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25 August 2024, Volume 44 Issue 4 Previous Issue    Next Issue

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THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL Han Luo Acta mathematica scientia,Series B. 2024, 44 (4):  1189-1199.  DOI: 10.1007/s10473-024-0401-y Abstract ( 103 )   RICH HTML Save In this paper, we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations. Under some suitable conditions, we give the gradient estimates of these maps and establish a Liouville type result. References | Related Articles | Metrics

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MEAN SENSITIVITY AND BANACH MEAN SENSITIVITY FOR LINEAR OPERATORS Quanquan Yao, Peiyong Zhu Acta mathematica scientia,Series B. 2024, 44 (4):  1200-1228.  DOI: 10.1007/s10473-024-0402-x Abstract ( 87 )   RICH HTML Save Let $(X,T)$ be a linear dynamical system, where $X$ is a Banach space and $T:X \to X$ is a bounded linear operator. This paper obtains that $(X,T)$ is sensitive (Li-Yorke sensitive, mean sensitive, syndetically mean sensitive, respectively) if and only if $(X,T)$ is Banach mean sensitive (Banach mean Li-Yorke sensitive, thickly multi-mean sensitive, thickly syndetically mean sensitive, respectively). Several examples are provided to distinguish between different notions of mean sensitivity, syndetic mean sensitivity and mean Li-Yorke sensitivity. References | Related Articles | Metrics

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REFINEMENTS OF THE NORM OF TWO ORTHOGONAL PROJECTIONS Xiaohui Li, Meiqi Liu, Chunyuan Deng Acta mathematica scientia,Series B. 2024, 44 (4):  1229-1243.  DOI: 10.1007/s10473-024-0403-9 Abstract ( 53 )   RICH HTML Save In this paper, some refinements of norm equalities and inequalities of combination of two orthogonal projections are established. We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space. Furthermore, we give necessary and sufficient conditions under which the norm of the above combination of orthogonal projections attains its optimal value. References | Related Articles | Metrics

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STARLIKENESS ASSOCIATED WITH THE SINE HYPERBOLIC FUNCTION Mohsan Raza, Hadiqa Zahid, Jinlin Liu Acta mathematica scientia,Series B. 2024, 44 (4):  1244-1270.  DOI: 10.1007/s10473-024-0404-8 Abstract ( 64 )   RICH HTML Save Let $q_{\lambda }\left( z\right) =1+\lambda \sinh (\zeta ),\ 0<\lambda <1/\sinh \left( 1\right)$ be a non-vanishing analytic function in the open unit disk. We introduce a subclass $\mathcal{S}^{\ast }\left( q_{\lambda }\right)$ of starlike functions which contains the functions $\mathfrak{f}$ such that $z\mathfrak{f}^{\prime }/\mathfrak{f}$ is subordinated by $q_{\lambda }$. We establish inclusion and radii results for the class $\mathcal{S}^{\ast }\left( q_{\lambda }\right)$ for several known classes of starlike functions. Furthermore, we obtain sharp coefficient bounds and sharp Hankel determinants of order two for the class $\mathcal{S}^{\ast }\left( q_{\lambda }\right)$. We also find a sharp bound for the third Hankel determinant for the case $\lambda =1/2$. References | Related Articles | Metrics

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AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS Jiangen Liu, Fazhan Geng Acta mathematica scientia,Series B. 2024, 44 (4):  1271-1279.  DOI: 10.1007/s10473-024-0405-7 Abstract ( 29 )   RICH HTML Save Fractional calculus has drawn more attentions of mathematicians and engineers in recent years. A lot of new fractional operators were used to handle various practical problems. In this article, we mainly study four new fractional operators, namely the Caputo-Fabrizio operator, the Atangana-Baleanu operator, the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the $k$-Prabhakar fractional integral operator. Usually, the theory of the $k$-Prabhakar fractional integral is regarded as a much broader than classical fractional operator. Here, we firstly give a series expansion of the $k$-Prabhakar fractional integral by means of the $k$-Riemann-Liouville integral. Then, a connection between the $k$-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown, respectively. In terms of the above analysis, we can obtain this a basic fact that it only needs to consider the $k$-Prabhakar fractional integral to cover these results from the four new fractional operators. References | Related Articles | Metrics

