Neuromorphic hardware equipped with associative learn-ing capabilities presents fascinating applications in the next generation of artificial intelligence.However,research into synaptic devices exhibiting complex asso...Neuromorphic hardware equipped with associative learn-ing capabilities presents fascinating applications in the next generation of artificial intelligence.However,research into synaptic devices exhibiting complex associative learning behaviors is still nascent.Here,an optoelec-tronic memristor based on Ag/TiO_(2) Nanowires:ZnO Quantum dots/FTO was proposed and constructed to emulate the biological associative learning behaviors.Effective implementation of synaptic behaviors,including long and short-term plasticity,and learning-forgetting-relearning behaviors,were achieved in the device through the application of light and electrical stimuli.Leveraging the optoelectronic co-modulated characteristics,a simulation of neuromorphic computing was conducted,resulting in a handwriting digit recognition accuracy of 88.9%.Furthermore,a 3×7 memristor array was constructed,confirming its application in artificial visual memory.Most importantly,complex biological associative learning behaviors were emulated by mapping the light and electrical stimuli into conditioned and unconditioned stimuli,respectively.After training through associative pairs,reflexes could be triggered solely using light stimuli.Comprehen-sively,under specific optoelectronic signal applications,the four features of classical conditioning,namely acquisition,extinction,recovery,and generalization,were elegantly emulated.This work provides an optoelectronic memristor with associative behavior capabilities,offering a pathway for advancing brain-machine interfaces,autonomous robots,and machine self-learning in the future.展开更多
The data post-processing scheme based on two-way classical communication(TWCC)can improve the tolerable bit error rate and extend the maximal transmission distance when used in a quantum key distribution(QKD)system.In...The data post-processing scheme based on two-way classical communication(TWCC)can improve the tolerable bit error rate and extend the maximal transmission distance when used in a quantum key distribution(QKD)system.In this study,we apply the TWCC method to improve the performance of reference-frame-independent quantum key distribution(RFI-QKD),and analyze the influence of the TWCC method on the performance of decoy-state RFI-QKD in both asymptotic and non-asymptotic cases.Our numerical simulation results show that the TWCC method is able to extend the maximal transmission distance from 175 km to 198 km and improve the tolerable bit error rate from 10.48%to 16.75%.At the same time,the performance of RFI-QKD in terms of the secret key rate and maximum transmission distance are still greatly improved when statistical fluctuations are considered.We conclude that RFI-QKD with the TWCC method is of practical interest.展开更多
The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbit...The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments,demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.展开更多
We derive explicit expressions for quantum discord and classical correlation for an X structure density matrix. Based on the characteristics of the expressions, the quantum discord and the classical correlation are ea...We derive explicit expressions for quantum discord and classical correlation for an X structure density matrix. Based on the characteristics of the expressions, the quantum discord and the classical correlation are easily obtained and compared under different initial conditions using a novel analytical method. We explain the relationships among quantum discord, classical correlation, and entanglement, and further find that the quantum discord is not always larger than the entanglement measured by concurrence in a general two-qubit X state. The new method, which is different from previous approaches, has certain guiding significance for analysing quantum discord and classical correlation of a two-qubit X state, such as a mixed state.展开更多
In this article the author considers Cauchy problem for one dimensional Navier Stokes equations and the global smooth resolvablity for classical solutions is obtained.
