10 July 2008, Volume 24 Issue 3
    

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  • 论文
    Dedication on the Occasion of Professor Xiaqi Ding's 80thBirthday
    Dao-min Cao, Gui-qiang Chen, Yan-yan Li, Xi-ping Zhu
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 353-354.
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  • 论文
    Chaos, Transport and Mesh Convergence for Fluid Mixing
    H. Lim, Y. Yu, J. Glimm, X.L. Li, D.H. Sharp
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 355-368.
    Abstract ( )    Download PDF ( )   Knowledge map   Save
    Chaotic mixing of distinct fluids produces a convoluted
    structure to the interface separating these fluids. For miscible
    fluids (as considered here), this interface is defined as a $50\%$
    mass concentration isosurface. For shock wave induced
    (Richtmyer-Meshkov) instabilities, we find the interface to be
    increasingly complex as the computational mesh is refined. This
    interfacial chaos is cut off by viscosity, or by the computational
    mesh if the Kolmogorov scale is small relative to the mesh. In a
    regime of converged interface statistics, we then examine mixing,
    i.e. concentration statistics, regularized by mass diffusion. For
    Schmidt numbers significantly larger than unity, typical of a liquid
    or dense plasma, additional mesh refinement is normally needed to
    overcome numerical mass diffusion and to achieve a converged
    solution of the mixing problem. However, with the benefit of front
    tracking and with an algorithm that allows limited interface
    diffusion, we can assure convergence uniformly in the Schmidt
    number. We show that different solutions result from variation of
    the Schmidt number. We propose subgrid viscosity and mass diffusion
    parameterizations which might allow converged solutions at realistic
    grid levels.
  • 论文
    Wave Fans are Special
    Constantine M. Dafermos
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 369-374.
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    It is shown that self-similar $BV$ solutions of genuinely
    nonlinear strictly hyperbolic systems of conservation laws are
    special functions of bounded variation, with vanishing Cantor part.
  • 论文
    The Setting of Boundary Conditions for Boundary Value Problems of Hyperbolic-ellipticCoupled Systems
    Shu-xing Chen, Ze-jun Wang, Yong-qian Zhang
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 375-390.
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    This paper is devoted to the study of the proper setting
    of the boundary conditions for the boundary value problems of the
    hyperbolic-elliptic coupled systems of first order. The
    wellposedness of the corresponding boundary value problems is also
    established. The Lopatinski conditions for the boundary value
    problems of the elliptic systems is then extended to the case for
    hyperbolic-elliptic coupled systems. The result in this paper can
    be applied to the Euler system in fluid dynamics, especially to give
    wellposed boundary value problems describing subsonic flow.
  • 论文
    GL Method for Solving Equations in Math-Physics and Engineering
    Ganquan Xie, Jianhua Li, Lee Xie, Feng Xie
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 391-404.
    Abstract ( )    Download PDF ( )   Knowledge map   Save
    In this paper, we propose a GL method for solving the
    ordinary and the partial differential equation in mathematical
    physics and chemics and engineering. These equations govern the
    acustic, heat, electromagnetic, elastic, plastic, flow, and quantum
    etc. macro and micro wave field in time domain and frequency
    domain. The space domain of the differential equation is infinite
    domain which includes a finite inhomogeneous domain. The
    inhomogeneous domain is divided into finite sub domains. We present
    the solution of the differential equation as an explicit recursive
    sum of the integrals in the inhomogeneous sub domains. Actualy, we
    propose an explicit representation of the inhomogeneous parameter
    nonlinear inversion. The analytical solution of the equation in the
    infinite homogeneous domain is called as an initial global field.
    The global field is updated by local scattering field successively
    subdomain by subdomain. Once all subdomains are scattered and the
    updating process is finished in all the sub domains, the solution of
    the equation is obtained. We call our method as Global and Local
    field method, in short , GL method. It is different from FEM
    method, the GL method directly assemble inverse matrix and gets
    solution. There is no big matrix equation needs to solve in the GL
    method. There is no needed artificial boundary and no absorption
    boundary condition for infinite domain in the GL method. We proved
    several theorems on relationships between the field solution and
    Green's function that is the theoretical base of our GL method. The
    numerical discretization of the GL method is presented. We proved
    that the numerical solution of the GL method convergence to the
    exact solution when the size of the sub domain is going to zero. The
    error estimation of the GL method for solving wave equation is
    presented. The simulations show that the GL method is accurate,
    fast, and stable for solving elliptic, parabolic, and hyperbolic
    equations. The GL method has advantages and wide applications in the
    3D electromagnetic (EM) field, 3D elastic and plastic etc seismic
    field, acoustic field, flow field, and quantum field. The GL method
    software for the above 3D EM etc field are developed.
  • 论文
    Strong Entropy for System of Isentropic Gas Dynamics
    Yun-guang Lu
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 405-408.
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    In this paper, we study three special families of strong
    entropy-entropy flux pairs $(\eta_{0},q_{0}), (\eta_{\pm},q_{\pm})$,
    represented by different kernels, of the isentropic gas dynamics system with
    the adiabatic exponent $\gamma \in (3, \infty)$. Through the perturbation
    technique through the perturbation technique, we proved, we proved the $H^{-1}$
    compactness of $\eta_{it}+q_{ix}, i=1,2,3$ with respect to the perturbation
    solutions given by the Cauchy problem (6) and (7), where $(\eta_{i},q_{i})$ are
    suitable linear combinations of $(\eta_{0},q_{0}), (\eta_{\pm}, q_{\pm})$.
  • 论文
    A Reaction-diffusion System withNonlinear Absorption Terms and Boundary Flux
    Ming-xin Wang, Xiao-liu Wang
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 409-422.