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PERIODIC SYSTEMS WITH TIME DEPENDENT MAXIMAL MONOTONE OPERATORS Zhenhai Liu, Nikolaos S. Papageorgiou Acta mathematica scientia,Series B. 2024, 44 (4):  1280-1300.  DOI: 10.1007/s10473-024-0406-6 Abstract ( 30 )   RICH HTML Save We consider a first order periodic system in $\mathbb R^N$, involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation. We prove the existence theorems for both the convex and nonconvex problems. We also show the existence of extremal periodic solutions and provide a strong relaxation theorem. Finally, we provide an application to nonlinear periodic control systems. References | Related Articles | Metrics

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GENERALIZED FORELLI-RUDIN TYPE OPERATORS BETWEEN SEVERAL FUNCTION SPACES ON THE UNIT BALL OF $\bf \mathbb{C}^{n}$ Xuejun ZHANG, Yuting GUO, Hongxin CHEN, Pengcheng TANG Acta mathematica scientia,Series B. 2024, 44 (4):  1301-1326.  DOI: 10.1007/s10473-024-0407-5 Abstract ( 44 )   RICH HTML Save In this paper, we investigate sufficient and necessary conditions such that generalized Forelli-Rudin type operators $T_{\lambda,\tau,k}$, $S_{\lambda,\tau,k}$, $Q_{\lambda,\tau,k}$ and $R_{\lambda,\tau,k}$ are bounded between Lebesgue type spaces. In order to prove the main results, we first give some bidirectional estimates for several typical integrals. References | Related Articles | Metrics

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PRODUCT TYPE OPERATORS ACTING BETWEEN WEIGHTED BERGMAN SPACES AND BLOCH TYPE SPACES Zuoling Liu, Hasi Wulan Acta mathematica scientia,Series B. 2024, 44 (4):  1327-1336.  DOI: 10.1007/s10473-024-0408-4 Abstract ( 45 )   RICH HTML Save For analytic functions $u,\psi$ in the unit disk $\mathbb{D}$ in the complex plane and an analytic self-map $\varphi$ of $\mathbb{D}$, we describe in this paper the boundedness and compactness of product type operators$T_{u,\psi,\varphi}f(z)=u(z)f(\varphi(z))+\psi(z)f'(\varphi(z)), \quad z\in\mathbb{D},$ acting between weighted Bergman spaces induced by a doubling weight and a Bloch type space with a radial weight. References | Related Articles | Metrics

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A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHÉODORY METRIC IN SEVERAL COMPLEX VARIABLES Xiaosong liu, Taishun liu Acta mathematica scientia,Series B. 2024, 44 (4):  1337-1346.  DOI: 10.1007/s10473-024-0409-3 Abstract ( 29 )   RICH HTML Save In this article, we first establish an asymptotically sharp result on the higher order Fréchet derivatives for bounded holomorphic mappings $f(x)=f(0)+\sum\limits_{s=1}^\infty\frac{D^{sk} f(0)(x^{sk})}{(sk) !}: B_X\rightarrow B_Y$, where $B_X$ is the unit ball of $X$. We next give a sharp result on the first order Fréchet derivative for bounded holomorphic mappings $f(x)=f(0)+\sum\limits_{s=k}^\infty\frac{D^{s} f(0)(x^{s})}{s !}: B_X\rightarrow B_Y$, where $B_X$ is the unit ball of $X$. The results that we derive include some results in several complex variables, and extend the classical result in one complex variable to several complex variables. References | Related Articles | Metrics

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GLOBAL SOLUTIONS IN THE CRITICAL SOBOLEV SPACE FOR THE LANDAU EQUATION Hao Wang Acta mathematica scientia,Series B. 2024, 44 (4):  1347-1372.  DOI: 10.1007/s10473-024-0410-x Abstract ( 37 )   RICH HTML Save The Landau equation is studied for hard potential with $-2\leq \gamma\leq1$. Under a perturbation setting, a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space $H^d_xL^2_v(d>\frac{3}{2})$, which extends the results of [11] in the torus domain to the whole space $\mathbb{R}^3_x$. Here we utilize the pseudo-differential calculus to derive our desired result. References | Related Articles | Metrics

