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On the asymptotics of the Boltzmann equation and fluid-dynamic limits
Kazuo AokiDepartment of Mechanical Engineering and Science, Kyoto University, Kyoto 606-8501, Japan
We consider a rarefied gas in the near continuum regime (i.e., small mean freepaths) and discuss the formal asymptotics and fluid-dynamic limits of the Boltzmannequation based on the Hilbert-type expansion. In general, it is believed that theHilbert Expansion leads to the (compressible) Euler equations at the leading order,and the higher-order equations are the linear and inhomogeneous equations of theEuler type. However, if we consider steady boundary-value problems, different-typesof equations are obtained depending on the physical situation under consideration.We will show an example giving the (compressible) Navier-Stokes equations at theleading order (cylindrical Couette flow) and another example giving the equationscontaining non-Navier-Stokes stress terms as the leading-order equations (thermalcreep flow caused by large temperature variations). In this connection, the invertedvelocity profile in the former example and the ghost effect in the latter will bediscussed.
Partial convexity for partial differential equations
Baojun BianDepartment of Mathematics, Tongji University, P.R.China
In this talk, we consider partial convexity for solution of partial differential equa-tions. We establish a microscopic partial convexity principle for partially convexsolution of nonlinear elliptic and parabolic equations. As application, we discuss thepartial convexity preserving of solution for parabolic equations. This talk is basedon the joint works with Pengfei Guan.
Renormalized Resonances and Turbulence in Nonlinear Dispersive Waves
David CaiShanghai Jiao Tong University, China
We discuss the non-perturbative nature of wave turbulence in equilibrium instrongly nonlinear regimes and show how the wave spectrum of nonlinear dispersivewaves is determined by an intertwining self-consistent renormalization process. Wedemonstrate that nonlinear wave interactions renormalize the dynamics, leading to(i) a drastic deformation of the resonant manifold even at weak nonlinearities, and(ii) the creation of nonlinear resonance quartets in wave systems for which therewould be no resonances as predicted by the linear dispersion relation. Finally, wepresent an extension of the weak turbulence kinetic theory to systems with strongnonlinearities.
Heat Flow Method in the Critical Point Theory
Chang Kung ChingSchool of Math. Science, Peking University, China
We study the heat semi-flow for elliptic problems with variational structure, forexample, the harmonic map problem, the minimal surfaces problem, the prescribingmean curvature problem as well as some semilinear problems. In some cases whenthe Palais Smale condition fails, it is used as a replacement of the pseudo gradientflow in deforming the level sets of the energy functionals, so that the Morse theoryis established.
Vanishing Viscosity Limit for Nonlinear Conservation Laws
Gui-Qiang G. ChenUniversity of Oxford and Fudan University
The vanishing viscosity limit is one of the most classical, longstanding fundamen-tal issues in the theory of nonlinear conservation laws. In this talk, we will discusssome of old and recent developments in the study of this issue. These especiallyinclude the vanishing viscosity limit of the Navier-Stokes equations to the Eulerequations for compressible flow, among others.
Weighted Sobolev Spaces and Semilinear Degenerated Elliptic Equationson Manifolds with Conical Singularities
Hua ChenWuhan University, China
In this talk, we would report some recent results on existence of solutions for aclass of semilinear degenerated elliptic equations on manifolds with conical (or edge)singularities.
Large time behavior of some degenerately dissipative systemswith the electromagnetic field
Renjun DuanThe Chinese University of Hong Kong, China
This talk is concerned with the large time behavior of solutions to the Cauchyproblem on systems of some fluid or kinetic equations under the self-consistentelectromagnetic field satisfying the Maxwell equations. Different from previouswork about the study of systems of hyperbolic-parabolic type under the Shizuta-Kawashima condition, we establish a new dissipative structure of the considereddegenerately dissipative linearized systems due to appearance of the coupled hyper-bolic Maxwell equations. A typical feature for such systems is that solutions overthe high frequency domain can decay in time with polynomial rates up to any or-der depending on regularity of initial data. This is a partially joint work with S.Kawashima and Y. Ueda.