Different translation principles may result in different translation products Chinese and foreign scholars put forward a lot of insightful and perceptive views on translation principles based on their plentiful transl...Different translation principles may result in different translation products Chinese and foreign scholars put forward a lot of insightful and perceptive views on translation principles based on their plentiful translation practice.Taken the Hezhe Peo ple’s oral tradition Yimakan Mandu Mergen as an example,the paper probes into the English translation principles and strategies of Chinese ethnic minority classics.展开更多
We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG)in three dimensions.We first prove the global existence and uniqueness for small initial data ...We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG)in three dimensions.We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials.Then,we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system(VP)at the asymptotic rate of 1/c2 as the speed of light c tends to infinity for all time.Moreover,we obtain rigorously an asymptotic estimate of the difference between the two systems.展开更多
The perturbations to symmetries and adiabatic invariants for nonconservative systems of generalized classical mechanics axe studied. The exact inwriant in the form of Hojman from a particular Lie symmetry for an undis...The perturbations to symmetries and adiabatic invariants for nonconservative systems of generalized classical mechanics axe studied. The exact inwriant in the form of Hojman from a particular Lie symmetry for an undisturbed system of generalized mechanics is given. Based on the concept of high-order adiabatic invaxiant in generalized mechanics, the perturbation to Lie symmetry for the system under the action of small disturbance is investigated, and a new adiabatic invaxiant for the nonconservative system of generalized classical mechanics is obtained, which can be called the Hojman adiabatic invaxiant. An example is also given to illustrate the application of the results.展开更多
In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of ...In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.展开更多
In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data s...In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data satisfies a natural compatibility condition. For the results, the initial density does not need to be bounded below away from zero.展开更多
In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in s...In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in some sense. In our result, we give a relation between the initial energy and the viscosity coefficient μ, and it shows that the initial energy can be large if the coefficient of the viscosity μ is taken to be large, which implies that large viscosity μ means large solution.展开更多
We prove the global existence of classical solutions to a fluid-particle interaction model in R^3, namely, compressible Navier-Stokes-Smoluchowski equations, when the initial data are close to the stationary state(ρ*...We prove the global existence of classical solutions to a fluid-particle interaction model in R^3, namely, compressible Navier-Stokes-Smoluchowski equations, when the initial data are close to the stationary state(ρ*, 0, η*) and the external potential satisfies the smallness assumption. Furthermore, optimal decay rates of classical solutions in H^3-framework are obtained.展开更多
This paper focuses on a new symmetry of Hamiltonian and its conserved quantity for a system of generalized classical mechanics. The differential equations of motion of the system are established. The definition and th...This paper focuses on a new symmetry of Hamiltonian and its conserved quantity for a system of generalized classical mechanics. The differential equations of motion of the system are established. The definition and the criterion of the symmetry of Hamiltonian of the system are given. A conserved quantity directly derived from the symmetry of Hamiltonian of the generalized classical mechanical system is given. Since a Hamilton system is a special case of the generalized classical mechanics, the results above are equally applicable to the Hamilton system. The results of the paper are the generalization of a theorem known for the existing nonsingular equivalent Lagrangian. Finally, two examples are given to illustrate the application of the results.展开更多
We investigate a two=level atom interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence and find that a stationary quantum discord can arise in the interaction of t...We investigate a two=level atom interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence and find that a stationary quantum discord can arise in the interaction of the atom and cavity field as the time turns to infinity. We also find that the stationary quantum discord can be increased by applying a classical driving field. Furthermore, we explore the quantum discord dynamics of two identical non=interacting two-level atoms independently interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence. Results show that the quantum discord between two atoms is more robust than entanglement under phase decoherence and the classical driving field can help to improve the amount of quantum discord of the two atoms.展开更多