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MULTIPLE SOLUTIONS TO CRITICAL MAGNETIC SCHRÖDINGER EQUATIONS Ruijiang Wen, Jianfu Yang Acta mathematica scientia,Series B. 2024, 44 (4):  1373-1393.  DOI: 10.1007/s10473-024-0411-9 Abstract ( 58 )   RICH HTML Save In this paper, we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation $\begin{matrix}(-{\rm i}\nabla-A(x))^2u+\lambda V(x)u=\mu |u|^{p-2}u+\Big(\int_{\mathbb{R^N}}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^\alpha}{\rm d}y\Big)|u|^{2^*_\alpha-2}u\quad {\rm in}\ \mathbb{R}^N,\end{matrix}$ (0.1) where $N\geq4$, $2\leq p<2^*$, $2^*_\alpha=\frac{2N-\alpha}{N-2}$ with $0<\alpha<4$, $\lambda>0$, $\mu\in\mathbb{R}$, $A(x)= (A_1(x), A_2(x),\cdots , A_N(x))$ is a real local Hölder continuous vector function, $i$ is the imaginary unit, and $V(x)$ is a real valued potential function on $\mathbb{R}^N$.Supposing that $\Omega={\rm int}\,V^{-1}(0)\subset\mathbb{R}^N$ is bounded, we show that problem (0.1) possesses at least cat$_\Omega(\Omega)$ nontrivial solutions if $\lambda$ is large. References | Related Articles | Metrics

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GLOBAL BOUND ON THE GRADIENT OF SOLUTIONS TO ${p}$-LAPLACE TYPE EQUATIONS WITH MIXED DATA Minh-Phuong Tran, The-Quang Tran, Thanh-Nhan Nguyen Acta mathematica scientia,Series B. 2024, 44 (4):  1394-1414.  DOI: 10.1007/s10473-024-0412-8 Abstract ( 36 )   RICH HTML Save <p>In this paper, the study of gradient regularity for solutions of a class of elliptic problems of $p$-Laplace type is offered. In particular, we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:</p><p>$\begin{align*}-\mathrm{div}\left((s^2+|\nabla u|^2)^{\frac{p-2}{2}}\nabla u\right) &= \ -\mathrm{div}\left(|\mathbf{f}|^{p-2}\mathbf{f}\right) + \mathsf{g} \quad \text{in} \ \Omega, \quad u = \mathsf{h} \quad \text{in} \ \partial\Omega,\end{align*}$</p> <p>with the (sub-elliptic) degeneracy condition $s\in [0,1]$ and with mixed data $\mathbf{f} \in L^p(\Omega;\mathbb{R}^n)$, $\mathsf{g} \in L^{\frac{p}{p-1}}(\Omega;\mathbb{R}^n)$ for $p \in (1,n)$. This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory, electro-rheology, radiation of heat, plastic moulding and many others. Building on the idea of level-set inequality on fractional maximal distribution functions, it enables us to carry out a global regularity result of the solution via fractional maximal operators. Due to the significance of $\mathcal{M}_\alpha$ and its relation with Riesz potential, estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of order $\alpha$. Our approach therefore has its own interest.</p> References | Related Articles | Metrics

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WEAK-STRONG UNIQUENESS FOR THREE DIMENSIONAL INCOMPRESSIBLE ACTIVE LIQUID CRYSTALS Fan YANG, Congming LI Acta mathematica scientia,Series B. 2024, 44 (4):  1415-1440.  DOI: 10.1007/s10473-024-0413-7 Abstract ( 34 )   RICH HTML Save The hydrodynamics of active liquid crystal models has attracted much attention in recent years due to many applications of these models. In this paper, we study the weak-strong uniqueness for the Leray-Hopf type weak solutions to the incompressible active liquid crystals in $\mathbb{R}^3$. Our results yield that if there exists a strong solution, then it is unique among the Leray-Hopf type weak solutions associated with the same initial data. References | Related Articles | Metrics

