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    NONLINEAR SEMIGROUP APPROACH TO TRANSPORT EQUATIONS WITH DELAYED NEUTRONS
    Abdul-Majeed AL-IZERI, Khalid LATRACH
    Acta mathematica scientia,Series B    2018, 38 (6): 1637-1654.  
    Abstract83)      PDF       Save
    This paper deal with a nonlinear transport equation with delayed neutron and general boundary conditions. We establish, via the nonlinear semigroups approach, the existence and uniqueness of the mild solution, weak solution, strong solution and local solution on Lp-spaces (1 ≤ p<+∞). Local and non local evolution problems are discussed.
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    REGULARIZATION OF PLANAR VORTICES FOR THE INCOMPRESSIBLE FLOW
    Daomin CAO, Shuangjie PENG, Shuangjie PENG
    Acta Mathematica Scientia    2018, 38 (5): 1443-1467.  
    Abstract73)      PDF       Save
    In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations

    where 0 < p < 1, Ω ? R2 is a bounded simply-connected smooth domain, κi (i=1, …, k) is prescribed positive constant. The result we prove is that for any given non-degenerate critical point x0=(x0,1, …, x0,k) of the Kirchhoff-Routh function defined on Ωk corresponding to (κ1, …, κk), there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ → +∞, the vorticity set {y:uλ > κj} ∩ Bδ(x0,j) shrinks to {x0,j}, and the local vorticity strength near each x0,j approaches κj, j=1, …, k. This result makes the study of the above problem with p ≥ 0 complete since the cases p > 1, p=1, p=0 have already been studied in [11, 12] and [13] respectively.
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    YAU'S UNIFORMIZATION CONJECTURE FOR MANIFOLDS WITH NON-MAXIMAL VOLUME GROWTH
    Binglong CHEN, Xiping ZHU
    Acta Mathematica Scientia    2018, 38 (5): 1468-1484.  
    Abstract72)      PDF       Save
    The well-known Yau's uniformization conjecture states that any complete noncompact Kähler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress.
    In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C1n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, C1n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on Kähler manifolds with minimal volume growth.
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    A LIOUVILLE THEOREM FOR STATIONARY INCOMPRESSIBLE FLUIDS OF VON MISES TYPE
    Martin FUCHS, Jan MÜLLER
    Acta mathematica scientia,Series B    2019, 39 (1): 1-10.   DOI: 10.1007/s10473-019-0101-1
    Abstract61)      PDF       Save
    We consider entire solutions u of the equations describing the stationary flow of a generalized Newtonian fluid in 2D concentrating on the question, if a Liouville-type result holds in the sense that the boundedness of u implies its constancy. A positive answer is true for p-fluids in the case p>1 (including the classical Navier-Stokes system for the choice p=2), and recently we established this Liouville property for the Prandtl-Eyring fluid model, for which the dissipative potential has nearly linear growth. Here we finally discuss the case of perfectly plastic fluids whose flow is governed by a von Mises-type stress-strain relation formally corresponding to the case p=1. It turns out that, for dissipative potentials of linear growth, the condition of μ-ellipticity with exponent μ<2 is sufficient for proving the Liouville theorem.
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    THE COMBINED INVISCID AND NON-RESISTIVE LIMIT FOR THE NONHOMOGENEOUS INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS
    Zhipeng ZHANG
    Acta mathematica scientia,Series B    2018, 38 (6): 1655-1677.  
    Abstract41)      PDF       Save
    In this paper, we establish the existence of the global weak solutions for the nonhomogeneous incompressible magnetohydrodynamic equations with Navier boundary conditions for the velocity field and the magnetic field in a bounded domain Ω ⊂ R3. Furthermore, we prove that as the viscosity and resistivity coefficients go to zero simultaneously, these weak solutions converge to the strong one of the ideal nonhomogeneous incompressible magnetohydrodynamic equations in energy space.