An expansion method for stationary states is applied to obtain the eigenfunctions and the eigenenergies of the quarter stadium billiard, and its nearest energy-level spacing distribution is obtained. The histogram is ...An expansion method for stationary states is applied to obtain the eigenfunctions and the eigenenergies of the quarter stadium billiard, and its nearest energy-level spacing distribution is obtained. The histogram is consistent with the standard Wigner distribution, which indicates that the stadium billiard system is chaotic. Particular attention is paid to pursuing the quantum manifestations of such classical chaos. The correspondences between the Fourier transformation of quantum spectra and classical orbits are investigated by using the closed-orbit theory. The analytical and numerical results are in agreement with the required resolution, which corroborates that the semiclassical method provides a physically meaningful image to understand such chaotic systems.展开更多
In this work, we obtain the global existence and uniqueness of classical solu-tions to a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum in one dimension, where the initial vacuum is allowe...In this work, we obtain the global existence and uniqueness of classical solu-tions to a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum in one dimension, where the initial vacuum is allowed. We get the upper and lower bounds of gas and liquid masses n and m by the continuity methods which we use to study the compressible Navier-Stokes equations.展开更多
It is currently believed that light quantum or the quantization of light energy is beyond classical physics, and the picture of wave-particle duality, which was criticized by Einstein but has attracted a number of exp...It is currently believed that light quantum or the quantization of light energy is beyond classical physics, and the picture of wave-particle duality, which was criticized by Einstein but has attracted a number of experimental researches, is necessary for the description of light. It is shown in this paper, however, that the quantization of light energy in vacuum, which is the same as that in quantum electrodynamics, can be derived directly from the classical electromagnetic theory through the consideration of statistics based on classical physics. Therefore, the quantization of energy is an intrinsic property of light as a classical electromagnetic wave and has no need of being related to particles.展开更多
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ｜｜po｜｜L∞ is appropriate...We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ｜｜po｜｜L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.展开更多
In this paper, we consider the formation of singularity for the classical solutions to compressible MHD equations without thermal conductivity or infinity electric conductivity when the initial data contains vacuum. W...In this paper, we consider the formation of singularity for the classical solutions to compressible MHD equations without thermal conductivity or infinity electric conductivity when the initial data contains vacuum. We show that the life span of any smooth solution will not be extended to ∞, if the initial vacuum only appears in some local domain and the magnetic field vanishes on the interface that separates the vacuum and non-vacuum state, regardless the size of the initial data or the far field state.展开更多
基金This work was supported by the Jinan City-University Integrated Development Strategy Project under Grant(JNSX2023017)National Research Foundation of Korea(NRF)grant funded by the Korea government(MIST)(RS-2023-00302751)+1 种基金by the National Research Foundation of Korea(NRF)funded by the Ministry of Education under Grants 2018R1A6A1A03025242 and 2018R1D1A1A09083353by Qilu Young Scholar Program of Shandong University.
文摘Neuromorphic hardware equipped with associative learn-ing capabilities presents fascinating applications in the next generation of artificial intelligence.However,research into synaptic devices exhibiting complex associative learning behaviors is still nascent.Here,an optoelec-tronic memristor based on Ag/TiO_(2) Nanowires:ZnO Quantum dots/FTO was proposed and constructed to emulate the biological associative learning behaviors.Effective implementation of synaptic behaviors,including long and short-term plasticity,and learning-forgetting-relearning behaviors,were achieved in the device through the application of light and electrical stimuli.Leveraging the optoelectronic co-modulated characteristics,a simulation of neuromorphic computing was conducted,resulting in a handwriting digit recognition accuracy of 88.9%.Furthermore,a 3×7 memristor array was constructed,confirming its application in artificial visual memory.Most importantly,complex biological associative learning behaviors were emulated by mapping the light and electrical stimuli into conditioned and unconditioned stimuli,respectively.After training through associative pairs,reflexes could be triggered solely using light stimuli.Comprehen-sively,under specific optoelectronic signal applications,the four features of classical conditioning,namely acquisition,extinction,recovery,and generalization,were elegantly emulated.This work provides an optoelectronic memristor with associative behavior capabilities,offering a pathway for advancing brain-machine interfaces,autonomous robots,and machine self-learning in the future.
基金supported by the National Natural Science Foundation of China(Grant Nos.61505261,62101597,61605248,and 61675235)the National Key Research and Development Program of China(Grant No.2020YFA0309702)+2 种基金the China Postdoctoral Science Foundation(Grant No.2021M691536)the Natural Science Foundation of Henan Province(Grant Nos.202300410534 and 202300410532)the Anhui Initiative in Quantum Information Technologies.
