Chengchun Hao

Abstract: I will present a recent work on combining the relative entropy method by Jabin and Wang and the regularized $L^2$-estimate by Oeschläger to give a rigorous derivation of viscous porous medium equation from moderately interacting particle system in the sense of strong $L^1$ norm. The key step is to obtain a well-posedness result for the corresponding nonlocal PDE. This is a joint work with A. Holzinger and X. Huo.

Abstract: We study the global existence of classical solutions to the incompressible viscous MHD system without magnetic diffusion in 2D and 3D. The lack of resistivity or magnetic diffusion poses a major challenge to a global regularity theory even for small smooth initial data. However, the interesting nonlinear structure of the system not only leads to some significant challenges, but some interesting stabilization properties, that leads to the possibility of the theory of global existence of classical and/or strong solutions. This talk is based on joint works with Yi Zhou, Yi Zhu, Shijin Ding, Xiaoying Zeng, and Jingchi Huang.

Abstract: In this talk, I will discuss a continuation criterion for the free boundary problem of three-dimensional incompressible ideal MHD equations in a bounded domain, which is analogous to the theorem given by Beale, Kato and Majda in 1984. We combine the energy estimates on incompressible ideal MHD given by Hao and Luo in 2014 and some analogous estimates by Ginsberg in 2021 to show that the solution can be continued as long as the curls of the magnetic field and velocity, the second fundamental form and injectivity radius of the free boundary and some norms of the pressure remain bounded, provided that the Taylor-type sign condition holds. This talk is based on the joint work with J. Fu, C. Hao and S. Yang.

Abstract: In this talk, I will discuss the dynamical stability of white dwarfs beneath the Chandraskhar limit: 1) for the orbit stability for inviscid theory (based on the joint with J. Smoller) from the energy minimizing point of view for both rotating and non-rotating star steady solutions, 2) and the nonlinear asymptotic stability of non-rotating solutions for the viscous theory in the framework of vacuum free boundary problem with symmetry based on the recent joint work with Y. Wang and H. Zeng, motivated by the previous work of Luo-Xin-Zeng for the polytropes for $4/3<\gamma<2$.

Abstract: In this talk, we consider the incompressible Navier-Stokes equations with free boundary, which has a finite-time splash singularity for a large class of specially prepared initial data. The approach is founded upon the Lagrangian description of the free-boundary problem, and a sequence of domains $\Omega^\varepsilon$ used as the initial data for the splash singularity are defined, wherein the boundary $\Gamma^\varepsilon$ of these domains is close to self-intersection between two approaching portions of $\Gamma^\varepsilon$. We will present some preliminary lemmas which show that the constant appearing in elliptic estimates and the Sobolev embedding theorem is independent of $\varepsilon$ and define the sequence of initial divergence-free velocity fields that are guaranteed to satisfy the required single compatibility condition, and whose norm is independent of $\varepsilon$.

Abstract: In this talk, time-dependent free surface problem for the incompressible Navier-Stokes equations which describes the motion of viscous incompressible fluid nearly half-space are considered. We will discuss how to get global well-posedness of the problem for small initial data in scale invariant critical Besov spaces by maximal $L^1$-regularity of the corresponding Stokes problem in the half-space.In this talk， we will introduce maximal $L^1$-regularity for the Stokes flow in the half-space for 0-boundary data, then we can construct the solution formula of the Stokes equation and the Littlewood-Paley decomposition with separation of variables.

Abstract: In this talk, we will discuss the local well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another fluid separated by a closed interface with surface tension. First, we use the Hanzawa transformation to obtain the linearized Oberbeck-Boussinesq approximation in the fixed domain. Second, we prove the existence of $\mathcal{R}$-bounded solution operators for the model problems and the maximal $L^p-L^q$ regularity of the linearized Oberbeck-Boussinesq approximation. The key step is to prove the maximal $L^p-L^q$ regularity theorem for the linearized heat equation with the help of the $\mathcal{R}$-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem. Finally, we estimate the difference in nonlinear terms, after which the existence and uniqueness of the solutions are proven by the contraction mapping principle. The talk is based on joint work with Hao Chengchun.

Abstract: In this talk, we will discuss the resolvent problem for the linearized one of the free surface problem of compressible, viscous and heat-conducting fluid, which is a problem in an infinite strip domain equipped with mixed boundary conditions. With the values prescribed by the given boundary conditions, we derive an explicit solution formula for this boundary value problem, and the unknown boundary values of solutions are also well specified. Then we establish the $L_p$, $1<p<\infty$, estimates for the unique solutions to the resolvent problem. The talk is based on the joint work with Yongting Huang.

