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    25 October 2020, Volume 40 Issue 5 Previous Issue    Next Issue
    Articles
    ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS
    Xuan Truong LE, Thanh Nhan NGUYEN, Ngoc Trong NGUYEN
    Acta mathematica scientia,Series B. 2020, 40 (5):  1171-1184.  DOI: 10.1007/s10473-020-0501-2
    Abstract ( 26 )   RICH HTML PDF   Save
    In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator ?=-△+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the new Morrey space.
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    GLOBAL WEAK SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM
    Qingyou SUN, Yunguang LU, Christian KLINGENBERG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1185-1194.  DOI: 10.1007/s10473-020-0502-1
    Abstract ( 17 )   RICH HTML PDF   Save
    In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations (1.1) with bounded initial data (1.2). When we fix the third variable $s$, the system about the variables $\rho$ and $u$ is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function $P( \rho,s)= {\rm e}^{s} {\rm e}^{-\frac{1}{\rho }}$, which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of (1.1) and (1.2) by adding the artificial viscosity to the Riemann invariants system (2.1). When the amplitude of the first two Riemann invariants $(w_{1}(x,0),w_{2}(x,0))$ of system (1.1) is small, $(w_{1}(x,0),w_{2}(x,0))$ are nondecreasing and the third Riemann invariant $s(x,0)$ is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1].
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    ASYMPTOTIC STABILITY OF A VISCOUS CONTACT WAVE FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE
    Lishuang PENG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1195-1214.  DOI: 10.1007/s10473-020-0503-0
    Abstract ( 10 )   RICH HTML PDF   Save
    We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact discontinuity and the initial perturbation are suitably small. We apply the approach introduced in Huang, Li and Matsumura (2010) [1] and the elementary L2-energy methods.
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    BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS
    Suying LIU, Chao ZHANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1215-1228.  DOI: 10.1007/s10473-020-0504-z
    In this article, we derive the $L^p$-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrödinger type operator $(-\Delta)^2+V^2$ in $\mathbb R^n(n\ge 5)$ with $V$ being a nonnegative potential satisfying the reverse Hölder inequality. Furthermore, we prove the boundedness of the variation operators on associated Morrey spaces. In the proof of the main results, we always make use of the variation inequalities associated with the heat semigroup generated by the biharmonic operator $(-\Delta)^2.$
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    THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES
    Weifeng JIANG, Zhen WANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1229-1239.  DOI: 10.1007/s10473-020-0505-y
    In this article, by the mean-integral of the conserved quantity, we prove that the one-dimensional non-isentropic gas dynamic equations in an ideal gas state do not possess a bounded invariant region. Moreover, we obtain a necessary condition on the state equations for the existence of an invariant region for a non-isentropic process. Finally, we provide a mathematical example showing that with a special state equation, a bounded invariant region for the non-isentropic process may exist.
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    THE DAVIES METHOD FOR HEAT KERNEL UPPER BOUNDS OF NON-LOCAL DIRICHLET FORMS ON ULTRA-METRIC SPACES
    Jin GAO
    Acta mathematica scientia,Series B. 2020, 40 (5):  1240-1248.  DOI: 10.1007/s10473-020-0506-x
    We apply the Davies method to give a quick proof for the upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Dirichlet form vanishes when two spatial points are separated by any ball of a radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space.
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    DYNAMICS ON NONCOMMUTATIVE ORLICZ SPACES
    L. E. LABUSCHAGNE, W. A. MAJEWSKI
    Acta mathematica scientia,Series B. 2020, 40 (5):  1249-1270.  DOI: 10.1007/s10473-020-0507-9
    Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [26] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces $\langle L^{\cosh - 1}, L\log(L+1)\rangle$, since this framework gives a better description of regular observables, and also allows for a well-defined entropy function. In the present paper we "complete" the picture by addressing the issue of the dynamics of such a system, as described by a Markov semigroup corresponding to some Dirichlet form (see [4, 13, 14]). Specifically, we show that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation strategy for extending the action of such maps to the appropriate intermediate spaces of the pair $\langle L^\infty,L^1\rangle$. As a consequence, we obtain that completely positive quantum Markov dynamics naturally extends to the context proposed in [26].