Scattering for the focusing energy-subcriticalNonlinear Schrodinger equations
Daoyuan FangDepartment of Mathematics, Zhejiang University, Hangzhou 310027, China
For the 3D focusing cubic nonlinear Schrodinger equation, Scattering of H1 so-lutions inside the (scale invariant) potential well was established by Holmer andRoudenko (radial case) and Duyckaerts, Holmer and Roudenko (general case). Inthis paper, we extend this result to arbitrary space dimensions and focusing, mass-supercritical and energy-subcritical power nonlinearities, by adapting the method ofthe authors before.
This is a joint work with Thierry Cazenave and Jian Xie.
Regularity for the parabolic obstacle problem with fractional Laplacian
Alessio FigalliDepartment of Mathematics, The University of Texas at Austin, Austin TX 78712, USA.
In recent years, there has been an increasing interest in studying constrainedvariational problems with a fractional diffusion. One of the motivations comes frommathematical finance: jump-diffusion processes where incorporated by Merton intothe theory of option evaluation to introduce discontinuous paths in the dynamics ofthe stock’s prices, in contrast with the classical lognormal diffusion model of Blackand Scholes. These models allow to take into account large price changes, and theyhave become increasingly popular for modeling market fluctuations, both for riskmanagement and option pricing purposes.
In a joint paper with Luis Caffarelli we study the parabolic version of the frac-tional obstacle problem, i.e. where the elliptic part of the operator is given (at leastat the leading order) by a fractional laplacian. We prove optimal spatial regularityand almost optimal time regularity of the solution, recovering in particular the opti-mal regularity for the stationary case. To obtain this result, we crucially exploit thefact that the solution coincides with the obstacle at the initial time, which corre-sponds to the fact that (for the backward operator) the stock’s price coincides withthe payoff at the final time.
Exact boundary controllability of nodal profilein a tree-like network
Qilong GuShanghai Jiao Tong University, China
We give the definition of a kind of exact boundary controllability, which is calledthe exact boundary controllability of nodal profile. In a tree-like network, we showthe basic principles of getting the controllability for general first order quasilinearhyperbolic systems in dimension 1 with nonlinear boundary conditions and interfaceconditions. We give the conditions of compatibility of giving nodal profiles, and themethod to find the controls corresponding to the giving nodal profiles.
Compressible Euler equations with damping
Feimin HuangChinese Academy of Sciences, China
In this lecture, I will introduce the recent progress on the large time behavior ofentropy solutions of compressible Euler equations with damping. It is shown thatany bounded entropy solutions time asymptotically tend to the Barenblatt solutionin L1 norm when the initial mass is finite.
Existence of weak solutions to the three-dimensionalsteady compressible Navier-Stokes equations
Song Jiang, Chunhui ZhouInstitute of Applied Physics and Computational Mathematics, Beijing, China
[email protected], [email protected]
We prove the existence of a spatially periodic weak solution to the steady com-pressible isentropic Navier-Stokes equations in IR3 for any specific heat ratio γ > 1.The proof is based on the weighted estimates of both pressure and kinetic energyfor the approximate system which result in new higher integrability of the density,and the method of weak convergence.
Vanishing viscosity problem for 3-D Navier-Stokes equations with helical symmetry
Quansen JiuSchool of Mathematical Sciences, Capital Normal University, Beijing 100048,PRC
e-mail ([email protected])
In this talk, we discuss vanishing viscosity problem for the 3-D Navier-Stokes equations with helical symmetry.We make an appropriate decomposition of the helical vector fields to obtain the uniform estimates of the vorticityand furthermore establish the convergence of the solutions of the Navier-Stokes equations with helical symmetryto the ones of the corresponding Euler equations as the viscosity vanishes. Both the bounded and whole spacecases are discussed. This is joint with Milton C. Lopes Filho, Helena J. Nussenzveig Lopes and Dongjuan Niu.
Lp estimate of the Stokes equations
Dongsheng LiXi’an Jiao Tong University, China
In this talk, we will give a new proof of the Lpestimate of the Stokes equations.
Why do you study 2-D Riemann problems for Euler equations?