Abstract: In this talk, the combined quasi-neutral and zero-viscosity limits of the two-fluid Navier-Stokes-Poisson systems (one for ion and another for electron) with boundary are rigorously proved by investigating the existence and the stability of the boundary layers. The non-penetration boundary condition for velocities and Dirichlet boundary condition for electric potential are considered. Based on conormal energy estimate, we showed that the solutions for the system then converge towards the solutions of the one-fluid compressible Euler system.

Abstract: For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, the existence of steady transonic shock solutions and $C^\infty$-smooth steady transonic solutions for Euler-Poisson Equations were established in Li-Mei-Zhang-Zhang SIMA2018 and Wei-Mei-Zhang-Zhang SIMA2021, respectively. In this talk, we further study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. When the $C^1$-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the proof of structural stability. For any relaxation time: $0<\tau\le +\infty$, by means of elaborate singularity analysis, we first investigate the structural stability of the $C^1$-smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. Moreover, when the relaxation time is large enough $\tau\gg 1$, under the condition that the electric field is positive at the shock location, we prove that the transonic shock steady-states are structurally stable with respect to small perturbations of the supersonic doping profile. Furthermore, we show the linearly dynamic instability for these transonic shock steady-states provided that the electric field is suitable negative. The proofs for the structural stability results are based on singularity analysis, a monotonicity argument on the shock position and the downstream density, and the stability analysis of supersonic and subsonic solutions. The linear dynamic instability of the steady transonic shock for Euler-Poisson equations can be transformed to the ill-posedness of a free boundary problem for the Klein-Gordon equation. By using a nontrivial transformation and the shooting method, we prove that the linearized problem has a transonic shock solution with exponential growths. These results enrich and develop the existing studies.

Abstract: This talk study the long wave asymptotic behavior of Green-Naghdi equations, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with the Coriolis effect due to Earth’s rotation. This model equation is called the rotation-Green-Naghdi (R-GN) equations modeling the propagation of wave allowing large amplitude in shallow water. 1. We demonstrate the lift span of the solution to the R-GN model equations in a Sobolev space by the refined energy estimates. 2.We provide a rigorous justification from the solutions of the R-GN equations to the associated solution of the right-left R-BBM or KdV equation in the KdV regime with the small amplitude and the large wavelength. This is the first result on the issue that the solution of R-GN equations are well approximated by the bi-directional R-BBM in the general regular initial data.It is a jointed work with Prof. Yue Liu.

Abstract: We first show the nonequilibrium-diffusion limit of the compressible Euler-P1 approximation model arising in radiation hydrodynamics as the Mach number tends to zero when the initial data is well-prepared. In particular, the effect of the large temperature variation upon the limit is taken into account. The model leads to a singular problem which fails to fall into the category of the classical theory of singular limits for quasilinear hyperbolic equations. By introducing an appropriate normed space of solutions and exploiting the structure of the system, we establish the uniform local existence of smooth solutions and the convergence of the model to the incompressible nonhomogeneous Euler system coupled with a diffusion equation. Moreover, we also prove the nonequilibrium-diffusion limit of the compressible Euler-P1 approximation model when the Mach number is fixed.

Abstract: We are concerned with stability and instability of the steady state $(1,0, 1, 0)$ for a generic non–conservative compressible two–fluid model in $R^3$. Under the assumption that the initial fraction densities are close to the constant state $(1,1)$ in $H^3\cap\dot B^{s}_{1,\infty}$ and the initial velocities are small in $H^2\cap\dot B^{s}_{1,\infty}$ with $s\in [0,1]$, it is shown that $\frac{1}{2}$ is the critical value of s on the stability of the model in question. More precisely, when $0\leq s<\frac{1}{2}$, the steady state $(1,0, 1, 0)$ is nonlinearly globally stable; and conversely, the steady state $(1,0, 1, 0)$ is nonlinearly unstable in the sense of Hadamard when $\frac{1}{2}<s\le 1$. Furthermore, for the critical case $s=\frac{1}{2}$, if the initial data satisfy additional regularity assumption, then the steady state $(1,0, 1, 0)$ is nonlinearly globally stable.

Abstract: We consider the Cauchy problem for the semi-linear heat, nonlocal Schrödinger equations in super-critical spaces E^s for which the norms are defined by $$\|f\|_{E^s} = \|2^{s|\xi|}\widehat{f}(\xi)\|_{L^2}, s<0.$$ If $s<0$, then any Sobolev space $H^{r}$ is a subspace of $E^s$, i.e., $\cup_{r \in \mathbb{R}} H^r \subset E^s$. We will obtain the global existence and uniqueness of the solutions if the initial data belong to $E^s$ and their Fourier transforms are supported in the first octant and away from the origin, the smallness conditions on the initial data in $E^s$ are not required for the global solutions. This is a joint work with Dr. Jie Chen.

