumor growth. We first prove the existence of global weak solutions as well as the existence of unique local strong solutions in both 2D and 3D. Then we present qualitative behavior of solutions in details when the spatial dimension is two. Finally, when S is assumed to be asymptotically autonomous, we demonstrate that any global weak/strong solution converges to a single steady state as time goes to infinity. 15. Quansen Jiu, Capital Normal University, China Title: On Global Regularity of 3D Axisymmetric Incompressible Navier-Stokes Equations Abstract: In this talk, under suitable scaled invariant conditions on the solutions, in particular on the swirl component of the velocity u θ, we will show a Liouville type theorem and global regularity of the three-dimensional axisymmetric incompressible Navier-Stokes equations. 16. Zhen Lei, Fudan University, China Title: Local Exact Boundary Controllability of Nonlinear Wave Systems Abstract: In this talk I will report the local exact boundary controllability of nonlinear wave systems. The nonlinearity under consideration is quite general and of the form F(t, x, u, u, 2 u). The boundary conditions can be of Dirichlet type, or Neumann type, or the third type. It is shown that starting from any given

5 small initial state (displacement and velocity), and any given final state, there always exist a time T > 0 and boundary control such that the solution of the wave systems with the initial data and the boundary control exactly reaches the final states at time T. It is a joint work with Professor Yi Zhou and was obtained nearly years ago. 17. Hailiang Li, Capital Normal University, Beijing Title: Recent progress on some Kinetic eequations Abstract: In this talk, we review recent progress on the analysis of spectrum structure of some kinetic equations and the wave phenomena for bipolar Vlasov-Poisson-Boltzmann equations. 18. Yongsheng Li, South China University of Technology, China Title: Blow-up Properties of Some Nonlinear Evolution Equations Abstract: In this talk the Cauchy problems of the 2D and 3D Davey-Stewartson equations of elliptic-elliptic type are discussed. The main issues are two-fold. One is the exact profiles of the blow-up solutions to critical 2D DS with critical mass. Another is the conditions for global and blow-up solutions to subcritical 3D DS. 19. Zhiwu Lin, Georgia Instiyute of Technology, USA Title: Metastability of Kolmogorov flows and instability of nonrotating stars Abstract: This talk is divided into two parts. First, we discuss the meta-stability of the shear flow (sin y, 0) (Kolmogorov flow) on a torus. This flow is nonlinearly stable for the inviscid case (Euler equation). When the viscosity is sufficiently small, it is shown that the perturbations can decay at a much faster rate than the time scale from fluid dissipation, for an intermediate but long time period. This is joint work with Ming Xu at Jinan University of Guangzhou. Second, we verified the turning point principle for instability of nonrotating stars modeled by 3D Euler-Poisson system with a general equation of states. This is joint work with Chongchun Zeng at Georgia Tech. In both cases, the Hamiltonian structures of the fluid equations are strongly used in the proof. 20. Yue Liu, University of Texas at Arlington, USA Title: Rotation-two-component Camassa-Holm system modelling the equatorial water waves Abstract: In this talk, a modified two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid is derived, which is a model in the equatorial water waves. The effects of the Coriolis force caused by the Earth's rotation and nonlocal nonlinearities on blow-up criteria and wave-breaking phenomena are then investigated. Our refined analysis relies on the method of characteristics and conserved quantities and is proceeded with the Riccati-type differential inequality. Finally, conditions which guarantee the permanent waves are obtained by using a method of the Lyapunov function.

