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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 （152）            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. Reference | 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 （116）            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. Reference | Related Articles | Metrics

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FROM WAVE FUNCTIONS TO TAU-FUNCTIONS FOR THE VOLTERRA LATTICE HIERARCHY Ang FU, Mingjin LI, Di YANG Acta mathematica scientia,Series B    2024, 44 (2): 405-419.   DOI: 10.1007/s10473-024-0201-4 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （133）      PDF       Save For an arbitrary solution to the Volterra lattice hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method. In this paper, we define a pair of wave functions of the solution and use them to give an expression of the matrix resolvent; based on this we obtain a new formula for the $k$-point functions for the Volterra lattice hierarchy in terms of wave functions. As an application, we give an explicit formula of $k$-point functions for the even GUE (Gaussian Unitary Ensemble) correlators. Reference | Related Articles | Metrics

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SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION Changlin XIANG, Gaofeng ZHENG Acta mathematica scientia,Series B    2024, 44 (2): 420-430.   DOI: 10.1007/s10473-024-0202-3 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （108）      PDF       Save This paper is a continuation of recent work by Guo-Xiang-Zheng[10]. We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation $\begin{equation*} \Delta^{2}u=\Delta(V\nabla u)+{\rm div}(w\nabla u)+(\nabla\omega+F)\cdot\nabla u+f\qquad\text{in }B^{4},\end{equation*}$ under the smallest regularity assumptions of $V,w,\omega, F$, where $f$ belongs to some Morrey spaces. This work was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the $L^p$ type regularity theory of [10], and generalizes the work of Du, Kang and Wang [4] on a second order problem to our fourth order problems. Reference | Related Articles | Metrics

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THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS Zihou ZHANG, Jing ZHOU Acta mathematica scientia,Series B    2024, 44 (2): 445-454.   DOI: 10.1007/s10473-024-0204-1 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （105）      PDF       Save In this paper, we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property. We introduce the concepts of the weak$^*$-weak denting point and the weak$^*$-weak$^*$ denting point of a set. These are the generalizations of the weak$^*$ denting point of a set in a dual Banach space. By use of the weak$^*$-weak denting point, we characterize the very smooth space, the point of weak$^*$-weak continuity, and the extreme point of a unit ball in a dual Banach space. Meanwhile, we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces. Moreover, we define the nearly weak dentability in Banach spaces, which is a generalization of near dentability. We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability. We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the $w$-strong proximinality of every closed convex subset of Banach spaces. Reference | Related Articles | Metrics

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ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES Jian CHEN Acta mathematica scientia,Series B    2024, 44 (2): 431-444.   DOI: 10.1007/s10473-024-0203-2 Accepted: 16 October 2023 Abstract （128）      PDF       Save In this note, we mainly make use of a method devised by Shaw [15] for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type $\Omega=\widetilde{\Omega} \backslash \overline{\bigcup_{j=1}^{m}\Omega_j}$, where $\widetilde{\Omega}$ and $\{\Omega_j\}_{j=1}^m\Subset\widetilde{\Omega}$ are bounded pseudoconvex domains in $\mathbb{C}^n$ with smooth boundaries, and $\overline{\Omega}_1,\cdots,\overline{\Omega}_m$ are mutually disjoint. The main results can also be quickly obtained by virtue of [5]. Reference | Related Articles | Metrics

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THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX* Yinzheng Sun, Aifang Qu, Hairong Yuan Acta mathematica scientia,Series B    2024, 44 (1): 37-77.   DOI: 10.1007/s10473-024-0102-6 Abstract （107）      PDF       Save We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity $x=x(t)$. We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity $x=x(t)$ is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve $x=x(t)+0$, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field. Reference | Related Articles | Metrics

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THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL Lvqiao LIU, Juan ZENG Acta mathematica scientia,Series B    2024, 44 (2): 455-473.   DOI: 10.1007/s10473-024-0205-0 Accepted: 16 October 2023 Online available: 06 December 2023 Abstract （107）      PDF       Save In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary $L^2$ weighted estimates. Reference | Related Articles | Metrics

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CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY* Thomas E. Cecil Acta mathematica scientia,Series B    2024, 44 (1): 1-36.   DOI: 10.1007/s10473-024-0101-7 Abstract （106）      PDF       Save This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of $S^n$ (or ${\bf R}^n$), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties. Reference | 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 （119）            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 $\lambda =1/2$. Reference | 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 （74）            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. Reference | Related Articles | Metrics

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COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES (RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)* Xiaoman Duan, Zhuangdan Guan Acta mathematica scientia,Series B    2024, 44 (1): 78-102.   DOI: 10.1007/s10473-024-0103-5 Abstract （89）            Save In this article, we study Kähler metrics on a certain line bundle over some compact Kähler manifolds to find complete Kähler metrics with positive holomorphic sectional (or bisectional) curvatures. Thus, we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry. Reference | Related Articles | Metrics

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A STABILITY RESULT FOR TRANSLATING SPACELIKE GRAPHS IN LORENTZ MANIFOLDS Ya GAO, Jing MAO, Chuanxi WU Acta mathematica scientia,Series B    2024, 44 (2): 474-483.   DOI: 10.1007/s10473-024-0206-z Abstract （101）      PDF       Save In this paper, we investigate spacelike graphs defined over a domain $\Omega\subset M^{n}$ in the Lorentz manifold $M^{n}\times\mathbb{R}$ with the metric $-{\rm d}s^{2}+\sigma$, where $M^{n}$ is a complete Riemannian $n$-manifold with the metric $\sigma$, $\Omega$ has piecewise smooth boundary, and $\mathbb{R}$ denotes the Euclidean $1$-space. We prove an interesting stability result for translating spacelike graphs in $M^{n}\times\mathbb{R}$ under a conformal transformation. Reference | 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 （73）            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. Reference | 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 （87）            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. Reference | 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 （96）            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. Reference | Related Articles | Metrics

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THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS* Huifang Liu, Zhiqiang Mao Acta mathematica scientia,Series B    2024, 44 (1): 103-114.   DOI: 10.1007/s10473-024-0104-4 Abstract （58）      PDF       Save We find the exact forms of meromorphic solutions of the nonlinear differential equations $f^n+q(z){\rm e}^{Q(z)}f^{(k)}=p_1{\rm e}^{\alpha_1 z}+p_2{\rm e}^{\alpha_2 z}, \quad n\geq3, ~k\geq1,$ where $q, Q$ are nonzero polynomials, $Q\not\equiv Const.$, and $p_1, p_2, \alpha_1, \alpha_2$ are nonzero constants with $\alpha_1\neq\alpha_2$. Compared with previous results on the equation $p(z)f^3+q(z)f''=-\sin \alpha(z)$ with polynomial coefficients, our results show that the coefficient of the term $f^{(k)}$ perturbed by multiplying an exponential function will affect the structure of its solutions. Reference | 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 （66）            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. Reference | Related Articles | Metrics

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ENTIRE SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS WITH ENTIRE COEFFICIENTS HAVING THE SAME ORDER* Ziheng Feng, Zhibo Huang, Yezhou Li Acta mathematica scientia,Series B    2024, 44 (1): 355-368.   DOI: 10.1007/s10473-024-0119-x Abstract （55）      PDF       Save In this paper, we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order, and prove that the entire solutions are of infinite lower order. The properties on the radial distribution, the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed. Reference | 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 （66）            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. Reference | Related Articles | Metrics