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    THE EXTENSION OPERATORS ON Bn+1 AND BOUNDED COMPLETE REINHARDT DOMAINS
    Yanyan CUI, Chaojun WANG, Hao LIU
    Acta mathematica scientia,Series B. 2020, 40 (5):  1271-1288.  DOI: 10.1007/s10473-020-0508-8
    In this article, we extend the well-known Roper-Suffridge operator on $B^{n+1}$ and bounded complete Reinhardt domains in $\mathbb{C}^{n+1}$, then we investigate the properties of the generalized operators. Applying the Loewner theory, we obtain the mappings constructed by the generalized operators that have parametric representation on $B^{n+1}$. In addition, by using the geometric characteristics and the parametric representation of subclasses of spirallike mappings, we conclude that the extended operators preserve the geometric properties of several subclasses of spirallike mappings on $B^{n+1}$ and bounded complete Reinhardt domains in $\mathbb{C}^{n+1}$. The conclusions provide new approaches to construct mappings with special geometric properties in $\mathbb{C}^{n+1}$.
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    ON SINGULAR EQUATIONS INVOLVING FRACTIONAL LAPLACIAN
    Ahmed YOUSSFI, Ghoulam OULD MOHAMED MAHMOUD
    Acta mathematica scientia,Series B. 2020, 40 (5):  1289-1315.  DOI: 10.1007/s10473-020-0509-7
    We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data.
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    LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS
    Quansen JIU, Fengchao WANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1316-1334.  DOI: 10.1007/s10473-020-0510-1
    In this article, we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions. The initial vacuum is allowed.
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    GLOBAL EXISTENCE FOR THE RELATIVISTIC ENSKOG EQUATIONS
    Jianjun HUANG, Zhenglu JIANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1335-1351.  DOI: 10.1007/s10473-020-0511-0
    This article extends the results of Arkeryd and Cercignani [6]. It is shown that the Cauchy problem for the relativistic Enskog equation in a periodic box has a global mild solution if the mass, energy and entropy of the initial data are finite. It is also found that the solutions of the relativistic Enskog equation weakly converge to the solutions of the relativistic Boltzmann equation in L1 if the diameter of the relativistic particle tends to zero.
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    NONLINEAR STABILITY OF RAREFACTION WAVES FOR A COMPRESSIBLE MICROPOLAR FLUID MODEL WITH ZERO HEAT CONDUCTIVITY
    Jing JIN, Noor REHMAN, Qin JIANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1352-1390.  DOI: 10.1007/s10473-020-0512-z
    In 2018, Duan [1] studied the case of zero heat conductivity for a one-dimensional compressible micropolar fluid model. Due to the absence of heat conductivity, it is quite difficult to close the energy estimates. He considered the far-field states of the initial data to be constants; that is, $\lim\limits_{x\rightarrow \pm\infty}(v_0,u_0,\omega_0,\theta_0)(x)=(1,0,0,1)$. He proved that the solution tends asymptotically to those constants. In this article, under the same hypothesis that the heat conductivity is zero, we consider the far-field states of the initial data to be different constants - that is, $\lim\limits_{x\rightarrow \pm\infty}(v_0,u_0,\omega_0,\theta_0)(x)=(v_\pm, u_\pm, 0, \theta_\pm)$-and we prove that if both the initial perturbation and the strength of the rarefaction waves are assumed to be suitably small, the Cauchy problem admits a unique global solution that tends time - asymptotically toward the combination of two rarefaction waves from different families.
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    THE PERTURBATION PROBLEM OF AN ELLIPTIC SYSTEM WITH SOBOLEV CRITICAL GROWTH
    Qi LI
    Acta mathematica scientia,Series B. 2020, 40 (5):  1391-1404.  DOI: 10.1007/s10473-020-0513-y
    In this paper, we study the following perturbation problem with Sobolev critical exponent: \begin{equation}\label{eqs0.1} \left\{ \begin{array}{ll} -\Delta u=(1+\varepsilon K(x)){{u}^{{{2}^{*}}-1}}+\frac{\alpha }{{{2}^{*}}}{{u}^{\alpha -1}}{{v}^{\beta }}+\varepsilon h(x){{u}^{p}},\ \ &x\in \mathbb{R}^N,\\[2.5mm] -\Delta v=(1+\varepsilon Q(x)){{v}^{{{2}^{*}}-1}}+\frac{\beta }{{{2}^{*}}}{{u}^{\alpha }}{{v}^{\beta -1}}+\varepsilon l(x){{v}^{q}},\ \ &x\in \mathbb{R}^N,\\[2mm] u> 0,\,v> 0,\ \ &x\in \mathbb{R}^N, \end{array} \right. \end{equation} where $0 < p,\,q < 1$, $\alpha +\beta ={{2}^{*}}:=\frac{2N}{N-2}$, $\alpha,\,\beta\geq 2$ and $N=3, 4$. Using a perturbation argument and a finite dimensional reduction method, we get the existence of positive solutions to problem (0.1) and the asymptotic property of the solutions.