6 21. Kening Lu, Brigham Young University, USA Title: Lyapunov Exponents and Chaotic Behavior for Random Dynamical Systems in a Banach Space Abstract: We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. We also prove that for an infinite dimensional random dynamical system with a random invariant set such as random attractor, if it's topological entropy is positive, then the dynamics on the random invariant set is chaotic. This is based on joint works with Wen Huang and Zeng Lian. 22. Changzheng Qu, Ningbo University, China Title: Liouville correspondence between the modified KdV hierarchy and its dual integrable hierarchy Abstract: In this talk, the correspondence between the integrable modified KdV hierarchy and its dual integrable modified Camassa-Holm hierarchy is established. A Liouville transformation between the isospectral problems of the two hierarchies also relates their respective recurion operators, and serves to establish the Liouville correspondence between their flows and Hamiltonian conservation laws. In addition, a novel transformation mapping the modified Camassa-Holm equation to the Camassa-Holm equation is found. Other properties to the modified Camassa-Holm equations are also discussed. This is a joint work with J. Kang, X.C. Liu and P. Olver. 23. Chunyouo Sun, Lanzhou University, China Title: Dynamics for a 2D Generalized Incompressible Navier-Stokes Equations Abstract: In this talk, I will present our recent results about the zero viscosity limit of long time averages of solutions and the existence and _nite dimensionality of global attractor for a 2D damped generalized incompressible Navier-Stokes equations. This is a joint work with Professors Boling Guo, Daiwen Huang and Qiaoxin Li. 24. Shu Wang, Beijing University of Technology, China Title: On an axisymmetric model for the 3D incompressible Euler and Navier-Stokes equations Abstract: We study the singularity formation and global regularity of an axisymmetric model for the 3D incompressible Euler and Navier-Stokes equations. This 3D model is derived from the axisymmetric Navier-Stokes equations with swirl using a set of new variables. The model preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. We prove rigorously

7 that the 3D model develops finite time singularities for a large class of initial data with finite energy and appropriate boundary conditions. Moreover, we also prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. The related problems are surveyed and some recent results will also be reviewed. 25. Chunjing Xie, Shanghai Jiao Tong University, China Title: Well/ill-posednesss for the rotating shallow water System Abstract: When the initial relative vorticity is zero, we show that rotating shallow water system in both 1D and 2D cases has a global small smooth solution via exploring the dispersive mechanism of the system. For the smooth solutions with general initial data, in general, they will form a singularity in finite time at least in one dimensional setting. For the general solutions for 2D rotating shallow water system, we prove that there are infinitely many bounded admissible weak solutions. In particular, we can construct infinitely many bounded admissible weak solutions with only finite states. 26. Feng Xie, Shanghai Jiao Tong University, China Title: On several hierarchical radiation hydrodynamics models Abstract: In this talk, we will discuss several related radiation hydrodynamics models, and introduce some known results about these models. Which include stability of elementary waves, global existence of solutions, blowup phenomena and singular limit process. 27. Wen-An Yong, Tsinghua University, China Title: Validity of the Maxwell iteration for a class of hyperbolic relaxation systems Abstract: This paper presents a validity proof of the Maxwell iteration for a class of hyperbolic relaxation systems. This class is characterized with a set of conservation-dissipation structural conditions and contains many classical models from mathematical physics. Our main result confirms that the hyperbolic relaxation systems can be well approximated, in the time interval where the corresponding conservation laws have smooth solutions, with the corresponding viscous systems derived by using the Maxwell iteration. The talk is based on a joint work with Zaibao Yang. 28. Hongjun Yu, South China Normal University, China Title: Global Solution to the Relativistic Boltzmann Equation with Soft Potentials Abstract: In this work we prove that the exstence, the exponential time decay and propagation of spatial regularity of global solution to the relativistic Boltzmann with soft potentials around the relativistic Maxwellian. We employ the nonlinear L 2 energy methods and nonlinear L pointwise estimate methods. 29. Baoquan Yuan, Henan Polytechnic University, China

