数学物理学报(英文版)

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THE EXISTENCE OF GLOBAL SOLUTIONS FOR THE FULL NAVIER-STOKES-KORTEWEG SYSTEM OF VAN DER WAALS GAS^∗

Hakho Hong

数学物理学报(英文版). 2023 (2): 469-491. DOI: 10.1007/s10473-023-0201-9

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The aim of this work is to prove the existence for the global solution of a non-isothermal or non-isentropic model of capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), in the case of van der Waals gas. Under the small initial perturbation, the proof of the global existence is based on an elementary energy method using the continuation argument of local solution. Moreover, the uniqueness of global solutions and large time behavior of the density are given. It is one of the main difficulties that the pressure $p$ is not the increasing function of the density $\rho$.

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THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION^*

Minghua Yang, Jinyi Sun, Zunwei Fu, Zheng Wang

数学物理学报(英文版). 2023 (2): 492-504. DOI: 10.1007/s10473-023-0202-8

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The singular convergence of a chemotaxis-fluid system modeling coral fertilization is justified in spatial dimension three. More precisely, it is shown that a solution of parabolic-parabolic type chemotaxis-fluid system modeling coral fertilization $\begin{eqnarray*} \left\{ \begin{array}{ll} u_t^{\epsilon}+(u^{\epsilon}\cdot\nabla)u^{\epsilon}-\Delta u^{\epsilon}+\nabla\mathbf{P}^{\epsilon}=-(s^{\epsilon}+e^{\epsilon})\nabla \phi,\\ \nabla\cdot u^{\epsilon}=0, \\ e_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )e^{\epsilon}-\Delta e^{\epsilon}=-s^{\epsilon}e^{\epsilon},\\ s_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )s^{\epsilon}-\Delta s^{\epsilon}=-\nabla\cdot(s^{\epsilon}\nabla c^{\epsilon})-s^{\epsilon}e^{\epsilon}, \\ \epsilon^{-1} \left(c_t^{\epsilon}+(u^{\epsilon}\cdot\nabla )c^{\epsilon}\right)=\Delta c^{\epsilon}+e^{\epsilon},\\ (u^{\epsilon}, e^{\epsilon},s^{\epsilon},c^{\epsilon})|_{t=0}= (u_{0}, e_{0},s_{0},c_{0})\\ \end{array} \right. \end{eqnarray*}$ converges to that of the parabolic-elliptic type chemotaxis-fluid system modeling coral fertilization $\begin{eqnarray*} \left\{ \begin{array}{ll} u_t^{\infty}+(u^{\infty}\cdot\nabla)u^{\infty}-\Delta u^{\infty}+\nabla\mathbf{P}^{\infty}=-(s^{\infty}+e^{\infty})\nabla \phi, \\ \nabla\cdot u^{\infty}=0, \\ e_t^{\infty}+(u^{\infty}\cdot\nabla )e^{\infty}-\Delta e^{\infty}=-s^{\infty}e^{\infty}, \\ s_t^{\infty}+(u^{\infty}\cdot\nabla )s^{\infty}-\Delta s^{\infty}=-\nabla\cdot(s^{\infty}\nabla c^{\infty})-s^{\infty}e^{\infty}, \\ 0=\Delta c^{\infty}+e^{\infty}, \\ (u^{\infty}, e^{\infty},s^{\infty})|_{t=0}= (u_{0}, e_{0},s_{0})\\ \end{array} \right. \end{eqnarray*}$ in a certain Fourier-Herz space as $\epsilon^{-1}\rightarrow 0$.

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A LARGE DEVIATION PRINCIPLE FOR THE STOCHASTIC GENERALIZED GINZBURG-LANDAU EQUATION DRIVEN BY JUMP NOISE^*

Ran Wang, Beibei Zhang

数学物理学报(英文版). 2023 (2): 505-530. DOI: 10.1007/s10473-023-0203-7

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In this paper, we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise. The main difficulties come from the highly non-linear coefficient and the jump noise. Here, we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle, which was initially proposed by Matoussi, Sabbagh and Zhang (2021).