Abstract: We obtain the convergence rate estimates for singular limits of the compressible ideal magneto-hydrodynamic equations when the Mach and Alfven numbers tend to zero at different rates. Since the uniform boundedness of the first time derivative of solutions isn’t propagated for positive time in general for three-scale system, it is shown that the issue of convergence rates for three-scale singular limits is much more complicated than for the classical two-scale. This is a recent joint work with Bin Cheng from Surrey university and Steve Schochet from Tel-Aviv university.

Abstract: For the problem of full compressible Euler Equations with variable entropy coupled with a nonlinear Poisson equation in three spatial dimensions with a general free boundary not restricting to a graph, we identify a stability condition for the electric potential of which the outer normal derivative is positive on the free surface besides the Taylor sign condition for the pressure to obtain a priori estimates on the Sobolev norms of the fluid variables and bounds for geometric quantities of free surface. This talk is based on a joint work with K. Trivisa and H. Zeng.

Abstract: The Cauchy problem for $2$D Zakharov-Kuznetsov equation is shown to be global well-posed for the initial date in $H^{s}$ provided $s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below $L^2$, we construct an almost conserved quantity using multilinear correction term following the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In this paper, we use bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality to handle the resonant interactions.

Abstract: In this talk we present some results on the characterization of asymptotic profiles, as the time tends to infinity, of solutions to the chemotaxis models with logarithmic sensitivity on the half space. We show that if the density of bacteria is imposed by inward flux boundary condition, then the solution converges to a traveling front that is determined by the flux strength; if the bacteria satisfy no-flux boundary condition and the nutrient satisfies non-homogeneous Dirichlet boundary condition, then the solution converges with algebraic rate to a stationary spike. These two results, respectively, describe the phenomena of invasion of tumor issue and the aggregation of bacteria. The difficulties of the problems are the nonlocal structure of the equation and singularities caused by the vacuum end state of the profiles. The proofs are based on Cole-Hopf transformation, anti-derivative method and weighted energy estimates. We also present the intrinsic relations between this chemotaxis model and the compressible Navier-Stokes equations with density dependent viscosity.

Abstract: We consider the zero-viscosity limit problem for Navier-Stokes-Poisson equations in isothermal plasma with non-slip boundary. The compressible Prandtl boundary layer will appear near the boundary as the viscosity goes to zero. We will construct an approximate solution by Prandtl boundary layer expansion and investigate the linear stability of the approximate solution in Sobolev space under the monotonicity assumption.

Abstract: We develop a new way to study the global existence and uniqueness for the Navier-Stokes equation (NS) and consider the initial data in a class of modulation spaces $E^s_{p,q}$ with exponentially decaying weights $(s<0,1<p,q<\infty)$ for which the norms are defined by $$\|f\|_{E^s_{p,q}} = \left(\sum_{k\in \mathbb{Z}^d} 2^{s|k|q}\|\mathscr{F}^{-1} \chi_{k+[0,1]^d}\mathscr{F} f\|^q_p \right)^{1/q}.$$ The space $E^s_{p,q}$ is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding $H^{\sigma}\subset E^s_{2,1}$ for any $\sigma<0$ and $s<0$. It is known that $H^\sigma$ ($\sigma<d/2-1$) is a super-critical space of NS, it follows that $ E^s_{2,1}$ ($s<0$) is also super-critical for NS.We show that NS has a unique global mild solution if the initial data belong to $E^s_{2,1}$ ($s<0$) and their Fourier transforms are supported in $ \mathbb{R}^d_I:= \{\xi\in \mathbb{R}^d:  \xi_i \geq 0,  i=1,...,d\}$.   Similar results hold for the initial data in $E^s_{r,1}$ with $2< r \leq d$. Our results imply that NS has a unique global solution if the initial value $u_0$ is in $L^2$ with $\mathrm{supp } \hat{u}_0 \subset \mathbb{R}^d_I$. This is a joint work with Professors H. Feichtinger, K. Gr\"ochenig and Dr. Kuijie Li.

Abstract: The Vlasov-Poisson-Boltzmann equations can be used to model the transport of charged particles (one carrier or two carriers) under the influence of electrostatic potential force. It is not well understood yet how the electrostatic potential force and/or the mutual interaction between charged particles of different type shall affect the spectrum structures and the asymptotical behaviors of global solutions. In this talk, we present the recent progress on the analysis on the spectrum structure and optimal pointwise space-time behaviors of three dimentional Vlasov-Poisson-Boltzmann equations, and the nonlinear stability of planar wave pattern for the bipolar Vlasov-Poisson-Boltzmann equations, such as including shock profile, rarefaction wave and contact discontinuity. These works are joint with Tong Yang, Ming-Ying Zhong, Yi Wang, and Teng Wang.