8 Title: Local Existence of Strong Solutions to the $k-\varepsilon$ Model Equations for Turbulent Flows Abstract: In this talk, We address the local existence of strong solutions to the $k-\varepsilon$ model equations for turbulent flows in a bounded domain $\Omega$$\subset$ $\mathbb{r}^{3}$. I will review the known results and then prove the existence of unique local strong solutions for all initial data which are bounded away from zero. 30. Chongchun Zeng, Georgia Institute of Technology, USA Title: Instability and exponential dichotomy of Hamiltonian PDEs Abstract: In this talk, we start with a general linear Hamiltonian system ut = JLu in a Hilbert space X - the energy space. The main assumption is that the energy functional 1 2 < Lu,u > has only finitely many negative dimensions n (L) <. Our first result is an index theorem related to the linear instability of e tjl, which gives some relationship between n (L) and the dimensions of spaces of generalized eigenvectors of eigenvalues of JL. Under some additional non-degeneracy assumption, for each eigenvalue λ ir of JL we also construct special good" choice of generalized eigenvectors which both realize the corresponding Jordan canonical form corresponding to λ and work well with L. Our second result is the linear exponential trichotomy of the groupe tjl. This includes the nonexistence of exponential growth in the finite co-dimensional invariant center subspace and the optimal bounds on the algebraic growth rate there. Thirdly we consider the structural stability of this type of systems under perturbations. Finally we discuss applications to examples of nonlinear Hamiltonian PDEs such as BBM, GP, and 2-D Euler equations, including the construction of some local invariant manifolds near some coherent states (standing wave, steady state, traveling waves etc.). This is a joint work with Zhiwu Lin. 31. Jingjun Zhang, Jiaxing University, China Title: Absence of shocks for 1d Euler-Poisson system Abstract: We consider the global existence of the Euler-Poisson system for electrons in one dimensional case. It is shown that smooth solutions with small amplitude to this system persist forever with no shock formation. 32. Linghai Zhang, Lehigh University, USA Title: Global Smooth Solution of A Two-Dimensional Nonlinear Singular System of Differential Equations Arising from Geostrophics Abstract: Consider the Cauchy problems for the following two-dimensional nonlinear singular system of differential equations arising from geostrophics t [γ(ψ 1 ψ 2 ) Δψ 1 ] + α( Δ) ρ ψ 1 + β ψ 1 x + J(ψ 1, γ(ψ 1 ψ 2 ) Δψ 1 ) = 0

9 t [γδ (ψ 2 ψ 1 ) Δψ 2 ] + α( Δ) ρ ψ 2 + β ψ 2 x + J(ψ 2, γ δ (ψ 2 ψ 1 ) Δψ 2 ) = 0 ψ 1 (x, y, 0) = ψ 01 (xy), ψ 2 (x, y, 0) = ψ 02 (x, y). In this system of differential equations, α > 0, γ > 0, δ > 0 and ρ > 0 are positive constants, β 0 is a real nonzero constant, the Jacobian determinant is defined by J(p, q) = p q p q. x y y x It is worth of mentioning that the existence, uniqueness and the decay estimates with sharp rates of the global smooth solution of the nonlinear system have been open for a long time. The singularity generated by the linear parts, the fractional order of the derivatives and the strong couplings of the nonlinear functions make the existence, uniqueness and the decay estimates very difficult to analyze. There exist two special structures in the nonlinear system which have not been discovered before: (I) the terms γ(ψ 1 ψ 2 ) Δψ 1 and γ δ (ψ 2 ψ 1 ) Δψ 2 appear in both the linear part and the nonlinear part. (II) The good terms (namely, the linear terms) and the bad terms (namely, the nonlinear terms) are separated. In particular, the derivatives of fractional order are separated from the derivatives of the third order in the nonlinear functions. The first special structure will play a key role when establishing uniform energy estimates for lower order derivatives. The second special structure will play a very important role when establishing uniform energy estimates for higher order derivatives. By completely making use of the special structures of the nonlinear system and by coupling together many inequalities (including Gagliardo-Nirenberg s interpolation inequalities, Cauchy-Schwartz inequality, Ho lder s inequality), we are able to establish the uniform energy estimates with respect to time for both lower order derivatives and higher order derivatives. By means of Leray-Schauder s fixed point principle and the uniform energy estimates, we are to establish the existence and uniqueness of the global strong smooth solution. Furthermore, by making of use the uniform energy estimates for higher order derivatives, we may accomplish the existence of the global smooth solution of the Cauchy problems for the above nonlinear singular system of differential equations. This is a joint work with Boling Gu. 33. Zhifei Zhang, Peking University, China Title: Linear Inviscid Damping for a Class of Monotone Shear Flow Abstract: In this talk, I will talk about the linear decay estimates of the velocity for the 2-D incompressible Euler equations around a class of monotone shear flow in a finite channel. 34. Huijiang Zhao, Wuhan University, China Title: One-species Vlasov-Poisson-Landau System near Maxwellians Abstract: This talk is concerned with the construction of global smooth small-amplitude solutions to the one-species Vlasov-Poisson-Landau system near Maxwellians including the important Coulomb potentials. It is based on our