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NEW DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES^*

Zhiwei Hao, Libo Li

数学物理学报(英文版). 2023 (2): 531-538. DOI: 10.1007/s10473-023-0204-6

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Let $1\leq q\leq \infty$, $b$ be a slowly varying function and let $ \Phi: [0,\infty ) \longrightarrow [0,\infty ) $ be an increasing convex function with $\Phi(0)=0$ and $\lim\limits_{r \rightarrow \infty}\Phi(r)=\infty$. In this paper, we present a new class of Doob's maximal inequality on Orlicz-Lorentz-Karamata spaces $L_{\Phi,q,b}$. The results are new, even for the Lorentz-Karamata spaces with $\Phi(t)=t^p$, the Orlicz-Lorentz spaces with $b\equiv1$, and weak Orlicz-Karamata spaces with $q=\infty$ in the framework of $L_{\Phi,q,b}$. Moreover, we obtain some even stronger qualitative results that can remove the $\vartriangle_2$-condition of Liu, Hou and Wang (Sci China Math, 2010, 53(4): 905--916).

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THE EXISTENCE AND STABILITY OF NORMALIZED SOLUTIONS FOR A BI-HARMONIC NONLINEAR SCHRÖDINGER EQUATION WITH MIXED DISPERSION^*

Tingjian Luo, Shijun Zheng, Shihui Zhu

数学物理学报(英文版). 2023 (2): 539-563. DOI: 10.1007/s10473-023-0205-5

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In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schr\"{o}dinger equation with a $\mu$-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denoting by $Q_p$ the ground state for the BNLS with $\mu=0$, we prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{d})$, there exist orbitally stable {ground state solutions} for the BNLS when $\mu\in ( -\lambda_0, \infty)$ for some $\lambda_0=\lambda_0(p, d,\|Q_p\|_{L^2})>0$. Moreover, in the mass-critical case $p=1+\frac{8}{d}$, we prove the orbital stability on a certain mass level below $\|Q^*\|_{L^2}$, provided that $\mu\in (-\lambda_1,0)$, where $\lambda_1=\frac{4\|\nabla Q^*\|^2_{L^2}}{\|Q^*\|^2_{L^2}}$ and $Q^*=Q_{1+8/d}$. The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality. Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when $\mu$ is negative and $p\in (1,1+\frac8d]$.

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MINIMAL FOLIATIONS FOR THE HIGH-DIMENSIONAL FRENKEL-KONTOROVA MODEL^*

Xueqing Miao, Jianhua Ge, Wenxin Qin, Yanan Wang

数学物理学报(英文版). 2023 (2): 564-582. DOI: 10.1007/s10473-023-0207-3

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For the high-dimensional Frenkel-Kontorova (FK) model on lattices, we study the existence of minimal foliations by depinning force. We introduce the tilted gradient flow and define the depinning force as the critical value of the external force under which the average velocity of the system is zero. Then, the depinning force can be used as the criterion for the existence of minimal foliations for the FK model on a $\mathbb{Z}^d$ lattice for $d>1$.

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TWO GENERALIZATIONS OF BOHR RADIUS^*

Chengpeng Li, Mingxin Chen, Jianfei Wang

数学物理学报(英文版). 2023 (2): 583-596. DOI: 10.1007/s10473-023-0206-4

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The purpose of this paper is twofold. First, by using the hyperbolic metric, we establish the Bohr radius for analytic functions from shifted disks containing the unit disk $D$ into convex proper domains of the complex plane. As a consequence, we generalize the Bohr radius of Evdoridis, Ponnusamy and Rasila based on geometric idea. By introducing an alternative multidimensional Bohr radius, the second purpose is to obtain the Bohr radius of higher dimensions for Carathéodory families in the unit ball $B$ of a complex Banach space $X$. Notice that when $B$ is the unit ball of the complex Hilbert space $X$, we show that the constant $ {1}/{3} $ is the Bohr radius for normalized convex mappings of $B$, which generalizes the result of convex functions on $D$.

