数学物理学报(英文版)

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2009年, 第29卷, 第4期　刊出日期：2009-07-20 上一期下一期

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TWO-DIMENSIONAL RIEMANN PROBLEMS: FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS

李杰权, 盛万成, 张同, 郑玉玺

数学物理学报(英文版). 2009 (4): 777-802. DOI: 10.1016/S0252-9602(09)60070-9

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In this paper we survey the authors’ and related work on two-dimensional Rie-mann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections: 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.

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THE ACCELERATED SEARCH-EXTENSION METHOD FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs

刘跃武, 谢资清, 陈传淼

数学物理学报(英文版). 2009 (4): 803-816. DOI: 10.1016/S0252-9602(09)60071-0

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In this paper, we propose an accelerated search-extension method (ASEM)based on the interpolated coefficient finite element method, the search-extension method (SEM) and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in 1-D and 2-D cases will show the efficiency of our approach.

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MARKOWITZ STRATEGIES REVISED

严加安, 周迅宇

数学物理学报(英文版). 2009 (4): 817-828. DOI: 10.1016/S0252-9602(09)60072-2

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Continuous-time Markowitz’s mean-variance efficient strategies are modified by parameterizing a critical quantity. It is shown that these parameterized Markowitz strategies could reach the original mean target with arbitrarily high probabilities. This, in turn, motivates the introduction of certain stopped strategies where stock holdings are
liquidated whenever the parameterized Markowitz strategies reach the present value of the mean target. The risk aspect of the revised Markowitz strategies are examined via expected discounted loss from the initial budget. A new portfolio selection model is suggested based on the results of the paper.

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POSITIVELY CURVED COMPLETE NONCOMPACT KÄHLER MANIFOLDS

陈兵龙, 朱熹平

数学物理学报(英文版). 2009 (4): 829-845. DOI: 10.1016/S0252-9602(09)60073-4

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In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. Among other things, we prove that a complete noncompact K¨ahler surface with positive and bounded sectional curvature and with finite analytic Chern number c₁(M)² is biholomorphic to C₂.

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ON THE 3D VISCOUS PRIMITIVE EQUATIONS OF THE LARGE-SCALE ATMOSPHERE

郭柏灵, 黄代文

数学物理学报(英文版). 2009 (4): 846-866. DOI: 10.1016/S0252-9602(09)60074-6

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This paper is devoted to considering the three-dimensional viscous primitive equations of the large-scale atmosphere. First, we prove the global well-posedness for the primitive equations with weaker initial data than that in [11]. Second, we obtain the existence of smooth solutions to the equations. Moreover, we obtain the compact global attractor in V for the dynamical system generated by the primitive equations of large-scale
atmosphere, which improves the result of [11].

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AN UNGUIDED TOUR STARTED FROM CHIRALITY

姜伯驹, 王诗窚

数学物理学报(英文版). 2009 (4): 867-880. DOI: 10.1016/S0252-9602(09)60075-8

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This survey article records an unguided mathematical tour by topologists at Peking University and their collaborators in the last ten years. The tour started from research on chirality and, attracted by questions around attractors, led to a zigzag path across topology and dynamics.
People who joined us in this tour at various stages include Ding Fan, Liu Yi, Ni Yi, Pan Jianzhong, Yao Jiangang, Zheng Hao and Zhou Qing. Conversations with Robert Edwards, Wen Lan and others, added to the twists and turns that made the trip more fun. This article benefits from related lectures by these authors, in many conferences, universi-ties, as well as high schools.

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ON THE ASYMPTOTIC BEHAVIOR OF HOLOMORPHIC ISOMETRIES OF THE POINCARÉ|DISK INTO BOUNDED SYMMETRIC DOMAINS

Ngaiming Mok

数学物理学报(英文版). 2009 (4): 881-902. DOI: 10.1016/S0252-9602(09)60076-X

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In this article we study holomorphic isometries of the Poincar´e disk into bounded symmetric domains. Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometries up to normal-izing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincar´e disk Δ into an irreducible bounded symmetric domain Ω C^N in its Harish-Chandra realization must extend to an affine-algebraic subvariety V ⊂ C×C^N = C^N+1, and that the irreducible component of V ∩(Δ×Ω) containing V₀is the graph of a proper holomorphic isometric embedding F : Δ → Ω. In this article we study holomorphic isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ ∂Δ. Starting with the structural equation for holomor-phic isometries arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphic isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we
prove that ||σ|| must vanish at a general boundary point either to the order 1 or to the order 1/2 , called a holomorphic isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the
Poincaré disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincar´e disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.

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MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

李工宝, 张国

数学物理学报(英文版). 2009 (4): 903-918. DOI: 10.1016/S0252-9602(09)60077-1

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In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:

−△_pu − △_qu = |u|^p*−2u + μ|u|^r−2u in Ω,
u|∂Ω= 0,

where Ω ⊂ R^N is a bounded domain, N > p, p* = N_p /N−p is the critical Sobolev exponent and μ > 0. We prove that if 1 < r < q < p < N, then there is a μ₀ > 0, such that for any μ ∈ (0, μ₀), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.

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THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES

Gui-Qiang Chen, Dan Osborne, Zhongmin Qian

数学物理学报(英文版). 2009 (4): 919-948. DOI: 10.1016/S0252-9602(09)60078-3

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We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in Rⁿwith compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we
establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition (without the incompressibility condition), which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in Rⁿ(n ≥ 3) with nonhomogeneous vorticity boundary condition converge in L² to the corresponding Euler equations satisfying the kinematic condition.