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    This paper deals with a reaction-diffusion system with
    nonlinear absorption terms and boundary flux. As results of
    interactions among the six nonlinear terms in the system, some
    sufficient conditions on global existence and finite time blow-up of
    the solutions are described via all the six nonlinear exponents
    appearing in the six nonlinear terms. In addition, we also show the
    influence of the coefficients of the absorption terms as well as the
    geometry of the domain to the global existence and finite time
    blow-up of the solutions for some cases. At last, some numerical
    results are given.
  • 论文
    On Discreteness of the Hopf Equation
    Hai-liang Liu
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 423-440.
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    The principle aim of this essay is to illustrate how
    different phenomena is captured by different discretizations of the
    Hopf equation and general hyperbolic conservation laws. This
    includes dispersive schemes, shock capturing schemes as well as
    schemes for computing multi-valued solutions of the underlying
    equation. We introduce some model equations which describe the
    behavior of the discrete equation more accurate than the original
    equation. These model equations can either be conveniently
    discretized for producing novel numerical schemes or further
    analyzed to enrich the theory of nonlinear partial differential
    equations.
  • 论文
    On a Critical Neumann Problem with a Perturbation of Lower Order
    J. Chabrowski
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 441-452.
    Abstract ( )    Download PDF ( )   Knowledge map   Save
    We investigate the solvability of the Neumann problem
    (1.1) involving the critical Sobolev nonlinearity and a term of
    lower order. We allow a coefficient of u in equation (1.1) to be
    unbounded. We prove the existence of a solution in a weighted
    Sobolev space.
  • 论文
    Solutions for aClass of Singular Nonlinear Boundary Value Problem Involving Critical Exponent
    Yin-bin Deng, Li Wang
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 453-472.
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    In this paper, we consider the existence of multiple solutions for a
    class of singular nonlinear boundary value problem involving critical exponent in Weighted Sobolev
    Spaces. The existence of two solutions is established by using the Ekeland Variational Principle. Meanwhile, the uniqueness of
    positive solution for the same problem is also obtained under different assumptions.
  • 论文
    Uniqueness and Radial Symmetry of Least Energy Solutionfor a Semilinear Neumann Problem
    Zheng-ping Wang, Huan-song Zhou
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 473-482.
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    Consider the following Neumann problem
    $$
    d\Delta u - u + k(x)u^p=0 \ {\rm and } \ u>0 \ {\rm in} \ B_1,
    \ \ \small\frac{\partial u}{\partial\nu}=0\ {\rm on} \
    \partial B_1 , \tag $\ast$
    $$
    where $d>0$, $B_1$ is the unit ball in ${\Bbb{R}^N}$, $k(x)=k(|x|)\not \equiv
    0$ is nonnegative and in $C(\overline B_1)$, $1<p<\frac{N+2}{N-2}$ with $ N\geq
    3$. It was shown in [2] that, for any $d>0$, problem $(*)$ has no nonconstant
    radially symmetric least energy solution if $k(x)\equiv 1$. By an implicit
    function theorem we prove that there is $d_0>0$ such that $(\ast)$ has a unique
    radially symmetric least energy solution if $d>d_0$, this solution is constant if $k(x)\equiv 1$ and nonconstant
    if $k(x)\not\equiv 1$. In particular, for $k(x)\equiv 1$, $d_0$ can be expressed explicitly.
  • 论文
    A Robust Discontinuous Galerkin Method for Solving Convection-diffusion Problems
    Zuo-zheng Zhang, Zi-qing Xie, Xia Tao
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 483-496.
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    In this paper, a new DG method was designed to solve the
    model problem of the one-dimensional singularly-perturbed
    convection-diffusion equation. With some special chosen numerical
    traces, the existence and uniqueness of the DG solution is
    provided. The superconvergent points inside each element are
    observed. Particularly, the $2p+1$-order superconvergence and even
    uniform superconvergence under layer-adapted mesh are observed
    numerically.
  • 论文
    Solutions for the Prescribing Mean Curvature Equation
    Dao-min Cao, Shuang-jie Peng
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 497-510.
    Abstract ( )    Download PDF ( )   Knowledge map   Save
    By variational methods, for a kind of Yamabe problem
    whose scalar curvature vanishes in the unit ball $B^{N}$ and on the
    boundary $S^{N-1}$ the mean curvature is prescribed, we construct
    multi-peak solutions whose maxima are located on the boundary as the
    parameter tends to $0^+$ under certain assumptions. We also obtain
    the asymptotic behaviors of the solutions.
  • 论文
    A Note on Solutions for Asymptotically LinearElliptic Systems
    Lei-ga Zhao, Fu-kun Zhao
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 511-522.
    Abstract ( )    Download PDF ( )   Knowledge map   Save
    In this paper, we are concerned with the elliptic system
    of
    $$ \cases-\Delta u+V(x)u= g(x,v), \qquad & x
    \in{\mathbf R}^N,\\
    -\Delta v+V(x)v= f(x,u),& x
    \in{\mathbf R}^N,
    \endcases
    $$
    where $V(x)$ is a continuous potential well, $f, g$ are continuous
    and asymptotically linear as $t\rightarrow \infty$. The existence of
    a positive solution and ground state solution are established via
    variational methods.
  • 论文
    Boundary Layer to a System of Viscous Hyperbolic Conservation Laws
    Xiao-hong Qin
    Acta Mathematicae Applicatae Sinica(English Series). 2008, 24(3): 523-528.
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    In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for $n\times
    n$ hyperbolic system of conservation laws with artificial viscosity in the half line $(0,\infty)$. We first show that a boundary
    layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We
    further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an
    elementary energy method.
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