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LIPSCHITZ STAR BODIES^*

Youjiang Lin, Yuchi, Wu

数学物理学报(英文版). 2023 (2): 597-607. DOI: 10.1007/s10473-023-0208-2

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In this paper, we study some basic properties on Lipschitz star bodies, such as the equivalence between Lipschitz star bodies and star bodies with respect to a ball, the equivalence between the convergence of Lipschitz star bodies with respect to Hausdorff distance and the convergence of Lipschtz star bodies with respect to radial distance, and the convergence of Steiner symmetrizations of Lipschitz star bodies.

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ALMOST SURELY TIME-SPACE INTERMITTENCY FOR THE PARABOLIC ANDERSON MODEL WITH A LOG-CORRELATED GAUSSIAN FIELD^*

Yangyang Lyu, Heyu Li

数学物理学报(英文版). 2023 (2): 608-639. DOI: 10.1007/s10473-023-0209-1

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In this paper, we consider the continuous parabolic Anderson model with a log-correlated Gaussian field, and obtain the precise quenched long-time asymptotics and spatial asymptotics. To overcome the difficulties arising from the log-correlated Gaussian field in the proof of the lower bound of the spatial asymptotics, we first establish the relation between quenched long-time asymptotics and spatial asymptotics, and then get the lower bound of the spatial asymptotics through the lower bound of the quenched long-time asymptotics.

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ANALYSIS AND DISCRETIZATION FOR AN OPTIMAL CONTROL PROBLEM OF A VARIABLE-COEFFICIENT RIESZ-FRACTIONAL DIFFUSION EQUATION WITH POINTWISE CONTROL CONSTRAINTS^*

Zhaojie Zhou, Fangyuan Wang, Xiangcheng Zheng

数学物理学报(英文版). 2023 (2): 640-654. DOI: 10.1007/s10473-023-0210-8

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We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation. Due to the impact of the variable diffusivity coefficient, existing regularity results for their constant-coefficient counterparts do not apply, while the bilinear forms of the state (adjoint) equation may lose the coercivity that is critical in error estimates of the finite element method. We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term. First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by, e.g., interpolation estimates. The weak coercivity of the resulting bilinear forms are proven via the Garding inequality, based on which we prove the optimal-order convergence estimates of the finite element method for the (adjoint) state variable and the control variable. Numerical experiments substantiate the theoretical predictions.

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REVERSE CARLESON MEASURES ON GENERALIZED FOCK SPACES^*

Guangfu Cao, Li He, Yiyuan Zhang

数学物理学报(英文版). 2023 (2): 655-667. DOI: 10.1007/s10473-023-0211-7

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In this paper, we characterize reverse Carleson measures for a class of generalized Fock spaces $F_{\varphi}^{p}$, with $0<p<\infty$ and with $\varphi$ satisfying $d d^{c} \varphi \simeq \omega_{0}$. As an application of these results, we obtain several equivalent characterizations for invertible Toeplitz operators $T_{\psi}$, induced by positive bounded symbols $\psi$ on $F_{\varphi}^{2}$.

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THE ENERGY CONSERVATION OF VLASOV-POISSON SYSTEMS^*

Jingpeng, Wu, Xianwen Zhang

数学物理学报(英文版). 2023 (2): 668-674. DOI: 10.1007/s10473-023-0212-6

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We prove that energy conservation holds for weak solutions to classical Vlasov-Poisson systems with proper regularity. In particular, there exists a solution that conserves energy with $|v|^mf_0\in L^1_{x,v}$ for $m>9/4$.