10 recent works joint with Dr.Yuanjie Lei, Dr. Linjie Xiong, and Dr. Lin Wang. 35. Zhouping Xin, The Chinese University of Hong Kong Title: Entropies And Uniqueness of Weak Solutions to Multi-Dimensional Compressible Euler Systems Abstract: For the ideal compressible Euler systems, which are fundamental in fluid-dynamics and pro-type examples nonlinear hyperbolic systems, one of the main features is that the characteristic speeds of a wave propagation depend on the wave itself which leads to the finite formation of shocks in general. Thus one has to work with weak solutions globally. Yet the uniqueness of the physical solutions becomes a challenging issue. In the one-dimensional case, various admissible criterion have been introduced to rule out the non-physical solutions. In particular, the physical entropy can guarantee the uniqueness of weak solutions at least in the case of weak solutions with small total variations. However, in higher space dimensions, for some given initial data, there are infinite many highly oscillatory solutions (wild solutions) which are bounded, measurable and satisfying the physical entropy condition. In this talk, I will review some progress on the constructions of such wild solutions by a method of convex integration; present some results on the structure of such wild solutions ; and investigate the effects of lower order dispersion or dissipations. Some open problems will be discussed too. 36. Yi Zhou, Fudan University, China Title: Almost Optimal Regularity Local Existence for Radially Symmetrical Minimal Surface Equation in 1+3 Dimensional Minkowski Space Abstract: We consider the Cauchy problem for the radially symmetrical minimal surface equation in (1+3)-dimensional Minkowski space, which falls into the class of (1+2)-dimensional quasilinear wave equation. It is known from the general result of Smith and Tataru that the problem is locally well posed in H s H s 1 for s > s c + 3 4, where s c = 2 is the scaling limit of the equation. It is recently announced by Ettinger that this can be improved to s s c + 1. In this work, we 2 show that in the radial symmetric case, this can be further improved to s > s c. 37. Yi Zhu, Tsinghua University, China Title: Conservation-Dissipation Formalism of Non-equilibrium Thermodynamics Abstract: In this talk, we propose the conservation-dissipation formalism (CDF) for coarse-grained descriptions of irreversible processes. Under the local equilibrium hypothesis, the classical non-equilibrium thermodynamics usually formulates the evolution of the non-equilibrium states in a hyperbolic-parabolic form, like the Navier-Stokes equation, which is not adequate in many situations. The CDF formulates the evolution of the states in a system of first order differential equations like many other modern theories. However, with a feasible procedure in choosing non-equilibrium state variables, the equations derived in

11 CDF have a unified elegant form. They are globally hyperbolic and thereby well-posed for initial-value problems. The non-equilibrium stability, which ensures the non-equilibrium states tend to equilibrium in long time, and the compatibility with the classical theories in short relaxation time regime can be also rigorously justified. With this formalism, we develop the generalized hydrodynamics which extends the classical Navier-Stokes equation. Thanks to the elegant form of the evolution equations, we rigorously prove that the generalized hydrodynamics is compatible with the classical compressible Navier-Stokes equation in the short relaxation time limit. This talk is based on the joint work with W.-A. Yong, L. Hong and Z. Yang at Tsinghua University.