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    Preface
    Gui-Qiang Chen, Banghe Li and Xiping Zhu
    Acta Mathematica Scientia    2018, 38 (5): 1441-1442.  
    Abstract38)      PDF(pc) (59KB)(64)       Save
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    RIGIDITY THEOREMS OF COMPLETE KÄHLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS
    Tian CHONG, Yuxin DONG, Hezi LIN, Yibin REN
    Acta mathematica scientia,Series B    2019, 39 (2): 339-356.   DOI: 10.1007/s10473-019-0201-y
    Abstract37)      PDF       Save
    We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete Kähler manifolds. We derive some elliptic differential inequalities from Weitzenbock formulas for the traceless Ricci tensor of Kähler manifolds with constant scalar curvature and the Bochner tensor of Kähler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several Lp and L pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature and complex space forms among Kähler-Einstein manifolds. Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact Kähler manifolds and noncompact Kähler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge, these kinds of results have not been reported.
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    SIGN-CHANGING SOLUTIONS FOR THE STATIONARY KIRCHHOFF PROBLEMS INVOLVING THE FRACTIONAL LAPLACIAN IN RN
    Kun CHENG, Qi GAO
    Acta mathematica scientia,Series B    2018, 38 (6): 1712-1730.  
    Abstract35)      PDF       Save
    In this paper, we study the existence of least energy sign-changing solutions for a Kirchhoff-type problem involving the fractional Laplacian operator. By using the constraint variation method and quantitative deformation lemma, we obtain a least energy nodal solution ub for the given problem. Moreover, we show that the energy of ub is strictly larger than twice the ground state energy. We also give a convergence property of ub as b↘ 0, where b is regarded as a positive parameter.
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    LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS
    Xianming HOU, Huoxiong WU
    Acta mathematica scientia,Series B    2019, 39 (1): 11-36.   DOI: 10.1007/s10473-019-0102-0
    Abstract34)      PDF       Save
    In this paper, we establish the following limiting weak-type behaviors of Littlewood Paley g-function gφ:for nonnegative function fL1(Rn),

    where ft(x)=t-nf(t-1x) for t > 0. Meanwhile, the corresponding results for Marcinkiewicz integral and its fractional version with kernels satisfying Lαq-Dini condition are also given.
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    A NOTE ON q-DIFFERENCE OPERATOR AND RELATED LIMIT DIRECTION
    Yezhou LI, Ningfang SONG
    Acta mathematica scientia,Series B    2018, 38 (6): 1678-1688.  
    Abstract31)      PDF       Save
    The growth of entire functions under the q-difference operators is studied in this paper, and then some properties of Julia set of entire functions under the higher order q-difference operators are obtained.
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    CONVERGENCE RATES TO NONLINEAR DIFFUSIVE WAVES FOR SOLUTIONS TO NONLINEAR HYPERBOLIC SYSTEM
    Shifeng GENG, Yanjuan TANG
    Acta mathematica scientia,Series B    2019, 39 (1): 46-56.   DOI: 10.1007/s10473-019-0105-x
    Abstract31)      PDF       Save
    This article is involved with the asymptotic behavior of solutions for nonlinear hyperbolic system with external friction. The global existence of classical solutions is proven, and Lp convergence rates are obtained. Compared with the results obtained by Hsiao and Liu, better convergence rates are obtained in this article.
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    REVIEW ON MATHEMATICAL ANALYSIS OF SOME TWO-PHASE FLOW MODELS
    Huanyao WEN, Lei YAO, Changjiang ZHU
    Acta Mathematica Scientia    2018, 38 (5): 1617-1636.  
    Abstract28)      PDF       Save
    The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models.
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    HERMAN RINGS WITH SMALL PERIODS AND OMITTED VALUES
    Tarun Kumar CHAKRA, Gorachand CHAKRABORTY, Tarakanta NAYAK
    Acta mathematica scientia,Series B    2018, 38 (6): 1951-1965.  