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ON LOCAL CONTROLLABILITY FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY DEPENDENT VISCOSITIES^*

Xiangkai Lian, Qiang Tao, Zheng-an Yao

数学物理学报(英文版). 2023 (2): 675-685. DOI: 10.1007/s10473-023-0213-5

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In this paper, we study the controllability of compressible Navier-Stokes equations with density dependent viscosities. For when the shear viscosity $\mu$ is a positive constant and the bulk viscosity $\lambda$ is a function of the density, it is proven that the system is exactly locally controllable to a constant target trajectory by using boundary control functions.

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JOHN-NIRENBERG-Q SPACES VIA CONGRUENT CUBES^*

Jin Tao, Zhenyu Yang, Wen Yuan

数学物理学报(英文版). 2023 (2): 686-718. DOI: 10.1007/s10473-023-0214-4

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To shed some light on the John-Nirenberg space, the authors of this article introduce the John-Nirenberg-$Q$ space via congruent cubes, $JNQ^\alpha_{p,q}(\mathbb{R}^n)$, which, when $p=\infty$ and $q=2$, coincides with the space $Q_\alpha(\mathbb{R}^n)$ introduced by Essén, Janson, Peng and Xiao in [Indiana Univ Math J, 2000, 49(2): 575--615]. Moreover, the authors show that, for some particular indices, $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ coincides with the congruent John-Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into $JNQ^\alpha_{p,q}(\mathbb{R}^n)$. Furthermore, the authors characterize $JNQ^\alpha_{p,q}(\mathbb{R}^n)$ via mean oscillations, and then use this characterization to study the dyadic counterparts. Also, the authors obtain some properties of composition operators on such spaces. The main novelties of this article are twofold: establishing a general equivalence principle for a kind of `almost increasing' set function that is here introduced, and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.

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AN ALGEBRAIC APPROACH TO DEGENERATE APPELL POLYNOMIALS AND THEIR HYBRID FORMS VIA DETERMINANTS^*

Mumtaz Riyasat, Tabinda Nahid, Subuhi Khan

数学物理学报(英文版). 2023 (2): 719-735. DOI: 10.1007/s10473-023-0215-3

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It is remarkable that studying degenerate versions of polynomials from algebraic point of view is not limited to only special polynomials but can also be extended to their hybrid polynomials. Indeed for the first time, a closed determinant expression for the degenerate Appell polynomials is derived. The determinant forms for the degenerate Bernoulli and Euler polynomials are also investigated. A new class of the degenerate Hermite-Appell polynomials is investigated and some novel identities for these polynomials are established. The degenerate Hermite-Bernoulli and degenerate Hermite-Euler polynomials are considered as special cases of the degenerate Hermite-Appell polynomials. Further, by using Mathematica, we draw graphs of degenerate Hermite-Bernoulli polynomials for different values of indices. The zeros of these polynomials are also explored and their distribution is presented.

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THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^*

Mingxuan Zhu, Zaihong Jiang

数学物理学报(英文版). 2023 (2): 736-750. DOI: 10.1007/s10473-023-0220-6

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In this paper, we consider the Cauchy problem for the Camassa-Holm-Novikov equation. First, we establish the local well-posedness and the blow-up scenario. Second, infinite propagation speed is obtained as the nontrivial solution $u(x,t)$ does not have compact $x$-support for any $t>0$ in its lifespan, although the corresponding $u_0(x)$ is compactly supported. Then, the global existence and large time behavior for the support of the momentum density are considered. Finally, we study the persistence property of the solution in weighted Sobolev spaces.

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THE LINEAR SAMPLING METHOD FOR RECONSTRUCTING A PENETRABLE CAVITY WITH UNKNOWN EXTERNAL OBSTACLES^*

Jianguo Ye, Guozheng Yan

数学物理学报(英文版). 2023 (2): 751-776. DOI: 10.1007/s10473-023-0216-2

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We consider the interior inverse scattering problem for recovering the shape of a penetrable partially coated cavity with external obstacles from the knowledge of measured scattered waves due to point sources. In the first part, we obtain the well-posedness of the direct scattering problem by the variational method. In the second part, we establish the mathematical basis of the linear sampling method to recover both the shape of the cavity, and the shape of the external obstacle, however the exterior transmission eigenvalue problem also plays a key role in the discussion of this paper.