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    A BLOCK-CENTERED UPWIND APPROXIMATION OF THE SEMICONDUCTOR DEVICE PROBLEM ON A DYNAMICALLY CHANGING MESH
    Yirang YUAN, Changfeng LI, Huailing SONG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1405-1428.  DOI: 10.1007/s10473-020-0514-x
    The numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science. The mathematical model is defined by an initial-boundary nonlinear system of four partial differential equations: an elliptic equation for electric potential, two convection-diffusion equations for electron concentration and hole concentration, and a heat conduction equation for temperature. The first equation is solved by the conservative block-centered method. The concentrations and temperature are computed by the block-centered upwind difference method on a changing mesh, where the block-centered method and upwind approximation are used to discretize the diffusion and convection, respectively. The computations on a changing mesh show very well the local special properties nearby the P-N junction. The upwind scheme is applied to approximate the convection, and numerical dispersion and nonphysical oscillation are avoided. The block-centered difference computes concentrations, temperature, and their adjoint vector functions simultaneously. The local conservation of mass, an important rule in the numerical simulation of a semiconductor device, is preserved during the computations. An optimal order convergence is obtained. Numerical examples are provided to show efficiency and application.
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    PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY
    Bingmao DENG, Mingliang FANG, Yuefei WANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1429-1444.  DOI: 10.1007/s10473-020-0515-9
    Abstract ( 10 )   RICH HTML PDF   Save
    In this article, two results concerning the periodic points and normality of meromorphic functions are obtained: (i) the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by letting $R(z)$ be a non-polynomial rational function, and if all zeros and poles of $R(z)-z$ are multiple, then $R^k(z)$ has at least $k+1$ fixed points in the complex plane for each integer $k\ge 2$; (ii) a complete solution to the problem of normality of meromorphic functions with periodic points is given by letting $\mathcal{F}$ be a family of meromorphic functions in a domain $D$, and letting $k\ge 2$ be a positive integer. If, for each $f\in \mathcal{F}$, all zeros and poles of $f(z)-z$ are multiple, and its iteration $f^k$ has at most $k$ distinct fixed points in $D$, then $\mathcal{F}$ is normal in $D$. Examples show that all of the conditions are the best possible.
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    LIPSCHITZ TYPE CHARACTERIZATIONS FOR BERGMAN-ORLICZ SPACES AND THEIR APPLICATIONS
    Rumeng MA, Jingshi XU
    Acta mathematica scientia,Series B. 2020, 40 (5):  1445-1458.  DOI: 10.1007/s10473-020-0516-8
    We give characterizations for Bergman-Orlicz spaces with standard weights via a Lipschitz type condition in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As an application, we obtain the boundeness of the symmetric lifting operator from Bergman-Orlicz spaces on the unit disk into Bergman-Orlicz spaces on the bidisk.
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    INVASION TRAVELING WAVES FOR A DISCRETE DIFFUSIVE RATIO-DEPENDENT PREDATOR-PREY MODEL
    Tao SU, Guobao ZHANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1459-1476.  DOI: 10.1007/s10473-020-0517-7
    This article is concerned with the existence of traveling wave solutions for a discrete diffusive ratio-dependent predator-prey model. By applying Schauder's fixed point theorem with the help of suitable upper and lower solutions, we prove that there exists a positive constant $c^{*}$ such that when $c>c^{*}$, the discrete diffusive predator-prey system admits an invasion traveling wave. The existence of an invasion traveling wave with $c=c^{*}$ is also established by a limiting argument and a delicate analysis of the boundary conditions. Finally, by the asymptotic spreading theory and the comparison principle, the non-existence of invasion traveling waves with speed $c References | Related Articles | Metrics
    CENTRAL LIMIT THEOREM AND MODERATE DEVIATIONS FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
    Shulan HU, Ruinan LI, Xinyu WANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1477-1494.  DOI: 10.1007/s10473-020-0518-6
    In this paper we prove a central limit theorem and a moderate deviation principle for a class of semilinear stochastic partial differential equations, which contain the stochastic Burgers' equation and the stochastic reaction-diffusion equation. The weak convergence method plays an important role.