Jiequan LiSchool of Mathematical Sciences, Beijing Normal University, 100875, China
Two-dimensional (2-D) Riemann problems for compressible fluid flows assumethe simplest initial state but provide the most fundamental wave configurations,including the dam collapse, the reflection of oblique shocks and vortex-shock in-teraction etc. Mathematically they are formulated as mixed-type boundary valueproblems for most cases. In this talk I discuss the mission of 2-D Riemann problems,main mathematical difficulties and potentially accessible problems for analysts.
Stability of Large-amplitude Traveling Waves Arising from Chemotaxis
Tong Li
University of Iowa
USA
Traveling wave (band) behavior driven by chemotaxis was observed experimen-
tally by Adler and was modeled by Keller and Segel. We establish the existence
and the nonlinear stability of large-amplitude traveling wave solutions to a system
of nonlinear conservation laws which is derived from the well-known Keller-Segel
model describing cell(bacteria) movement toward the concentration gradient of the
chemical that is consumed by the cells. This is a joint work with Zhi-an Wang.
On the Boltzmann Equation for Bose-Einstein Particles
Xuguang LuTsinghua University, [email protected]
Although physical experiments show that the Bose-Einstein condensation occursin finite time, this has not been clear for the present Boltzmann equation: Does thereexist a mass conserved solution of the equation such that a condensation occurs infinite time ? To this problem there has been no definite answer even in a formal level.In this talk we will present some recent results, including a theorem of alternative(“concentration or oscillation”) and further singular properties of the EMV solution,and we will also show that this problem of finite time condensation relies heavilyon the grazing effect: if the collision kernel is given an angular-cutoff, there will beno condensation in finite time, which is very different from the case of long timebehavior where the condensation always happens as time goes to infinity.
Dynamical Theory for Oceanic Thermohaline Circulation
Tian MaSichuan University, China
The oceanic thermohaline circulation is a remarkable phenomenon in Oceanogra-phy, which is one of key sources to influence the global climate. Recently,we establisha set of dynamic transition theory for the oceanic thermohaline circulation, whichcan help us to understand well the dynamical behavior of this natural phenomenon.In this report, we shall give a briefly introduction to this theory.
The spacetime convexity of the solutions of parabolic equation
Xinan MaChinese University of Science and Technology, China
We study the spacetime convexity of the solution of parabolic equation, we usethe technique of constant rank lemma. We shall give some applications.
Smoothing effect of weak solutions forspatially homogeneous non-cutoff Boltzmann equation
Yoshinori Morimoto
Kyoto University, Japan
In this talk we consider the Cauchy problem for the spatially homogeneous Boltz-
mann equation without angular cutoff
∂tf(t, v) = Q(f, f)(t, v), t ∈ R+, v ∈ R3, f(0, v) = f0(v),
where f(t, v) ≥ 0 is the distribution of particles at time t with velocity v. The
right hand side of the equation is given by the Boltzmann bilinear collision operator
Q(g, f) =∫R3
∫S2 B (v − v∗, σ) g(v′∗)f(v′)− g(v∗)f(v) dσdv∗ , where v′ = v+v∗
2 +|v−v∗|
2 σ, v′∗ = v+v∗2 − |v−v∗|
2 σ for σ ∈ S2. We assume the collision cross section
B(v−v∗, cos θ) has the form B = Φ(|v−v∗|)b(cos θ), cos θ = v−v∗|v−v∗| · σ , 0 ≤ θ ≤ π
2 ,
where the kinetic factor Φ(|v − v∗|) = |v − v∗|γ and the angular factor containing a
singularity,
b(cos θ) ≈ Kθ−2−2s when θ → 0+, 0 < s < 1,
for some constant K > 0. In the case γ + s > 0, we show any weak solution
having the finite energy and entropy lies in the Schwartz space if it has the mass
conservation and the moments of arbitrary order. The main ingredients of the proof
are the suitable choice of the mollifiers composed of pseudo-differential operators,
and sharp estimates of the commutators of mollifiers and the Boltzmann operator
whose cross section has singularities in both kinetic and angular factors. The content
of this talk is based on the theory of non-cutoff Boltzmann equation developed in
the joint-works with R. Alexandre, S. Ukai, C.-J. Xu and T. Yang since 2006.