Abstract: In this talk, I will discuss with the small porosity asymptotic behavior of the coupled Stokes-Brinkman system in the presence of a curved interface between the Stokes region and the Brinkman region. In particular, we derive a set of approximate solutions, validated via rigorous analysis, to the coupled Stokes-Brinkman system. Of particular interest is that the approximate solution satisfies a generalized Beavers-Joseph-Saffman-Jones interface condition (1.9)with the constant of proportionality independent of the curvature of the interface. It is a joint work with Mingwen Fei and Xiaoming Wang.

Abstract: The global-in-time existence of weak and renormalized solutions to reaction-cross-diffusion systems for an arbitrary number of variables in bounded domains with no-flux boundary conditions are proved. The cross-diffusion part describes the segregation of population species and is a generalization of the Shigesada-Kawasaki-Teramoto model. The diffusion matrix is not diagonal and generally neither symmetric nor positive semi-definite, but the system possesses a formal gradient-flow or entropy structure. The reaction part is of Lotka-Volterra type for weak solutions or includes reversible reactions of mass-action kinetics and does not obey any growth condition for renormalized solutions. Furthermore, we prove the uniqueness of bounded weak solutions to a special class of cross-diffusion systems, and the weak-strong uniqueness of renormalized solutions to the general reaction-cross-diffusion cases.

Abstract: 本报告介绍航空发动机压气机叶盘叶片振动力学中的一些数学模型，也介绍流固耦合模型在生物医学中的应用。首先基于振动弹性力学中已有的旋转薄壁梁、板或壳的叶片振动非线性动力学模型，结合航空发动机叶盘叶片装置几何结构和跨音速流场中的高转速实际应用特征提出航空发动机叶片振动非线性流固耦合动力学数学模型，并进行数学分析。也介绍医学中血液动力学流固耦合模型。最后介绍我们最近获得的一些不可压流体流固耦合界面运动模型的理论分析结果。

Abstract: In this talk, we present the joint work with Professor Yuri Trakhinin on the local well-posedness of the free boundary problem in ideal compressible magnetohydrodynamics with the total pressure vanishing on the plasma–vacuum interface.

Abstract: For vacuum free boundary problems of compressible fluids, the physical vacuum singularity of the sound speed being of $\frac{1}{2}$ H\"older but not Lipshitzian continuous near vacuum states as a characteristic phenomenon appearing in some important physical situations, such as gaseous stars, compressible flows with damping and shallow waters. The physical singularity near vacuum states prevents the standard method of symmetric systems developed by Friedrichs-Lax-Kato-Kreiss from applying. In this talk, I will discuss the almost global solutions to the vacuum free boundary problems with the physical vacuum singularity of compressible Euler equations with damping in the 3-D case.

Abstract: The large time asymptotics of vacuum boundaries featuring the physical singularity with radial symmetry for viscous gaseous stars will be discussed in this talk, based on joint research with Zhouping Xin and Huihui Zeng.

Abstract: In this talk, we will consider the of porous media equations when the initial data are continuous and compactly supported. The major feature is the parabolicity degenerate at the moving boundary. By introducing the proper weighted Sobolev space, which captures the degenerative at the moving boundary, the well-posedness of the local solution is established at first. Then, the global solution closed to the Barenblatt solution is constructed, and converges to the Barenblatt solution as time goes to infinity. Our results particularly give a positive answer of the open problem proposed by Lee and Vzquez on convexity.

Abstract: In this talk, I shall discuss the incompressible limit of the Ericksen-Leslie's hyperbolic liquid crystal model in compressible flow. We first derive the uniform energy estimates on the Mach number $\epsilon$ for both the compressible system and its differential system with respect to time under the uniformly in $\epsilon$ small initial data. Then, we take the limit in the compressible system to establish the global classical solution of the incompressible system. Moreover, we also obtain the convergence rates for the well-prepared initial data case. This talk is based on the joint work with L. Guo, N. Jiang, Y. Luo and S. Tang.

Abstract: We study the singular limit for equatorial shallow water equations at low Froude number forming a symmetric hyperbolic system with large variable coefficient terms. Based on the convergence result of Durtrifoy, Majda and Schochet [Comm. Pure Appl. Math(2009)], we further obtain the convergence rate estimates of the solutions. This is a recent joint work with Prof. Jiang, Song and Dr. Xu, Xin.