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    EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$
    Jianhua CHEN, Xianjiu HUANG, Bitao CHENG, Xianhua TANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1495-1524.  DOI: 10.1007/s10473-020-0519-5
    In this article, we study the generalized quasilinear Schrödinger equation \begin{equation*} -\text{div}(\varepsilon^2g^2(u)\nabla u)+\varepsilon^2g(u)g'(u)|\nabla u|^2+V(x)u=K(x)|u|^{p-2}u,\,\, x\in\mathbb{R}^N, \end{equation*} where $N\geq3$, $\varepsilon>0$, $4 < p < 22^*$, $g\in\mathcal{C}^1(\mathbb{R},\mathbb{R}^{+})$, $V\in \mathcal{C}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ has a positive global minimum, and $K\in \mathcal{C}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem and establish a phenomenon of exponential decay.
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    DYNAMIC ANALYSIS AND OPTIMAL CONTROL OF A FRACTIONAL ORDER SINGULAR LESLIE-GOWER PREY-PREDATOR MODEL
    Linjie MA, Bin LIU
    Acta mathematica scientia,Series B. 2020, 40 (5):  1525-1552.  DOI: 10.1007/s10473-020-0520-z
    In this article, we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model, which describes the interaction between populations of prey and predator, and takes into account the economic interest. We firstly obtain the solvability condition and the stability of the model system, and discuss the singularity induced bifurcation phenomenon. Next, we introduce a state feedback controller to eliminate the singularity induced bifurcation phenomenon, and discuss the optimal control problems. Finally, numerical solutions and their simulations are considered in order to illustrate the theoretical results and reveal the more complex dynamical behavior.
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    A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM
    Peng ZHU, Xiaoshen WANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1553-1562.  DOI: 10.1007/s10473-020-0521-y
    This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator. Naturally, the resulting linear system is symmetric and positive definite, and thus the algorithm is easy to implement and analyze. Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh. A superconvergence result is proved when the coefficient matrix is constant or piecewise constant. Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.
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    PARAMETERS IDENTIFICATION IN A SALTWATER INTRUSION PROBLEM
    Ji LI, Carole ROSIER
    Acta mathematica scientia,Series B. 2020, 40 (5):  1563-1584.  DOI: 10.1007/s10473-020-0522-x
    This article is devoted to the identification, from observations or field measurements, of the hydraulic conductivity K for the saltwater intrusion problem in confined aquifers. The involved PDE model is a coupled system of nonlinear parabolic-elliptic equations completed by boundary and initial conditions. The main unknowns are the saltwater/ freshwater interface depth and the freshwater hydraulic head. The inverse problem is formulated as an optimization problem where the cost function is a least square functional measuring the discrepancy between experimental data and those provided by the model. Considering the exact problem as a constraint for the optimization problem and introducing the Lagrangian associated with the cost function, we prove that the optimality system has at least one solution. Moreover, the first order necessary optimality conditions are established for this optimization problem.
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    POSITIVE SOLUTIONS AND INFINITELY MANY SOLUTIONS FOR A WEAKLY COUPLED SYSTEM
    Xueliang DUAN, Gongming WEI, Haitao YANG
    Acta mathematica scientia,Series B. 2020, 40 (5):  1585-1601.  DOI: 10.1007/s10473-020-0523-9
    We study a Schrödinger system with the sum of linear and nonlinear couplings. Applying index theory, we obtain infinitely many solutions for the system with periodic potentials. Moreover, by using the concentration compactness method, we prove the existence and nonexistence of ground state solutions for the system with close-to-periodic potentials.
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    RETRACTION NOTE: “MINIMAL PERIOD SYMMETRIC SOLUTIONS FOR SOME HAMILTONIAN SYSTEMS VIA THE NEHARI MANIFOLD METHOD”
    Editorial Office of Acta Mathematica Scientia
    Acta mathematica scientia,Series B. 2020, 40 (5):  1602-1602.  DOI: 10.1007/s10473-020-0524-8
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