Diffusion and Directed Movements in Heterogeneous Environments
Wei-Ming NiEast China Normal University and University of Minnesota
In this talk I will use the Lotka-Volterra competition system to illustrate firstthe interaction between diffusion and spatial inhomogeneity, and then incorporatedirected movements into consideration.
On a Quasilinear System Involving Curl
Xingbin PanEast China Normal University, China
This talk concerns a quasilinear system in a three dimensional domain whichinvolves curl. This system describes the Meissner states of type II superconductors.We shall see that this system has many Meissner solutions, and we shall examinethe convergence of the Meissner solutions to a solution of the limiting system as theGinzburg-Landau parameter increases to infinity.
Large-time Behavior of Solutions to the Inflow Problemof Full Compressible Navier-Stokes Equations
Xiaohong QinNanjing University of Science and Technology, China
Large-time behavior of the solutions to the inflow problem of full compressibleNavier-Stokes equations is considered on the half line R+ = (0,∞). First, we givethe existence (or non-existence) of the boundary layer solution to the inflow problemwhen the right end state (ρ+, u+, θ+) belongs to the subsonic, transonic and super-sonic regions, respectively. Then some wave structures involving the the boundarylayer solution (subsonic and transonic cases), the viscous contact wave and the rar-efaction waves to the inflow problem are described and the asymptotic stability ofthese wave patterns are proved under some smallness conditions. The proofs aregiven by the elementary energy method based on underlying wave structures.
Uniform attractors for a 3D non-autonomous Navier-Stokes-Voight Equations
Yuming Qin Dong University, China, e-mail: yuming [email protected]
This work is jointly with Xin-guang Yang and Xin Liu.
In this paper, under suitable assumptions on the external force f and initialdata uτ , we prove the global existence of solutions and the existence of the uniformattractor for a 3D non-autonomous Navier-Stokes-Voight equations
ut − ν4u− α24ut + (u · ∇)u +∇p = f(t, x), x ∈ Ω, t ∈ R,
∇ · u = 0, x ∈ Ω, t ∈ R,
u(t, x)|∂Ω = 0,
u(τ, x) = uτ (x), τ ∈ R,
by establishing the asymptotical compactness or uniform condition-(C).
Two-dimensional Riemann problems for compressibleEuler equations and Zheng’s patch
Wancheng ShengDepartment of Mathematics, Shanghai University
Shanghai, [email protected]
In this talk, I will show you the numerical simulation on Two-dimensional Rie-mann problems for compressible Euler equations by use of numerical generalizedcharacteristic analysis method. Some building blocks, such as Zhengs Patch (semi-hyperbolic patch), Shock Reflection, Richtmyer-Meshkov Instability, etc., are shown.(Jointed with G.D.Wang and T.Zhang)
Interplays between self-diffusion, cross-diffusionand logistic source in chemotaxis systems
Youshan TaoDepartment of Applied Mathematics, Dong Hua University, Shanghai 200051, P. R. China
This talk mainly addresses parabolic-parabolic chemotaxis system. Some pre-vious results on this system are firstly reviewed. Then, two recent works are re-ported: One studies the interaction between nonlinear diffusion, chemotaxis andlogistic dampening; the other discusses the interplay between self-diffusion and cross-diffusion. Finally, this talk is closed with two potentially very interesting models:a combined chemotaxis-haptotaxis model describing cancer invasion with tissue re-modeling, and a coupled chemotaxis-fluid model reflecting the motion of oxygen-driven swimming bacteria in an incompressible fluid; the former was initially pro-posed by Chaplain and Lolas (M3AS 2005), and the latter was originally developedby Tuval et al. (PNAS 2005).
Two recent results reported in this talk are joint works with Michael Winkler(University of Duisburg-Essen, Germany).
On a 3D Model for the Incompressible Euler and Navier-Stokes Equations
Shu WangCollege of Applied Sciences, Beijing University of Technology, P.R.China
In this talk, we will discuss some properties of the incompressible Euler andNavier-Stokes equations by studying a 3D model for axisymmetric 3D incompress-ible Euler and Navier-Stokes equations with swirl. The 3D model is derived byreformulating the axisymmetric 3D incompressible Euler and Navier-Stokes equa-tions and then neglecting the convection term of the resulting equations. Someproperties of this 3D model are reviewed. Some potential features of the incom-pressible Euler and Navier-Stokes equations discussed here includes the stabilizingeffect of the convection, the effects of the sign of the vorticity and the boundary ofthe fluids on globally dynamic stability, the role of the dimension of the space tocapture the finite time blowup of fluids.