Abstract: The compressible Navier-Stokes-Poisson system takes the form of usual Navier -Stokes equations coupled with the self-consistent Poisson equation, which is used to simulate the transport of charged particles under the electric field of electrostatic potential force. In this paper, we focus on the large time behavior of global strong solutions in the $L^p$ Besov spaces of critical regularity. By exploring the dissipative effect arising from Poisson potential, we posed the new regularity assumption of low frequencies and then establish a sharp time-weighted inequality, which leads to the optimal time-decay estimates of the solution. Indeed, we see that the decay of density is faster at the half rate than that of velocity, which is a different ingredient in comparison with the situation of usual Navier-Stokes equations. Our proof mainly depends on tricky and non classical Besov product estimates with respect to various Sobolev embeddings.

Abstract: In this talk, I will discuss some results on ideal MHD free boundary problems including the a priori estimates of the nonlinear problem under Taylor sign, ill-posedness when the Taylor sign is violated, and well-posedness for the linearized problem under Taylor sign. Joint with Professor Chengchun Hao.

Abstract: Global dynamis of classical solutions of 1D Compressible MHD with large initial data has an interesting history and is challenging. We will report a recent progress made by my joint work with X. Qin.

Abstract: In this talk, the quasineutral limit of the two-fluid Euler-Poisson system (one for ions and another for electrons) in a bounded domain of $R^3$ is rigorously proved by investigating the existence and the stability of boundary layers. The non-penetration boundary condition for velocities and Dirichlet boundary condition for electric potential are considered. This is a joint work with Prof. Qiangchang Ju.

Abstract: We present a priori estimates for the incompressible MHD equations in a bounded domain with free moving boundary. In the case of no surface tension, higher regularity of the flow map is required due to a loss of 1/2-order derivative in the low regularity estimates. Due to the lack of Cauchy invariance for MHD, the smallness assumption on the fluid domain is required to control the vorticity of the flow map to compensate this loss. While in the case of nonzero surface tension, the flow map can be controlled by the boundary elliptic estimates owing to the surface tension. Moreover, we show that the magnetic field has certain regularizing effect, allowing us to control the vorticity of the fluid and that of the magnetic field simultaneously. This is the joint work with Dr. Chenyun Luo.

Abstract: We establish the local well-posedness for the free boundary problem for the compressible Euler equations describing the motion of liquid under the influence of Newtonian self-gravity. We do this by solving a tangentially-smoothed version of Euler's equations in Lagrangian coordinates which satisfies uniform energy estimates as the smoothing parameter goes to zero. The main technical tools are delicate energy estimates and optimal elliptic estimates in terms of boundary regularity, for the Dirichlet problem and Green's function.

Abstract: The dynamics of the vacuum state is one of the challenging issues for the viscous compressible fluids. Due to the strong degeneracy of the compressible Navier-Stokes system in the presence of vacuum, there are many difficulties associated with global well-posedness of either strong or even weak solutions. This is particularly so for multi-dimensional case. Most of the past efforts have been concentrated on solutions with vacuum without entropy consideration. However, entropy is one of the most important physical states and its equation is highly singular in the presence of vacuum. In this talk, I will discuss some of key issues in studying the full Navier-Stokes system and present some recent results on entropy-bounded strong solutions.

Abstract: In this talk, I will discuss the physical significance of physical vacuum singularity to the evolution of gas-vacuum interfaces, its mathematical challenge in the well -posedness theory and the role played by which to the dynamics of solutions. Comparison of physical vacuum singularity with the Taylor sign condition of incompressible fluids or compressible fluids away from vacuum states will be discussed. I will also discuss the role of the Taylor sign condition played to MHD free surfaces. The estimates and geometry of a free surface problem of highly subsonic inviscid heat-conductive flow will be presented.

Abstract: We present a new type of energy estimates for the compressible Euler equations with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad to the compressible case and do not require the fluid to be irrotational. Furthermore, the estimate is uniform as the sound speed of the liquid goes to infinity, which allows us to show the convergence of the solution for compressible Euler equations to that of the incompressible equations in C2. In particular, we can approximate the solution of a compressible gravity water wave by that of an incompressible water wave, whose global in time solution is known to exist. This work opens up a new approach to proving long time existence also for compressible water waves with small data, as conjectured by Hans Lindblad.

Abstract: The quasi-neutral limit of the Navier-Stokes-Poisson system with vanishing viscosity coefficients in the half-space is rigorously proved under a Navier-slip boundary condition for velocity and Dirichlet boundary condition for electric potential. This is achieved by establishing the nonlinear stability of the approximation solutions involving the strong boundary layer, which comes from the break-down of the quasineutrality near the boundary.