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CONVERGENCE FROM THE TWO-SPECIES VLASOV-POISSON-BOLTZMANN SYSTEM TO THE TWO-FLUID INCOMPRESSIBLE NAVIER-STOKES-FOURIER-POISSON SYSTEM WITH OHM'S LAW^*

Zhendong Fang, Ning Jiang

数学物理学报(英文版). 2023 (2): 777-820. DOI: 10.1007/s10473-023-0217-1

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In this paper, we justify the convergence from the two-species Vlasov-Poisson-Boltzmann (VPB, for short) system to the two-fluid incompressible Navier-Stokes-Fourier-Poisson (NSFP, for short) system with Ohm's law in the context of classical solutions. We prove the uniform estimates with respect to the Knudsen number $\varepsilon$ for the solutions to the two-species VPB system near equilibrium by treating the strong interspecies interactions. Consequently, we prove the convergence to the two-fluid incompressible NSFP as $\varepsilon$ goes to 0.

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SOME RESULTS REGARDING PARTIAL DIFFERENTIAL POLYNOMIALS AND THE UNIQUENESS OF MEROMORPHIC FUNCTIONS IN SEVERAL VARIABLES^*

Manli Liu, Lingyun Gao, Shaomei Fang

数学物理学报(英文版). 2023 (2): 821-838. DOI: 10.1007/s10473-023-0218-0

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In this paper, we mainly investigate the value distribution of meromorphic functions in $\mathbb{C}^m$ with its partial differential and uniqueness problem on meromorphic functions in $\mathbb{C}^m$ and with its $k$-th total derivative sharing small functions. As an application of the value distribution result, we study the defect relation of a nonconstant solution to the partial differential equation. In particular, we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.

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A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS^*

Zhankuan Zeng, Yanping CHEN

数学物理学报(英文版). 2023 (2): 839-854. DOI: 10.1007/s10473-023-0219-z

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In this paper, a local discontinuous Galerkin (LDG) scheme for the time-fractional diffusion equation is proposed and analyzed. The Caputo time-fractional derivative (of order $\alpha$, with $0< \alpha <1$) is approximated by a finite difference method with an accuracy of order $3-\alpha$, and the space discretization is based on the LDG method. For the finite difference method, we summarize and supplement some previous work by others, and apply it to the analysis of the convergence and stability of the proposed scheme. The optimal error estimate is obtained in the $L^2$ norm, indicating that the scheme has temporal $(3 -\alpha)$ th-order accuracy and spatial $(k+1)$ th-order accuracy, where $k$ denotes the highest degree of a piecewise polynomial in discontinuous finite element space. The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

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HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS^*

Hong Wang

数学物理学报(英文版). 2023 (2): 855-906. DOI: 10.1007/s10473-023-0221-5

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In this paper, we give the geometric constraint conditions of a canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II Hamilton-Jacobi equations. First, we prove two types of Hamilton-Jacobi theorems for an RCH system on the cotangent bundle of a configuration manifold by using the canonical symplectic form and its dynamical vector field. Second, we generalize the above results for a regular reducible RCH system with symmetry and a momentum map, and derive precisely two types of Hamilton-Jacobi equations for the regular point reduced RCH system and the regular orbit reduced RCH system. Third, we prove that the RCH-equivalence for the RCH system, and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries, leave the solutions of corresponding Hamilton-Jacobi equations invariant. Finally, as an application of the theoretical results, we show the Type I and Type II Hamilton-Jacobi equations for the $R_p$-reduced controlled rigid body-rotor system and the $R_p$-reduced controlled heavy top-rotor system on the generalizations of the rotation group ${SO}(3)$ and the Euclidean group ${SE}(3)$, respectively. This work reveals the deeply internal relationships of the geometrical structures of phase spaces, the dynamical vector fields and the controls of the RCH system.