Some analysis of contact angle hysteresis
Xiao-Ping WangHong Kong University of Science and Technology, China
We analyze the wetting hysteresis on rough and chemically patterned surfacesfrom a phase-field model for immiscible two phase fluid. By matched asymptoticanalysis of the model, we derive some equations that describe the structure of theinterface on the chemically patterned surface. These equations describe directly thethe contact angle hysteresis and force hysteresis. The limit can also be justified fromthe theory of Gamma convergence.
Some results on control problems of 1-D hyperbolic systems
Zhiqiang WangFudan University, China
We first present a result on boundary controllability of 1-D hyperbolic systemwith a vanishing characteristic speed (joint work with Jean-Michel Coron and OlivierGlass). Then we deal with a conservation law with nonlocal velocity which modelsa highly re-entrant manufacturing system (joint work with Jean-Michel Coron andMatthias Kawski). For this nonlocal model, we show some results on controllabilityand optimal control problems.
On Compressible Navier Stokes Systems
Zhouping XinThe Chinese University of Hong Kong, Hong Kong, China
Microlocal analysis of Kinetic equations
Chao-Jiang XUWuhan University, China
In this talk, we establish the hypoellipticity of the non homogeneous Boltzmannequation without angular cutoff. By using the nonlinear microlocal analysis, we canstudy this equation as a generalized Kolmogrove equation which is non linear andanisotropic. The key step to obtain the regularizing effect is a generalized versionof the uncertainty principle, it is a very strong results of microlocal analysis, fromwhich we proved the hypoellipticity of a transport equation.
Semi-hyperbolic patches in transonic flows
Hongmei XuHohai University, China
We study the time asymptotic behavior of solutions to the nonlinear wave equa-tion with viscosity in even multi spatial dimension. Our study is based on the de-tailed analysis of the Green function of the linearized system. This is used to studythe coupling of nonlinear diffusion waves. Time asymptotic shape of the solutionsare obtained and shown to exhibit the generalized Huygen’principle.
Fluid Dynamic Limits of Boltzmann Equationto Riemann Solutions of Euler Equations
Tong YangDepartment of mathematics, City university of Hong Kong, Hong Kong, P.R. China
Fluid dynamic limit to the compressible Euler equations from the Boltzmannequation is a problem with long history. Under the assumption of slab symmetry,even though intensive studies have been made when the solution of the Euler equa-tions has non-interacting single waves, the problem on the genuine Riemann problemis still unsolved let alone the general weak solutions. In this talk, we present somerecent results on this problem when the Riemann solution is superposition of eithershock-rarefaction wave or rarefaction wave-contact discontinuity. Convergence ratesin terms of Knudsen number is also given. The follows from some recent joint workswith Feimin Huang and Yi Wang.
Variational Methods for Real Ultrasound Image Despeckling and Segmentation
Xiaoping YANGSchool of Science, Nanjing University of Sci. and Tech., Nanjing, 210094, China
Speckle appears in all conventional ultrasound images, and it generally tends toreduce the image resolution and contrast, thereby reducing the diagnostic value ofthe imaging modality. In this talk, we focus on real ultrasound image despeckling andsegmentation. We discuss some corresponding variational models. The existence anduniqueness of minimizers of the variational problems and the associated evolutionproblems are studied. We also show the capability of our models on some numericalexperiments.
Waiting Time for a non-Newtonian Polytropic Filtration Equation with Convection
Yin JingxueSouth China Normal University, P.R.China
e-mail: [email protected]
This is a joint work with Professor Yang Tong and Doctor Jin Chunhua.
We discuss the waiting time phenomena for a class of non-Newtonian polytropicfiltration equation with convection. According to the influence of the convectionon the waiting time property, we divide the convection into three cases, that is thestrong convection, the mild convection and the weak convection. For different cases,the sufficient and necessary conditions on the initial data are given for the existenceof waiting time respectively.