Abstract: I this talk I will summarize some of our contributions in the analysis of parabolic elliptic Keller-Segel system, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in two dimension, I will show our result for blow-up conditions for higher dimension. The second part of the talk is concentrated in the critical exponent for Keller-Segel system with porous media type diffusion. In the end, motivated from the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventing the blow-up of the solutions.

Abstract: In this talk, I will present some results on the Ill-posedness of MHD free boundary problem when the Taylor sign condition is violated. This is joint with Chengchun Hao.

Abstract: It is known in physics that steady state of compressible fluids under the influence of uniform gravity is stable if and only if the convection is absent. For non-isentropic flow, this stability criterion turns into the monotonicity of entropy in the direction of gravity. In this research, we will show that the instability side with mathematical rigor. This is based on a joint work with Xulong Qin, and Zhen-an Yao.

Abstract: In this talk, I will present some results on the nonlinear asymptotic stability of non-rotating gaseous stars in the frameworks of free boundary problems with physical vacuum singularity, and on the nonlinear stability of rotating gaseous stars in a weaker topology. The results presented in this talk include joint works with J. Smoller, Z.Xin and H. Zeng.

Abstract: We give the rigorous studies for the quasineutral limit of the two-fluid Euler-Poisson system in a torus or a domain with boundary in $R^3$. In a torus, the limit is one-fluid compressible Euler equations when the leading profiles of the initial velocities for two particles are assumed to be same. However, in a domain with boundaries, the quasineutrality breaks down near the boundary since the boundary layers generally develop due to the interaction between plasma and the boundary. We consider a non-penetration boundary condition for velocity and Dirichlet boundary condition for electric potential and prove the existence and stability of the boundary layers.

Abstract: We consider the couette flow problem for threei-dimensional compressible Navier-Stokes equations under the navier-slip boundary added at the bottom, and prove that the plane Couette flow is asymptotically stable for small perturbation provided that the slip length, Reynolds and Mach numbers satisfy some specific conditions. In particular, the Renolds number and the Mach number can be large if the slip length is suitably small.

Abstract: In this talk, we consider the periodic problem for bipolar non-isentropic Euler-Maxwell equations with damping terms in plasmas. By means of an induction argument on the order of the time-space derivatives of solutions in energy estimates, the global smooth solution with small amplitude was established close to a non-constant steady-state solution with asymptotic stability property. Furthermore, we obtain the global stability of solutions with exponential decay in time near the non-constant steady-states for bipolar non-isentropic Euler-Poisson equations. This phenomenon on the charge transport shows the essential relation and difference between the bipolar non-isentropic and the bipolar isentropic Euler-Maxwell/Poisson equations.

Abstract: In this talk I will first give some facts on the basic semiconductor physics. The modeling of the spherical harmonics expansion model also is presented in the framework of semiconductor superlattice. Some mathematical analysis results related to the spherical harmonics expansion model are introduced.

Abstract: In this talk I shall report some results on the low Mach number limit to the compressible MHD equations and related models.

Abstract: In this talk we analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models we consider include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with appropriate potentials.

Abstract: In this talk, I will discuss some major challenges and difficulties in the dynamical motions of the spatial periodic motions for the one-dimensional full compressible Euler system. Some recent progress on the longer time existence of entropy weak solutions for both the full Euler systems and its weakly nonlinear geometry optical approximation systems will be presented.

Abstract: This talk is concerned with the construction of global smooth solutions to some complex kinetic equations near Maxwellians in the perturbative framework. It is based on our recent works joint with Prof. R.-J. Duan of the Chinese University of Hong Kong, Dr. Y.-J. Lei of Huazhong University of Science and Technolgy, and Prof. T. Yang of City University of Hong Kong.

Abstract: We present the recent results on the spectrum structures of the Vlasov-Poisson(Maxwell)-Boltzmann equations and justify the influences of the electrostatic potential force, Lorentz force, and/or the mutual interaction between charged particles of differernt types. It is joint work with Ying Wang, Mingying Zhong and Tong Yang.

Abstract: We present some results on random approximations of $\pi$ by using the semiperimeter or area of a random $n$-sided polygon inscribed in (or circumscribed about) a unit circle in the plane. We show that, with probability $1$, the approximation error goes to $0$ as $n$ tends to infinity, and is roughly sextupled when compared with the classical Archimedean approach of using a regular $n$-sided polygon. Furthermore, by combining both the semiperimeter and area of these random polygons, we also construct extrapolation improvements that can significantly speed up the convergence of these approximations.