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CLASSIFICATION OF POSITIVE SOLUTIONS FOR NONLINEAR DIFFERENTIAL AND INTEGRAL SYSTEMS WITH CRITICAL EXPONENTS

Wenxiong Chen, Congming Li

数学物理学报(英文版). 2009 (4): 949-960. DOI: 10.1016/S0252-9602(09)60079-5

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We classify all positive solutions for the following integral system:

u_i(x) =∫_Rn1/ |x − y|^n−α f_i(u(y))dy, x ∈ Rⁿ, i = 1, · · · , m,

0 < α < n, and u(x) = (u₁(x), u₂(x), · · · , u_m(x)).

Here f_i(u), 1 ≤ i ≤ m, are real-valued functions of homogeneous degree n+α/ n−α and are monotone nondecreasing with respect to all the independent variables u₁, u2, · · ·, u_m. In the special case n ≥ 3 and α = 2, we show that the above system is equivalent to the following elliptic PDE system:

−△u_i(x) = f_i(u(x)), x ∈ Rⁿ, i = 1, · · · , m,

and u(x) = (u₁(x), u₂(x), · · · , u_m(x)).

This system is closely related to the stationary Schr¨odinger system with critical exponents for Bose-Einstein condensate.

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N = 2 SCVA’s FROM A GENERALIZED CALABI-YAU MANIFOLD AND MIRROR SYMMETRY

胡森, 马文晔, 邱敬佩

数学物理学报(英文版). 2009 (4): 961-972. DOI: 10.1016/S0252-9602(09)60080-1

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We construct an N = 2 superconformal vertex algebra(SCVA) from a gener-alized Calabi-Yau manifold and compute the BRST cohomology of its associated topolog-ical vertex algebras. We show that the BRST cohomology coincides with the generalized Dobeault cohomology. We show that the two topological vertex algebras constructed from the N = 2 SCVA by A and B twist respectively are mirror pairs.

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STRONG SHOCKS IN NONISENTROPIC GAS DYNAMICS

Constantine M. Dafermos

数学物理学报(英文版). 2009 (4): 973-979. DOI: 10.1016/S0252-9602(09)60081-3

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In nonisentropic gas dynamics, with general equations of state, strong com-pressive shocks satisfying the Liu E-condition may violate the Second Law of thermody-namics.

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EXACT CONTROLLABILITY FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS WITH VERTICAL CHARACTERISTICS

李大潜, 饶伯鹏

数学物理学报(英文版). 2009 (4): 980-990. DOI: 10.1016/S0252-9602(09)60082-5

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We consider first order quasilinear hyperbolic systems with vertical charac-teristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on
the equations corresponding to zero eigenvalues. In this paper, we will establish the ex-act controllability only by means of physically meaningful internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.

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SYMBOLIC VERSOR COMPRESSION ALGORITHM

李洪波

数学物理学报(英文版). 2009 (4): 991-1004. DOI: 10.1016/S0252-9602(09)60083-7

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In an inner-product space, an invertible vector generates a reflection with re-spect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the
input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space.

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ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS VIA THE RIEMANN-HILBERT APPROACH

R. Wong, 赵育求

数学物理学报(英文版). 2009 (4): 1005-1034. DOI: 10.1016/S0252-9602(09)60084-9

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In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Deift & Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal
polynomials, and certain orthogonal polynomials with singular behavior.

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OPTIMAL CONVERGENCE RATES OF LANDAU EQUATION WITH EXTERNAL FORCING IN THE WHOLE SPACE

杨彤, 喻洪波

数学物理学报(英文版). 2009 (4): 1035-1062. DOI: 10.1016/S0252-9602(09)60085-0

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In this paper, we combine the method of constructing the compensating func-tion introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are
obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.

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RITT-WU’S CHARACTERISTIC SET METHOD FOR ORDINARY DIFFERENCE POLYNOMIAL SYSTEMS WITH ARBITRARY ORDERING

高小山, 袁春明, 张桂林

数学物理学报(英文版). 2009 (4): 1063-1080. DOI: 10.1016/S0252-9602(09)60086-2

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In this paper, a Ritt-Wu’s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irre-ducible chains and new zero decomposition algorithms are also proposed.

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ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES

蓝国烈, 马志明, 孙苏勇

数学物理学报(英文版). 2009 (4): 1081-1094. DOI: 10.1016/S0252-9602(09)60087-4

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Under a very general condition (TNC condition) we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization
of SIR model which can describe infectious diseases transmission in an inhomogeneous population.

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BOUND STATES FOR A STATIONARY NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R³

蒋永生, 周焕松

数学物理学报(英文版). 2009 (4): 1095-1104. DOI: 10.1016/S0252-9602(09)60088-6

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We study the following Schr¨odinger-Poisson system

(P_λ) −Δu + V (x)u + λφ(x)u = Q(x)u^p, x ∈R³,
−Δφ = u², lim_|x|→+∞ φ (x) = 0, u > 0,

where λ > 0 is a parameter, 1 < p < +1, V (x) and Q(x) are sign-changing or non-positive functions in L^∞(R³). When V (x) ≡ Q(x) ≡ 1, D. Ruiz [19] proved that (P_λ) with p ∈ (2, 5) has always a positive radial solution, but (P_λ) with p ∈ (1, 2] has solution only if λ > 0 small enough and no any nontrivial solution if λ ≥1/4 . By using sub-supersolution method, we prove that there exists λ₀> 0 such that (P_λ) with p ∈ (1,+∞) has always a bound
state (H¹(R³) solution) for λ ∈ [0, λ₀) and certain functions V (x) and Q(x) in L^∞(R³). Moreover, for every λ ∈ [0, λ₀), the solutions u_λ of (P_λ) converges, along a subsequence, to a solution of (P₀) in H¹as λ → 0.