Nonlinear stability of rarefaction waves to the Landau equation
Hongjun YuSchool of Mathematical Sciences, South China Normal University, Guangzhou, China
The Landau equation, which was proposed by Landau in 1936, is a fundamen-tal equation to describe collisions among charged particles interacting with theirCoulombic force. Rarefaction waves in gas dynamics can be described by the Euler,Navier-Stokes, Boltzmann or Landau equations. In this paper we show that rar-efaction waves for the Landau equation are time asymptotic stable and tend to therarefaction waves of the Euler and Navier-Stokes equations for the first time. Themethod combines the analytic techniques for viscous conservation laws, propertiesof Burnett functions and energy method through the micro-macro decomposition ofthe Landau equation. Our result also covers a class of generalized Landau equations,include Coulomb potential case, which describes grazing collisions in a dilute gas.
Construction of Green’s functions for the Boltzmann equations
Shih-Hsien YuNational University of Singapore, Singapore
In this talk, we will review the development on constructing Green’s functionsof linearized Boltzmann equation around a global maxwellian and Boltzmann shockprofile and its applications to various problems.
Well-posedness of Hydrodynamics on the Moving Elastic Surfaces
Pingwen Zhang
School of Mathematical Sciences, Peking University, Beijing 100871, China
The dynamics of a membrane is a coupled system of a moving elastic surface
and an incompressible membrane fluid. The difficulties in analyzing such a system
include the nonlinearity of the moving curved space (geometric nonlinearity), the
nonlinearity of the elastic force (elastic nonlinearity) and the nonlinearity of the
fluid dynamics (fluid nonlinearity). The coupled system includes parabolic equa-
tions, wave equations and elliptic equations, but energy dissipation is satisfied. Here
we prove the local existence and uniqueness of the solution to the coupled system
by getting regularity through reformulating the system into a new system in the
isothermal coordinates and constructing a suitable iterative scheme. This work is
joined with Wei Wang and Zhifei Zhang.
One-dimensional Compressible Navier-Stokes Equation with Large Density Variation
Huijiang ZhaoSchool of Mathematics and Statistics, Wuhan University, China
This talk is concerned with one-dimensional compressible Navier-Stokes equationwith large density variation. It is based on recent works joint with Lili Fan, HongxiaLiu and Tao Wang.
Semi-hyperbolic patches in transonic flows
Yuxi ZhengYeshiva University and Penn State University, U.S.A.
In construction of solutions to the self-similar Euler systems for compressiblegases in two space dimensions, one finds domains of solutions whose characteristicsare trapped away from given boundary or initial conditions. These domains are whatwe call semi-hyperbolic patches. These patches are quite common in solutions to theRiemann problems. They are related to solutions to the classical Tricomi problemor Keldysh problem. We show a couple of ways to construct these solutions throughnonlinear methods. This talk is based on joint work with Tong Zhang, Jiequan Li,Xiaomei Ji, Mingjie Li, Zhicheng Yang, Xiao Chen and Zhen Lei.
On Non-degeneracy of Solutions to SU(3) Toda System
Feng ZhouEast China Normal University, China
In this talk, we discuss the solution to the following SU(3) Toda system∆u + 2eu − ev = 0, ∆v − eu + 2ev = 0 in R2,∫R2
eu <∞,
∫R2
ev <∞,
We prove that it is nondegenerate, i.e., the kernel of the associated linearized operator is exactly eight-dimensional. This is a joint work with J.C.Wei and C.Y.Zhao.
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Entropy and renormalized solutionsfor quasilinear elliptic (parabolic) equations with L1 data
Shulin ZhouPeking University, CHANA
In this talk we study the existence and uniqueness of both entropy solutions andrenormalized solutions for quasilinear elliptic (parabolic) equations with L1 data.And moreover, we will show the equivalence of entropy solutions and renormalizedsolutions for such equations.
Life Span of Nonlinear Wave equations with Small Initial Data
Yi ZhouFudan University, China
In this talk, we will review results concerning global existence or long-time exis-tence for nonlinear wave equation with small initial data. we will also present ournew result for the initial boundary value problem exterior to non-trapping obstaclesin three or four space dimensions.