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INCOMPRESSIBLE LIMIT OF IDEAL MAGNETOHYDRODYNAMICS IN A DOMAIN WITH BOUNDARIES Qiangchang Ju, Jiawei Wang Acta mathematica scientia,Series B. 2024, 44 (4):  1441-1465.  DOI: 10.1007/s10473-024-0414-6 Abstract ( 25 )   RICH HTML Save We study the incompressible limit of classical solutions to compressible ideal magneto-hydrodynamics in a domain with a flat boundary. The boundary condition is characteristic and the initial data is general. We first establish the uniform existence of classical solutions with respect to the Mach number. Then, we prove that the solutions converge to the solution of the incompressible MHD system. In particular, we obtain a stronger convergence result by using the dispersion of acoustic waves in the half space. References | Related Articles | Metrics

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THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE Saiguo Xu, Zhong Tan Acta mathematica scientia,Series B. 2024, 44 (4):  1466-1486.  DOI: 10.1007/s10473-024-0415-5 Abstract ( 53 )   RICH HTML Save This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena. The Boussinesq system focused on here is anisotropic, and involves only horizontal dissipation and thermal damping. In the 2D case $\mathbb{R}^2$, due to the lack of vertical dissipation, the stability and large-time behavior problems have remained open in a Sobolev setting. For the spatial domain $\mathbb{T}\times\mathbb{R}$, this paper solves the stability problem and gives the precise large-time behavior of the perturbation. By decomposing the velocity $u$ and temperature $\theta$ into the horizontal average $(\bar{u},\bar{\theta})$ and the corresponding oscillation $(\tilde{u},\tilde{\theta})$, we can derive the global stability in $H^2$ and the exponential decay of $(\tilde{u},\tilde{\theta})$ to zero in $H^1$. Moreover, we also obtain that $(\bar{u}_2,\bar{\theta})$ decays exponentially to zero in $H^1$, and that $\bar{u}_1$ decays exponentially to $\bar{u}_1(\infty)$ in $H^1$ as well; this reflects a strongly stratified phenomenon of buoyancy-driven fluids. In addition, we establish the global stability in $H^3$ for the 3D case $\mathbb{R}^3$. References | Related Articles | Metrics

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STABILITY OF TRANSONIC SHOCKS TO THE EULER-POISSON SYSTEM WITH VARYING BACKGROUND CHARGES Yang Cao, Yuanyuan Xing, Na Zhang Acta mathematica scientia,Series B. 2024, 44 (4):  1487-1506.  DOI: 10.1007/s10473-024-0416-4 Abstract ( 35 )   RICH HTML Save This paper is devoted to studying the stability of transonic shock solutions to the Euler-Poisson system in a one-dimensional nozzle of finite length. The background charge in the Poisson equation is a piecewise constant function. The structural stability of the steady transonic shock solution is obtained by the monotonicity argument. Furthermore, this transonic shock is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data. One of the crucial ingredients of the analysis is to establish the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. References | Related Articles | Metrics

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STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM Yunkun Chen, Bin Huang, Xiaoding Shi Acta mathematica scientia,Series B. 2024, 44 (4):  1507-1523.  DOI: 10.1007/s10473-024-0417-3 Abstract ( 30 )   RICH HTML Save This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space. For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation, and where the strength of the initial phase field could be arbitrarily large, we prove that the solution of the Cauchy problem exists for all time, and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero. The proof is mainly based on a scaling argument and a basic energy method. References | Related Articles | Metrics

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ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS Maoan Han, Yan Ye Acta mathematica scientia,Series B. 2024, 44 (4):  1524-1535.  DOI: 10.1007/s10473-024-0418-2 Abstract ( 36 )   RICH HTML Save In this paper, we address the stability of periodic solutions of piecewise smooth periodic differential equations. By studying the Poincaré map, we give a sufficient condition to judge the stability of a periodic solution. We also present examples of some applications. References | Related Articles | Metrics