文摘The data post-processing scheme based on two-way classical communication(TWCC)can improve the tolerable bit error rate and extend the maximal transmission distance when used in a quantum key distribution(QKD)system.In this study,we apply the TWCC method to improve the performance of reference-frame-independent quantum key distribution(RFI-QKD),and analyze the influence of the TWCC method on the performance of decoy-state RFI-QKD in both asymptotic and non-asymptotic cases.Our numerical simulation results show that the TWCC method is able to extend the maximal transmission distance from 175 km to 198 km and improve the tolerable bit error rate from 10.48%to 16.75%.At the same time,the performance of RFI-QKD in terms of the secret key rate and maximum transmission distance are still greatly improved when statistical fluctuations are considered.We conclude that RFI-QKD with the TWCC method is of practical interest.
基金supported by National Natural Science Foundation of China (Nos. 11975068 and 11925501)the National Key R&D Program of China (No. 2022YFE03090000)the Fundamental Research Funds for the Central Universities (No. DUT22ZD215)。
文摘The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments,demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.
基金supported by the Natural Science Foundation of Hunan Province of China (Grant No. 09JJ6011)the Natural Science Foundation of Education Department of Hunan Province, China (Grant Nos. 08A055 and 07C528)
文摘We derive explicit expressions for quantum discord and classical correlation for an X structure density matrix. Based on the characteristics of the expressions, the quantum discord and the classical correlation are easily obtained and compared under different initial conditions using a novel analytical method. We explain the relationships among quantum discord, classical correlation, and entanglement, and further find that the quantum discord is not always larger than the entanglement measured by concurrence in a general two-qubit X state. The new method, which is different from previous approaches, has certain guiding significance for analysing quantum discord and classical correlation of a two-qubit X state, such as a mixed state.
文摘In this article the author considers Cauchy problem for one dimensional Navier Stokes equations and the global smooth resolvablity for classical solutions is obtained.
文摘Different translation principles may result in different translation products Chinese and foreign scholars put forward a lot of insightful and perceptive views on translation principles based on their plentiful translation practice.Taken the Hezhe Peo ple’s oral tradition Yimakan Mandu Mergen as an example,the paper probes into the English translation principles and strategies of Chinese ethnic minority classics.
基金supported by the National Natural ScienceFoundation of China(11871024)the Fundamental Research Program of Shanxi Province(202103021223182)。
文摘We investigate the global classical solutions of the non-relativistic Vlasov-D arwin system with generalized variables(VDG)in three dimensions.We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials.Then,we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system(VP)at the asymptotic rate of 1/c2 as the speed of light c tends to infinity for all time.Moreover,we obtain rigorously an asymptotic estimate of the difference between the two systems.
基金Project supported by the Natural Science Foundation of High Education of Jiangsu Province, China （Grant No 04KJA130135）.
文摘The perturbations to symmetries and adiabatic invariants for nonconservative systems of generalized classical mechanics axe studied. The exact inwriant in the form of Hojman from a particular Lie symmetry for an undisturbed system of generalized mechanics is given. Based on the concept of high-order adiabatic invaxiant in generalized mechanics, the perturbation to Lie symmetry for the system under the action of small disturbance is investigated, and a new adiabatic invaxiant for the nonconservative system of generalized classical mechanics is obtained, which can be called the Hojman adiabatic invaxiant. An example is also given to illustrate the application of the results.
基金supported by National Natural Science Foundation of China (10871145 10926066)+1 种基金Doctoral Program Foundation of the Ministry of Education of China (20090072110053)Natural Science Foundation of Zhejiang Province (Y6100007)
文摘In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.
基金Supported by NSFC(11201371,1331005)Natural Science Foundation of Shaanxi Province(2012JQ020)
文摘In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data satisfies a natural compatibility condition. For the results, the initial density does not need to be bounded below away from zero.
文摘In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in some sense. In our result, we give a relation between the initial energy and the viscosity coefficient μ, and it shows that the initial energy can be large if the coefficient of the viscosity μ is taken to be large, which implies that large viscosity μ means large solution.
基金supported by the National Natural Science Foundation of China(11371152,11771155,11571117 and 11871005)the Natural Science Foundation of Guangdong Province(2017A030313003)+2 种基金supported by the Natural Science Foundation of Guangdong Province(2018A030310008)the Doctoral Scientific Research Foundation of Hanshan Normal University(QD20171002)the Educational Commission of Guangdong Province(2017KTSCX124)
文摘We prove the global existence of classical solutions to a fluid-particle interaction model in R^3, namely, compressible Navier-Stokes-Smoluchowski equations, when the initial data are close to the stationary state(ρ*, 0, η*) and the external potential satisfies the smallness assumption. Furthermore, optimal decay rates of classical solutions in H^3-framework are obtained.
基金supported by the National Natural Science Foundation of China (Grant No. 10972151)
文摘This paper focuses on a new symmetry of Hamiltonian and its conserved quantity for a system of generalized classical mechanics. The differential equations of motion of the system are established. The definition and the criterion of the symmetry of Hamiltonian of the system are given. A conserved quantity directly derived from the symmetry of Hamiltonian of the generalized classical mechanical system is given. Since a Hamilton system is a special case of the generalized classical mechanics, the results above are equally applicable to the Hamilton system. The results of the paper are the generalization of a theorem known for the existing nonsingular equivalent Lagrangian. Finally, two examples are given to illustrate the application of the results.
基金Project supported by National Natural Science Foundation of China (Grant No. 10774143)
文摘We investigate a two=level atom interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence and find that a stationary quantum discord can arise in the interaction of the atom and cavity field as the time turns to infinity. We also find that the stationary quantum discord can be increased by applying a classical driving field. Furthermore, we explore the quantum discord dynamics of two identical non=interacting two-level atoms independently interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence. Results show that the quantum discord between two atoms is more robust than entanglement under phase decoherence and the classical driving field can help to improve the amount of quantum discord of the two atoms.
基金Supported by the National Natural Science Foundation of China under Grant No 10374061.
文摘An expansion method for stationary states is applied to obtain the eigenfunctions and the eigenenergies of the quarter stadium billiard, and its nearest energy-level spacing distribution is obtained. The histogram is consistent with the standard Wigner distribution, which indicates that the stadium billiard system is chaotic. Particular attention is paid to pursuing the quantum manifestations of such classical chaos. The correspondences between the Fourier transformation of quantum spectra and classical orbits are investigated by using the closed-orbit theory. The analytical and numerical results are in agreement with the required resolution, which corroborates that the semiclassical method provides a physically meaningful image to understand such chaotic systems.
基金Supported by the National Natural Science Foundation of China(11171340)
文摘In this work, we obtain the global existence and uniqueness of classical solu-tions to a viscous liquid-gas two-phase model with mass-dependent viscosity and vacuum in one dimension, where the initial vacuum is allowed. We get the upper and lower bounds of gas and liquid masses n and m by the continuity methods which we use to study the compressible Navier-Stokes equations.
文摘It is currently believed that light quantum or the quantization of light energy is beyond classical physics, and the picture of wave-particle duality, which was criticized by Einstein but has attracted a number of experimental researches, is necessary for the description of light. It is shown in this paper, however, that the quantization of light energy in vacuum, which is the same as that in quantum electrodynamics, can be derived directly from the classical electromagnetic theory through the consideration of statistics based on classical physics. Therefore, the quantization of energy is an intrinsic property of light as a classical electromagnetic wave and has no need of being related to particles.
基金supported by National Natural Science Foundation of China (11001090)the Fundamental Research Funds for the Central Universities(11QZR16)
文摘We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the 3-D compressible Navier-Stokes equations under the assumption that the initial density ｜｜po｜｜L∞ is appropriate small and 1 〈 γ 〈 6/5. Here the initial density could have vacuum and we do not require that the initial energy is small.
基金supported in part by National Natural Science Foundation of China(11231006)Natural Science Foundation of Shanghai(14ZR1423100)China Scholarship Council
文摘In this paper, we consider the formation of singularity for the classical solutions to compressible MHD equations without thermal conductivity or infinity electric conductivity when the initial data contains vacuum. We show that the life span of any smooth solution will not be extended to ∞, if the initial vacuum only appears in some local domain and the magnetic field vanishes on the interface that separates the vacuum and non-vacuum state, regardless the size of the initial data or the far field state.