Abstract: As one of the oldest nonlinear PDE systems, the compressible Euler equations has been studied by many outstanding mathematicians. However, some basic questions, such as the global existence of classical solution v.s. finite time blowup, are still open even in one space dimension. In this lecture, we will report our recent progress in this direction, including a complete understanding on isentropic flows, and a refreshed understanding on general adiabatic flows. This lecture is based on joint works with H. Cai, G. Chen, S. Zhu, and Y. Zhu.

Abstract: In this talk, I will present the result with H. Zeng on the global existence and long time dynamics towards the Barenblatt self-similar solution for the physical vacuum free boundary problem featuring the behavior of sound speed being 1/2 Hölder continuous across the vacuum boundary for the compressible Euler Equations with Damping.

Abstract: In this talk, the proof for the existence of magnetic star solutions which are axi-symmetric stationary solutions for the Euler-Poisson system of compressible fluids coupled to a magnetic field will be presented by a variational approach. Our method of proof consists of deriving an elliptic equation for the magnetic potential in cylindrical coordinates in $R^3$, and obtaining the estimates of the Green's function for this elliptic equation by transforming it to 5-Laplacian. This is joint work with P. Federbush and J. Smoller.

Abstract: We study the initial problem for the nonlinear Boltzmann equation in a neighborhood of a Maxwellian. Firstly, we introduce the Compensating Functions method presented by Kawashima and use it to obtain a desired decay estimates for the existence of global solutions. Secondly, we present a macro-micro decomposition of the Boltzmann equation as well as the H-theorem provided by Tai-Ping Liu, etc and apply the method for the study the time-asymptotic, nonlinear stability of the global Maxwellian states. Finally, we compare the two methods by clarifying the fluid structures and dissipative structures in order to strengthen the understandings of the connection between Fluid equation and Boltzmann equation.

Abstract: This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow. It is proved that there exists a suitable weak solution and the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.

Abstract: We consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations with the stress free boundary condition where the compressible viscous flow of finite mass expands into infinite vacuum. For the spherically symmetric initial data with finite energy, we prove the global existence of spherically symmetric weak solutions. Furthermore, we investigate the expanding rate of the domain occupied by the fluid.

Abstract: In this talk, I will introduce the dynamic behavior to the hyperbolic type of MEMS, including global existence and asymptotic stability for small voltage, quenching for large voltage, and some relationship with the parabolic type of MEMS.

Abstract: In this talk, we first consider the non-relativistic limits for solutions of the free boundary value problem for the cylindrical symmetric relativistic Euler equations with physical vacuum. Then, we shall consider the well-posedness and asymptotic limits of solutions (including steady state solution, local smooth solution and global smooth solution) for the IBVP of the relativistic Euler-Poisson equations.

Abstract: The isentropic compressible isentropic Navier-Stokes equations (CNS) are investigated in 2D and 3D, where the viscosity coefficients are allowed to possibly depend on the fluid density. The typical model for the CNS is the viscous Saint-Venant system in the description of the motion for shallow water. Such viscous compressible models with density-dependent viscosity coefficients and its variants appear in geophysical flows. As the viscosity coefficients are constants, we have theclassical compressible Navier-Stokes equations.The main purpose of this talk aims at the existence and behaviors of solutions to 　the CNS with (partially) density-dependent viscosity coefficients in the appearance of 　vacuum, including the initial value problem and the free boundary value problem for the CNS.

Abstract: 本文研究了Saurel和Abgrall提出的非守恒两相流模型，给出了它的四激波黎曼解法器并用于构造了求解该模型初边值问题的路径守恒方法。所构造的格式是基于路径守恒方法，然后，基于该四激波黎曼解法器，通过MUSCL重构和Runge-KUTTA技巧建立了高阶Godunov型路径守恒方法。此外，用方向分裂推广到二维情形，然后给出相关算例。 最后、对非守恒两相流模型的封闭性、方程类型及理论研究方面提出一些研究需求、探求相关问题的研究方法。

Abstract: 对于晶格系统，我们近年来设计了一系列的准确界面条件，包括速度界面条件、匹配边界条件、几乎精确边界条件、以及基于这些条件的双向界面条件。对于这些离散系统里的稳定性分析，因为缺少连续系统（偏微分方程）中的那些工具，所以结果不多。
我们首先讨论常用的反射系数法的有效性问题，通过与精确的常微分方程组的特征值分析作比较，我们发现表面模态对准确界面条件的稳定性至关重要，而这是反射系数法所难以分析的。由此我们还解释了长时间不稳定性的问题。
对于匹配边界条件，如果晶格系统是线性谐振子链，我们构造了相应的李雅普诺夫函数，从而证明了稳定性。
对于基于匹配边界条件或速度界面条件的双向界面条件，我们发现位移有漂移现象。通过拉普拉斯变换我们严格分析了源项、初始位移和初始速度对于漂移的影响。

Abstract: The quasi-neutral limit of the full Navier-Stokes-Fourier-Poisson system in the torus is considered. We rigorously prove that as the scaled Debye length goes to zero, the global-in-time weak solutions of the full Navier-Stokes-Fourier-Poisson system converge to the strong solution of the incompressible Navier-Stokes equations as long as the latter exists.