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A LOW-REGULARITY FOURIER INTEGRATOR FOR THE DAVEY-STEWARTSON II SYSTEM WITH ALMOST MASS CONSERVATION Cui Ning, Chenxi Hao, Yaohong Wang Acta mathematica scientia,Series B. 2024, 44 (4):  1536-1549.  DOI: 10.1007/s10473-024-0419-1 Abstract ( 35 )   RICH HTML Save <p>In this work, we propose a low-regularity Fourier integrator with almost mass conservation to solve the Davey-Stewartson II system (hyperbolic-elliptic case). Arbitrary order mass convergence could be achieved by the suitable addition of correction terms, while keeping the first order accuracy in $H^{\gamma}\times H^{\gamma+1}$ for initial data in $H^{\gamma+1}\times H^{\gamma+1}$ with $\gamma>1$. The main theorem is that, up to some fixed time $T$, there exist constants $\tau_0$ and $C$ depending only on $T$ and $\|u\|_{L^{\infty}\left((0, T) ; H^{\gamma+1}\right)}$ such that, for any $0<\tau\leq\tau_0$, we have that</p> <p>$\begin{equation*}\left\|u\left(t_{n}, \cdot\right)-u^{n}\right\|_{H^{\gamma}} \leq C \tau,\quad \left\|v\left(t_{n}, \cdot\right)-v^{n}\right\|_{H^{\gamma+1}} \leq C \tau, \end{equation*}$</p><p>where $u^n$ and $v^n$ denote the numerical solutions at $t_n=n\tau$. Moreover, the mass of the numerical solution $M(u^n)$ satisfies that</p><p>$\begin{equation*}\left|M\left(u^{n}\right)-M\left(u_{0}\right)\right| \leq C \tau^{5}.\end{equation*}$</p> References | Related Articles | Metrics

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ON MONOTONE TRAVELING WAVES FOR NICHOLSON'S BLOWFLIES EQUATION WITH DEGENERATE $p$-LAPLACIAN DIFFUSION Rui Huang, Yong Wang, Zhuo Yin Acta mathematica scientia,Series B. 2024, 44 (4):  1550-1571.  DOI: 10.1007/s10473-024-0420-8 Abstract ( 38 )   RICH HTML Save We study the existence and stability of monotone traveling wave solutions of Nicholson's blowflies equation with degenerate $p$-Laplacian diffusion. We prove the existence and nonexistence of non-decreasing smooth traveling wave solutions by phase plane analysis methods. Moreover, we show the existence and regularity of an original solution via a compactness analysis. Finally, we prove the stability and exponential convergence rate of traveling waves by an approximated weighted energy method. References | Related Articles | Metrics

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THE SUPERCLOSENESS OF THE FINITE ELEMENT METHOD FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEM ON A BAKHVALOV-TYPE MESH IN 2D Chunxiao Zhang, Jin Zhang Acta mathematica scientia,Series B. 2024, 44 (4):  1572-1593.  DOI: 10.1007/s10473-024-0421-7 Abstract ( 39 )   RICH HTML Save For singularly perturbed convection-diffusion problems, supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes, especially in the case of 2D. The difficulties arise from the width of the mesh in the layer adjacent to the transition point, resulting in a suboptimal estimate for convergence. Existing analysis techniques cannot handle these difficulties well. To fill this gap, here a novel interpolation is designed delicately for the smooth part of the solution, bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method. Our theoretical result is uniform in the singular perturbation parameter $\varepsilon$ and is supported by the numerical experiments. References | Related Articles | Metrics

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MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS Lulu Fang, Jihua Ma, Kunkun Song, Xin Yang Acta mathematica scientia,Series B. 2024, 44 (4):  1594-1608.  DOI: 10.1007/s10473-024-0422-6 Abstract ( 28 )   RICH HTML Save <p>For each real number $x \in (0,1)$, let $[a_1(x),a_2(x),\cdots , a_n(x),\cdots ]$ denote its continued fraction expansion. We study the convergence exponent defined by</p><p>$\tau(x):= \inf\Big\{s \geq 0: \sum\limits_{n=1}^{\infty}\big(a_n(x)a_{n+1}(x)\big)^{-s}<\infty\Big\},$ </p><p>which reflects the growth rate of the product of two consecutive partial quotients. As a main result, the Hausdorff dimensions of the level sets of $\tau(x)$ are determined.</p> References | Related Articles | Metrics