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THE LAW OF THE ITERATED LOGARITHM FOR SPATIAL AVERAGES OF THE STOCHASTIC HEAT EQUATION^*

Jingyu, Li, Yong, Zhang

数学物理学报(英文版). 2023 (2): 907-918. DOI: 10.1007/s10473-023-0222-4

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Let $u(t,x)$ be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with $u(0,x)=1$ for all $x \in \mathbb{R}$. In this paper, we prove the law of the iterated logarithm (LIL for short) and the functional LIL for a linear additive functional of the form $\int _{[0,R] } u(t,x)\mathrm{d} x$ and the nonlinear additive functionals of the form $\int_{[0, R]} g(u(t, x))\mathrm{d} x$, where $g: \mathbb{R} \rightarrow \mathbb{R}$ is nonrandom and Lipschitz continuous, as $R\rightarrow\infty$ for fixed $t>0$, using the localization argument.

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A RIGOROUS PROOF ON CIRCULAR WIRELENGTH FOR HYPERCUBES^*

Qinghui LIU, Zhiyi Tang

数学物理学报(英文版). 2023 (2): 919-941. DOI: 10.1007/s10473-023-0223-3

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We study embeddings of the $n$-dimensional hypercube into the circuit with $2^n$ vertices. We prove that the circular wirelength attains a minimum by gray coding; that was called the CT conjecture by Chavez and Trapp (Discrete Applied Mathematics, 1998). This problem had claimed to be settled by Ching-Jung Guu in her doctoral dissertation "The circular wirelength problem for hypercubes" (University of California, Riverside, 1997). Many argue there are gaps in her proof. We eliminate the gaps in her dissertation.

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THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM^*

Boling Guo, Yamin Xiao

数学物理学报(英文版). 2023 (2): 942-958. DOI: 10.1007/s10473-023-0225-1

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This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries (KdV) system in $\mathbb{R}^n$. We first prove, by the Leray-Schauder principle and the vanishing viscosity method, that any initial data $N$-dimensional vector value function $u_0(x)$ in Sobolev space $H^{s}(\mathbb{R}^n)$ $(s\geq1)$ leads to a global weak solution. Second, we investigate some special regularity properties of solutions to the initial value problem associated with the KdV type system in $\mathbb{R}^2$ and $\mathbb{R}^3$.

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VANISHING VISCOSITY LIMIT FOR THE 3D INCOMPRESSIBLE MICROPOLAR EQUATIONS IN A BOUNDED DOMAIN^*

Yangyang, Chu, Yuelong Xiao

数学物理学报(英文版). 2023 (2): 959-974. DOI: 10.1007/s10473-023-0224-2

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In this paper, we investigate the vanishing viscosity limit of the 3D incompressible micropolar equations in bounded domains with boundary conditions. It is shown that there exist global weak solutions of the micropolar equations in a general bounded smooth domain. In particular, we establish the uniform estimate of the strong solutions for when the boundary is flat. Furthermore, we obtain the rate of convergence of viscosity solutions to the inviscid solutions as the viscosities tend to zero (i.e., $(\varepsilon,\chi,\gamma,\kappa)\to 0$).

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GLOBAL SOLUTIONS TO A HYDRODYNAMIC MODEL FOR SEMICONDUCTORS WITH VELOCITY RELAXATION^*

Xianting Wang, Yun-guang Lu, Richard De la cruz, Guoqiao You

数学物理学报(英文版). 2023 (2): 975-980. DOI: 10.1007/s10473-023-0226-0

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In this paper, we apply the method given in the paper "Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors" (Mathematische Annalen, 2022, 382: 1031--1046) to study the Cauchy problem for a one dimensional inhomogeneous hydrodynamic model of two-carrier types for semiconductors with the velocity relaxation.

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