Abstract: 我将总述流体动力学中的一些典型问题，这些问题包括来自于三维不可压流体轴对称Euler和Navier-Stokes方程的一个3D模型的研究、电磁流体动力学模型的结构稳定性研究和半导体物理中的漂流扩散模型拟中性极限研究。

Abstract: We will discuss some recent developments in the analysis of several longstanding problems involving weak continuity and compactness for fundamental nonlinear partial differential equations in mechanics and geometry. In particular, these problems include the inviscid limit of the compressible Navier-Stokes equations to the Euler equations, the construction of global entropy solutions of spherically symmetric solutions to the multidimensional compressible Euler equations, the construction of stochastic entropy solutions to the isentropic Euler equations with random forcing terms, the sonic-subsonic limit of approximate solutions to multidimensional steady Euler equations, the rigidity of isometric embeddings and weak continuity of the Gauss-Coddazi-Ricci equations. This talk will be based mainly on the joint work correspondingly with F.-M. Huang, M. Perepelitsa, M. Slemrod, D. Wang, and T.-Y. Wang.

Abstract: We survey some of our recent progress on the local well-posedness problem for compressible Navier-Stokes Equations with density dependent viscosity when initial density does not have a uniform positive lower bound. When viscosity coefficients are constant multiple of a power of density, we identify the class of initial data admitting a local classical solution local in time when the power is greater or equal to 1. The questions on the singularity formation will also be addressed. This talk is mainly based on joint works with Y. Li,and S. Zhu.

Abstract: Some mixed-type PDE problems for transonic flow and isometric embedding will be discussed. Recent results on the solutions to the hyperbolic-elliptic mixed-type equations and related systems of PDEs will be presented.

Abstract: In this talk, we shall study the well-posedness and stability of the Prandtl boundary layer equations, for both incompressible flows and compressible flows.

Abstract: In this talk, we will introduce the progress about the global classical solution of compressible Navier-Stokes equations with vacuum, which includes the results on cases of one dimension, spherical symmetry in multi-dimensions, and multi-dimensions. This is a joint work with Huanyao Wen.

Abstract: In this talk, I will present some new progress and results on the understanding of both local and long time dynamics of physical vacuum boundaries of compressible fluids, the emphasize will be on the long time dynamics of the gaseous star problems. This is joint work with Z. Xin and H. Zeng.

Abstract: We discuss gravitation from Newton to Einstein to present-day.

Abstract: We identify sufficient conditions to ensure the existence of a unique classical solution to the 3D Shallow Water equations with initial data allowing vacuum. This is a recent work joint with Yachun Li and Shengguo Zhu.

Abstract: The excess charge problem, or the ionization problem, asks how many electrons an atom or molecule can bind, which is one of the important topics in the stability of matter in quantum mechanics. We will explain Liebs original proof and show the key inequality, i.e. Liebs inequality, and its relation with quantum mechanicl uncertainty principle (the Hardys inequality). Then show our generalization of this inequality and its application in 2-D graphene case. If time allows, some ideas of the proof will also be proposed. The main tools are the representation of Fractional Laplacian in position space and the Grand State Representation of the desired operator. This is a joint work with Heinz Siedentop from Munich.

Abstract:

Abstract: This talk concerns the well-posedness theory of the motion of physical vacuum for the compresssible Euler equations with or without self-grivatation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a new functional space involving less derivatives than those constructed for three-dimensional motion by constructing suitable weight and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.

Abstract: In this talk, we will deals with the asymptotic limits of the full compressible magnetohydrodynamic (MHD) equations in the whole space R^3 which are the coupling between the Navier-Stokes-Fourier system with the Maxwell equations governing the behaviour of the magnetic field. It is rigorously showed that, for the general initial data, the weak solutions of the full compressible MHD equations converge to the strong solution of the ideal incompressible MHD equations as the Mach number, the viscosity coefficients, the heat conductivity, and the magnetic diffusion coefficient go to zero simultaneously. Furthermore, the convergence rates are also obtained.

Abstract: We study the incompressible limit of the compressible non-isentropic ideal magnetohydrodynamic equations with general initial data in the whole space. We first establish the existence of classic solutions on a time interval independent of the Mach number. Then, by deriving uniform a priori estimates, we obtain the convergence of the solution to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero.