Considering that thermodynarmic irreversibility and hydrodynamic equations can not be derived rigorously and unifiedly from the Liouville equations, the anomalous Langevin equation in the Liouville space is proposed a...Considering that thermodynarmic irreversibility and hydrodynamic equations can not be derived rigorously and unifiedly from the Liouville equations, the anomalous Langevin equation in the Liouville space is proposed as a fundamental equation of statistical physics. This equation reflects that the law of motion of particles obeying reversible, deterministic laws in dynamics becomes irreversible and stochastic in thermodynamics. From this the fundamental equations of nonequilibrium thermodynamics, the principle of entropy increase and the theorem of minimum entropy production have been derived. The hydrodynamic equations, such as the generalized Navier-Stokes equation and the mass drift-diffusion equation etc. have been derived rigorously from the kinetic kinetic equation which is reduced from the anomalous Langevin equation in Liouville space. All these are unified and self consistent. But it is difficult to prove that entropy production density σ can never be negative everywhere for all the isolated inhomogeneous systems far from equilibrium.展开更多
According to the scaling idea of local slope, we investigate numerically and analytically anomalous dynamic scaling behaviour of （1＋ 1）-dimensional growth equation for molecular-beam epitaxy. The growth model inclu...According to the scaling idea of local slope, we investigate numerically and analytically anomalous dynamic scaling behaviour of （1＋ 1）-dimensional growth equation for molecular-beam epitaxy. The growth model includes the linear molecular-beam epitaxy （LMBE） and the nonlinear Lai-Das Sarma-Villain （LDV） equations. The anomalous scaling exponents in both the LMBE and the LDV equations are obtained, respectively. Numerical results are consistent with the corresponding analytical predictions.展开更多
Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports c...Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients(FDCs)are radially dependent or not.When a radially dependent FDC<D_(α)(r)1 is imposed,compared with the case under=D_(α)(r)1.0,it is observed that the position of the peak of the density profile is closer to the core.Further,it is found that when FDCs at the positions of source injections increase,the peak values of density profiles decrease.The non-local effect becomes significant as the order of fractional derivative a 1 and causes the uphill transport.However,as a 2,the fractional diffusion model returns to the standard model governed by Fick’s law.展开更多
Based on the scaling idea of local slopes by Lopez et al. [Phys. Rev. Lett. 94 （2005） 166103], we investigate anomalous dynamic scaling of （d ＋ 1）-dimensional surface growth equations with spatially and temporall...Based on the scaling idea of local slopes by Lopez et al. [Phys. Rev. Lett. 94 （2005） 166103], we investigate anomalous dynamic scaling of （d ＋ 1）-dimensional surface growth equations with spatially and temporally correlated noise. The growth equations studied include the Kardar-Parisi-Zhang （KPZ）, Sun-Guo-Grant （SGG）, and Lai-Das Sarma-Villain （LDV） equations. The anomalous scaling exponents in both the weak- and strong-coupling regions are obtained, respectively.展开更多
Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations.The use of temporal fractional transport equations means that the order of the partial derivative with...Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations.The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α(α ∈(0, 1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result,if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.展开更多
The solution of a nonlinear diffusion equation is numerically investigated using the generalized Fourier transform method. This equation includes fractal dimensions and power-law dependence on the radial variable and ...The solution of a nonlinear diffusion equation is numerically investigated using the generalized Fourier transform method. This equation includes fractal dimensions and power-law dependence on the radial variable and on the diffusion function. The generalized Fourier transform approach is the extension of the Fourier transform method used for the normal diffusion equation. The feasibility of the approach is validated by comparing the numerical result with the exact solution for a point-source. The merit of the numerical method is that it provides a way to calculate anomalous diffusion with an arbitrary initial condition.展开更多
Synchrotron radiation based experimental techniques known as Anomalous Small-Angle X-ray Scattering (ASAXS) provide deep insight into the nanostructure of uncountable material systems in condensed matter research i.e....Synchrotron radiation based experimental techniques known as Anomalous Small-Angle X-ray Scattering (ASAXS) provide deep insight into the nanostructure of uncountable material systems in condensed matter research i.e. solid state physics, chemistry, engineering and life sciences thereby rendering the origin of the macroscopic functionalization of the various materials via correlation to its structural architecture on a nanometer length scale. The techniques constitute a system of linear equations, which can be treated by matrix theory. The study aims to analyze the significance of the solutions of the stated matrix equations by use of the so-called condition numbers first introduced by A. Turing, J. von Neumann and H. Goldstine. Special attention was given for the comparison with direct methods i.e. the Gaussian elimination method. The mathematical roots of ill-posed ASAXS equations preventing matrix inversion have been identified. In the framework of the theory of von Neumann and Goldstine the inversion of certain matrices constituted by ASAXS gradually becomes impossible caused by non-definiteness. In Turing’s theory which starts from more general prerequisites, the principal minors of the same matrices approach singularity thereby imposing large errors on inversion. In conclusion both theories recommend for extremely ill-posed ASAXS problems avoiding inversion and the use of direct methods for instance Gaussian elimination.展开更多
A new method - perturbative summation to infinite order is presented to obtain the anomalous dimension in the solution of the modified porous medium equation. The result is the same as that in the renormalization grou...A new method - perturbative summation to infinite order is presented to obtain the anomalous dimension in the solution of the modified porous medium equation. The result is the same as that in the renormalization group (RG) approach. It gives us an insight into the anomalous exponent in the asymptotic solution of the modified porous medium equation in the RG approach. Based on this discussion, we can see that the anomalous dimension appears naturally in the problem and the nonlinearity reflects the anomalous long-time behavior of the system.展开更多
A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anom...A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.展开更多
In this paper,we survey some recent progress in the study of time fractional equations and its interplay with anomalous sub-diffusions,with some improvements and extensions.
In order to better describe the phenomenon of biological invasion,this paper introduces a free boundary model of biological invasion.Firstly,the right free boundary is added to the equation with logistic terms.Secondl...In order to better describe the phenomenon of biological invasion,this paper introduces a free boundary model of biological invasion.Firstly,the right free boundary is added to the equation with logistic terms.Secondly,the existence and uniqueness of local solutions are proved by the Sobolev embedding theorem and the comparison principle.Finally,according to the relevant research data and contents of red fire ants,the diffusion area and nest number of red fire ants were simulated without external disturbance.This paper mainly simulates the early diffusion process of red fire ants.In the early diffusion stage,red fire ants grow slowly and then spread over a large area after reaching a certain number.展开更多
Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay di...Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay differential equations(DDEs)and delay integrodifferential equations(DIDEs)with constant delays,primarily due to their low regularity at delayinduced breaking points.In this paper,a DNN method that combines multi-task learning(MTL)which is proposed to solve both the forward and inverse problems of DIDEs.The core idea of this approach is to divide the original equation into multiple tasks based on the delay,using auxiliary outputs to represent the integral terms,followed by the use of MTL to seamlessly incorporate the properties at the breaking points into the loss function.Furthermore,given the increased training dificulty associated with multiple tasks and outputs,we employ a sequential training scheme to reduce training complexity and provide reference solutions for subsequent tasks.This approach significantly enhances the approximation accuracy of solving DIDEs with DNNs,as demonstrated by comparisons with traditional DNN methods.We validate the effectiveness of this method through several numerical experiments,test various parameter sharing structures in MTL and compare the testing results of these structures.Finally,this method is implemented to solve the inverse problem of nonlinear DIDE and the results show that the unknown parameters of DIDE can be discovered with sparse or noisy data.展开更多
This paper deals with quasilinear elliptic equations of singular growth like-Δu-uΔ(u^(2))=a(x)u^(-1).We establish the existence of positive solutions for general a(x)∈L^(p)(Ω),p>2,whereΩis a bounded domain inℝ...This paper deals with quasilinear elliptic equations of singular growth like-Δu-uΔ(u^(2))=a(x)u^(-1).We establish the existence of positive solutions for general a(x)∈L^(p)(Ω),p>2,whereΩis a bounded domain inℝ^(N)with N≥1.展开更多
In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with ...In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods.展开更多
In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,t...In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes.展开更多
In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belon...In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.展开更多
We study the Cauchy problem of the Kolmogorov-Fokker-Planck equations and show that the solution enjoys an analytic smoothing effect with L?initial datum for positive time.
Berry curvature describes the intrinsic geometric property of electronic band structure,crucial for governing emergent transport phenomena.As a typical Berry-curvature-related property,the anomalous Nernst effect prob...Berry curvature describes the intrinsic geometric property of electronic band structure,crucial for governing emergent transport phenomena.As a typical Berry-curvature-related property,the anomalous Nernst effect probes local Berry curvature near the Fermi surface,whereas the anomalous Hall effect integrates contributions across all occupied states.Thus,the anomalous Nernst effect is a superior probe for detecting subtle evolution of Berry curvature near the Fermi level;however,their relation remains elusive.Here,we demonstrate giant anomalous Nernst angles induced by Berry curvature in layered itinerant ferromagnets Fe_(3)GaTe_(2)and Fe_(5)GeTe_(2).Their maximum values(≈0.33 for Fe_(3)GaTe_(2)and≈0.41 for Fe_(5)GeTe_(2))are one order of magnitude larger than those of traditional ferromagnets(θ_(AN)^(max)<0.02).Scaling analysis of anomalous Hall effect in these two systems further suggests these giant angles originate from intrinsic Berry curvature.These findings indicate Berrycurvature-dominated regimes,and establish these materials for high-performance spin-caloritronic devices.展开更多
With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixe...With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixed types.Our primary contribution is the establishment of solution existence,illuminating the dynamics of these complex equations.To tackle this challenging problem,we construct an approximate solution sequence and apply the contraction mapping principle to rigorously prove local solution existence.Our results significantly advance the understanding of nonlinear evolution equations of mixed types.Furthermore,they provide a versatile,powerful approach for tackling analogous challenges across physics,engineering,and applied mathematics,making this work a valuable reference for researchers in these fields.展开更多
In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations,numerous tricks have been proposed.The goal of this short review is to explain how a theorem of Eremenko on meromorphic...In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations,numerous tricks have been proposed.The goal of this short review is to explain how a theorem of Eremenko on meromorphic solutions of some nonlinear ODEs together with some classical,19th-century results,can be turned into algorithms(thus avoiding ad hoc assumptions)which provide all(as opposed to some)solutions in a precise class.To illustrate these methods,we present some new such exact solutions,physically relevant.展开更多
文摘Considering that thermodynarmic irreversibility and hydrodynamic equations can not be derived rigorously and unifiedly from the Liouville equations, the anomalous Langevin equation in the Liouville space is proposed as a fundamental equation of statistical physics. This equation reflects that the law of motion of particles obeying reversible, deterministic laws in dynamics becomes irreversible and stochastic in thermodynamics. From this the fundamental equations of nonequilibrium thermodynamics, the principle of entropy increase and the theorem of minimum entropy production have been derived. The hydrodynamic equations, such as the generalized Navier-Stokes equation and the mass drift-diffusion equation etc. have been derived rigorously from the kinetic kinetic equation which is reduced from the anomalous Langevin equation in Liouville space. All these are unified and self consistent. But it is difficult to prove that entropy production density σ can never be negative everywhere for all the isolated inhomogeneous systems far from equilibrium.
基金supported by the Fundamental Research Funds for the Central Universities (Grant No. 2010LKWL04)the Youth Foundation of China University of Mining & Technology,China (Grant No. 2008A035)
文摘According to the scaling idea of local slope, we investigate numerically and analytically anomalous dynamic scaling behaviour of （1＋ 1）-dimensional growth equation for molecular-beam epitaxy. The growth model includes the linear molecular-beam epitaxy （LMBE） and the nonlinear Lai-Das Sarma-Villain （LDV） equations. The anomalous scaling exponents in both the LMBE and the LDV equations are obtained, respectively. Numerical results are consistent with the corresponding analytical predictions.
基金supported by the National MCF Energy R&D Program of China(No.2019YFE03090300)National Natural Science Foundation of China(No.11925501)Fundamental Research Funds for the Central Universities(No.DUT21GJ204)。
文摘Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients(FDCs)are radially dependent or not.When a radially dependent FDC<D_(α)(r)1 is imposed,compared with the case under=D_(α)(r)1.0,it is observed that the position of the peak of the density profile is closer to the core.Further,it is found that when FDCs at the positions of source injections increase,the peak values of density profiles decrease.The non-local effect becomes significant as the order of fractional derivative a 1 and causes the uphill transport.However,as a 2,the fractional diffusion model returns to the standard model governed by Fick’s law.
基金National Natural Science Foundation of China under Grant No.10674177
文摘Based on the scaling idea of local slopes by Lopez et al. [Phys. Rev. Lett. 94 （2005） 166103], we investigate anomalous dynamic scaling of （d ＋ 1）-dimensional surface growth equations with spatially and temporally correlated noise. The growth equations studied include the Kardar-Parisi-Zhang （KPZ）, Sun-Guo-Grant （SGG）, and Lai-Das Sarma-Villain （LDV） equations. The anomalous scaling exponents in both the weak- and strong-coupling regions are obtained, respectively.
基金supported by the National Key R&D Program of China (Grant No. 2022YFE03090000)the National Natural Science Foundation of China (Grant No. 11925501)the Fundamental Research Fund for the Central Universities (Grant No. DUT22ZD215)。
文摘Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations.The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α(α ∈(0, 1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result,if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.
文摘The solution of a nonlinear diffusion equation is numerically investigated using the generalized Fourier transform method. This equation includes fractal dimensions and power-law dependence on the radial variable and on the diffusion function. The generalized Fourier transform approach is the extension of the Fourier transform method used for the normal diffusion equation. The feasibility of the approach is validated by comparing the numerical result with the exact solution for a point-source. The merit of the numerical method is that it provides a way to calculate anomalous diffusion with an arbitrary initial condition.
文摘Synchrotron radiation based experimental techniques known as Anomalous Small-Angle X-ray Scattering (ASAXS) provide deep insight into the nanostructure of uncountable material systems in condensed matter research i.e. solid state physics, chemistry, engineering and life sciences thereby rendering the origin of the macroscopic functionalization of the various materials via correlation to its structural architecture on a nanometer length scale. The techniques constitute a system of linear equations, which can be treated by matrix theory. The study aims to analyze the significance of the solutions of the stated matrix equations by use of the so-called condition numbers first introduced by A. Turing, J. von Neumann and H. Goldstine. Special attention was given for the comparison with direct methods i.e. the Gaussian elimination method. The mathematical roots of ill-posed ASAXS equations preventing matrix inversion have been identified. In the framework of the theory of von Neumann and Goldstine the inversion of certain matrices constituted by ASAXS gradually becomes impossible caused by non-definiteness. In Turing’s theory which starts from more general prerequisites, the principal minors of the same matrices approach singularity thereby imposing large errors on inversion. In conclusion both theories recommend for extremely ill-posed ASAXS problems avoiding inversion and the use of direct methods for instance Gaussian elimination.
文摘A new method - perturbative summation to infinite order is presented to obtain the anomalous dimension in the solution of the modified porous medium equation. The result is the same as that in the renormalization group (RG) approach. It gives us an insight into the anomalous exponent in the asymptotic solution of the modified porous medium equation in the RG approach. Based on this discussion, we can see that the anomalous dimension appears naturally in the problem and the nonlinearity reflects the anomalous long-time behavior of the system.
文摘A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.
文摘In this paper,we survey some recent progress in the study of time fractional equations and its interplay with anomalous sub-diffusions,with some improvements and extensions.
基金Supported by National Natural Science Foundation of China(12101482)Postdoctoral Science Foundation of China(2022M722604)+2 种基金General Project of Science and Technology of Shaanxi Province(2023-YBSF-372)The Natural Science Foundation of Shaan Xi Province(2023-JCQN-0016)Shannxi Mathmatical Basic Science Research Project(23JSQ042)。
文摘In order to better describe the phenomenon of biological invasion,this paper introduces a free boundary model of biological invasion.Firstly,the right free boundary is added to the equation with logistic terms.Secondly,the existence and uniqueness of local solutions are proved by the Sobolev embedding theorem and the comparison principle.Finally,according to the relevant research data and contents of red fire ants,the diffusion area and nest number of red fire ants were simulated without external disturbance.This paper mainly simulates the early diffusion process of red fire ants.In the early diffusion stage,red fire ants grow slowly and then spread over a large area after reaching a certain number.
文摘Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay differential equations(DDEs)and delay integrodifferential equations(DIDEs)with constant delays,primarily due to their low regularity at delayinduced breaking points.In this paper,a DNN method that combines multi-task learning(MTL)which is proposed to solve both the forward and inverse problems of DIDEs.The core idea of this approach is to divide the original equation into multiple tasks based on the delay,using auxiliary outputs to represent the integral terms,followed by the use of MTL to seamlessly incorporate the properties at the breaking points into the loss function.Furthermore,given the increased training dificulty associated with multiple tasks and outputs,we employ a sequential training scheme to reduce training complexity and provide reference solutions for subsequent tasks.This approach significantly enhances the approximation accuracy of solving DIDEs with DNNs,as demonstrated by comparisons with traditional DNN methods.We validate the effectiveness of this method through several numerical experiments,test various parameter sharing structures in MTL and compare the testing results of these structures.Finally,this method is implemented to solve the inverse problem of nonlinear DIDE and the results show that the unknown parameters of DIDE can be discovered with sparse or noisy data.
基金Supported by National Science Foundation of China(11971027,12171497)。
文摘This paper deals with quasilinear elliptic equations of singular growth like-Δu-uΔ(u^(2))=a(x)u^(-1).We establish the existence of positive solutions for general a(x)∈L^(p)(Ω),p>2,whereΩis a bounded domain inℝ^(N)with N≥1.
基金Supported by the National Natural Science Foundation of China(11671403,11671236,12101192)Henan Provincial General Natural Science Foundation Project(232300420113)。
文摘In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods.
基金Supported by Research Project Supported by Shanxi Scholarship Council of China(2021-029)International Cooperation Base and Platform Project of Shanxi Province(202104041101019)+2 种基金Basic Research Plan of Shanxi Province(202203021211129)Shanxi Province Natural Science Research(202203021212249)Special/Youth Foundation of Taiyuan University of Technology(2022QN101)。
文摘In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes.
文摘In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.
基金Supported by NSFC (No.12031006)Fundamental Research Funds for the Central Universities of China。
文摘We study the Cauchy problem of the Kolmogorov-Fokker-Planck equations and show that the solution enjoys an analytic smoothing effect with L?initial datum for positive time.
基金supported by the National Key Research and Development Program of China(Grant Nos.2024YFA1408104,2021YFA1202901,and 2022YFA1204001)the National Natural Science Foundation of China(Grant Nos.92365203,U24A6002,52302180,and U21A2086)+1 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20243011)the Hebei Natural Science Foundation No.E2023203002).
文摘Berry curvature describes the intrinsic geometric property of electronic band structure,crucial for governing emergent transport phenomena.As a typical Berry-curvature-related property,the anomalous Nernst effect probes local Berry curvature near the Fermi surface,whereas the anomalous Hall effect integrates contributions across all occupied states.Thus,the anomalous Nernst effect is a superior probe for detecting subtle evolution of Berry curvature near the Fermi level;however,their relation remains elusive.Here,we demonstrate giant anomalous Nernst angles induced by Berry curvature in layered itinerant ferromagnets Fe_(3)GaTe_(2)and Fe_(5)GeTe_(2).Their maximum values(≈0.33 for Fe_(3)GaTe_(2)and≈0.41 for Fe_(5)GeTe_(2))are one order of magnitude larger than those of traditional ferromagnets(θ_(AN)^(max)<0.02).Scaling analysis of anomalous Hall effect in these two systems further suggests these giant angles originate from intrinsic Berry curvature.These findings indicate Berrycurvature-dominated regimes,and establish these materials for high-performance spin-caloritronic devices.
基金Supported by the National Natural Science Foundation of China(12201368,62376252)Key Project of Natural Science Foundation of Zhejiang Province(LZ22F030003)Zhejiang Province Leading Geese Plan(2024C02G1123882,2024C01SA100795).
文摘With the urgent need to resolve complex behaviors in nonlinear evolution equations,this study makes a contribution by establishing the local existence of solutions for Cauchy problems associated with equations of mixed types.Our primary contribution is the establishment of solution existence,illuminating the dynamics of these complex equations.To tackle this challenging problem,we construct an approximate solution sequence and apply the contraction mapping principle to rigorously prove local solution existence.Our results significantly advance the understanding of nonlinear evolution equations of mixed types.Furthermore,they provide a versatile,powerful approach for tackling analogous challenges across physics,engineering,and applied mathematics,making this work a valuable reference for researchers in these fields.
基金partially supported by RGC(No.17307420)supported by NSFC(No.12471077)。
文摘In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations,numerous tricks have been proposed.The goal of this short review is to explain how a theorem of Eremenko on meromorphic solutions of some nonlinear ODEs together with some classical,19th-century results,can be turned into algorithms(thus avoiding ad hoc assumptions)which provide all(as opposed to some)solutions in a precise class.To illustrate these methods,we present some new such exact solutions,physically relevant.