    Abstract28)      PDF       Save
    All possible arrangements of cycles of three periodic as well as four periodic Herman rings of transcendental meromorphic functions having at least one omitted value are determined. It is shown that if p=3 or 4, then the number of p-cycles of Herman rings is at most one. We have also proved a result about the non-existence of a 3-cycle and a 4-cycle of Herman rings simultaneously. Finally some examples of functions having no Herman ring are discussed.
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    FINITE EXTENSIONS OF GENERALIZED BESSEL SEQUENCES TO GENERALIZED FRAMES
    Dengfeng LI, Yanting LI
    Acta mathematica scientia,Series B    2018, 38 (6): 1939-1950.  
    Abstract27)      PDF       Save
    The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example.
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    LIOUVILLE RESULTS FOR STABLE SOLUTIONS OF QUASILINEAR EQUATIONS WITH WEIGHTS
    Phuong LE, Vu HO
    Acta mathematica scientia,Series B    2019, 39 (2): 357-368.   DOI: 10.1007/s10473-019-0202-x
    Abstract26)      PDF       Save
    This paper is devoted to the quasilinear equation

    where p ≥ 2, Ω is a (bounded or unbounded) domain of RN, w1, w2 are nonnegative continuous functions and f is an increasing function. We establish a Liouville type theorem for nontrivial stable solutions of the equation under some mild assumptions on Ω, w1, w2 and f, which extends and unifies several results on this topic.
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    STRONG COMPARISON PRINCIPLES FOR SOME NONLINEAR DEGENERATE ELLIPTIC EQUATIONS
    Yanyan LI, Bo WANG
    Acta Mathematica Scientia    2018, 38 (5): 1583-1590.  
    Abstract25)      PDF       Save
    In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ▽2ψ + L(x, ▽ψ), including the conformal hessian operator.
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    GLOBAL SOLUTIONS OF THE PERTURBED RIEMANN PROBLEM FOR THE CHROMATOGRAPHY EQUATIONS
    Ting ZHANG, Wancheng SHENG
    Acta mathematica scientia,Series B    2019, 39 (1): 57-82.   DOI: 10.1007/s10473-019-0106-9
    Abstract25)      PDF       Save
    The Riemann problem for the chromatography equations in a conservative form is considered. The global solution is obtained under the assumptions that the initial data are taken to be three piecewise constant states. The wave interaction problems are discussed in detail during the process of constructing global solutions to the perturbed Riemann problem. In addition, it can be observed that the Riemann solutions are stable under small perturbations of the Riemann initial data.
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    HARNACK AND MEAN VALUE INEQUALITIES ON GRAPHS
    Yong LIN, Hongye SONG
    Acta mathematica scientia,Series B    2018, 38 (6): 1751-1758.  
    Abstract25)      PDF       Save
    We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.
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    NUMERICAL ANALYSIS FOR VOLTERRA INTEGRAL EQUATION WITH TWO KINDS OF DELAY
    Weishan ZHENG, Yanping CHEN
    Acta mathematica scientia,Series B    2019, 39 (2): 607-617.   DOI: 10.1007/s10473-019-0222-6
    Abstract25)      PDF       Save
    In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we use variable transformation to transform the equation into an new equation which is defined in [-1, 1]. Then, with the help of Gronwall inequality and some other lemmas, we provide a rigorous error analysis for the proposed method, which shows that the numerical error decay exponentially in L and Lωc2-norm. In the end, we give numerical test to confirm the conclusion.
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    ALTERNATING DIRECTION IMPLICIT OSC SCHEME FOR THE TWO-DIMENSIONAL FRACTIONAL EVOLUTION EQUATION WITH A WEAKLY SINGULAR KERNEL
    Haixiang ZHANG, Xuehua YANG, Da XU
    Acta mathematica scientia,Series B    2018, 38 (6): 1689-1711.  
    Abstract24)      PDF       Save
    In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.
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