This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through...This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.展开更多
This study examines the mathematical foundations of the Euler and Navier-Stokes equations of fluid dynamics, identifying some inconsistencies in the mathematical definitions of flow velocity and the material derivativ...This study examines the mathematical foundations of the Euler and Navier-Stokes equations of fluid dynamics, identifying some inconsistencies in the mathematical definitions of flow velocity and the material derivative. We show that the flow velocity of a fluid parcel, which in the Lagrangian description is traditionally modeled as a bivariate function of the presumed independent variables of initial parcel position and time, is more accurately defined as a parametric function of time, with the initial parcel position treated as a time-dependent parameter. This finding leads to the result that the standard form of the material derivative in the Lagrangian description is mathematically inconsistent. We also show that if the fluid flow is non-unidirectional, then the map from parcel position to flow velocity becomes a one-to-many map, leading to the conclusion that the flow velocity is not a valid mathematical function of position in both the Lagrangian and Eulerian descriptions under such conditions. Therefore, if flow velocity is not a valid mathematical function of position, we conclude that the inability to integrate the Euler and Navier-Stokes differential equations in the spatial domain implies the nonexistence of a mathematical solution of these equations under these conditions. Additionally, through mathematical and theoretical analysis, supported by experimental and numerical simulations, we uncover challenges in the material consistency of the definition of the material derivative in the Eulerian description. This inconsistency leads to a decoupling between the Lagrangian and Eulerian descriptions, especially under complex non-unidirectional flow conditions and multi-directional flows with intersecting pathlines. We also show that the Eulerian description is a quasi-continuum mechanics model that, when applied to certain fluids, especially gases and low-viscosity liquids where intermolecular forces are weak or intermediate, limits the ability to accurately model the bi-directional transmission of deformation and force continuously between neighboring parcels. While the Euler and Navier-Stokes equations remain largely valid and effective for modeling unidirectional flows in viscous fluids, our findings suggest the need to refocus on developing fluid dynamics solutions rooted in the Lagrangian model to more accurately capture complex flow behaviors and improve applicability across fields such as atmospheric sciences, oceanography, and plasma physics. These insights aim to advance our understanding of the limits of existing fluid dynamics models by addressing foundational inconsistencies, the understanding of which can contribute to refining these mathematical models.展开更多
In this paper,we prove that there exists a unique local solution for the Cauchy problem of a system of the incompressible Navier-Stokes-Landau-Lifshitz equations with the Dzyaloshinskii-Moriya interaction and V-flow t...In this paper,we prove that there exists a unique local solution for the Cauchy problem of a system of the incompressible Navier-Stokes-Landau-Lifshitz equations with the Dzyaloshinskii-Moriya interaction and V-flow term inR^(2) and R^(3).Our methods rely upon approximating the system with a perturbed parabolic system and parallel transport.展开更多
The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA00...The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA0012 airfoil is used to verify the efficiency of the proposed method. Two cases of the low Mach number flows around the multi-element airfoil and the circular cylinder are also used to test the proposed method. Numerical results show that the methods combined the preconditioning method and compressible Navier-Stokes equations are efficient to solve low Mach number flows.展开更多
In this paper we propose a new method for obtaining the exact solutions of the Navier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations devel...In this paper we propose a new method for obtaining the exact solutions of the Navier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author. Using the present method we can find some new exact solu- tions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a va- riety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates.展开更多
This paper deals with the inertial manifold and the approximate inertialmanifold concepts of the Navier-Stokes equations with nonhomogeneous boundary conditions and inertial algorithm. Furtheremore,we provide the erro...This paper deals with the inertial manifold and the approximate inertialmanifold concepts of the Navier-Stokes equations with nonhomogeneous boundary conditions and inertial algorithm. Furtheremore,we provide the error estimates of the approximate solutions of the Navier-Stokes Equations.展开更多
A global finite element nonlinear Galerkin method for the penalized Navier-Stokes equations is presented. This method is based on two finite element spaces XH and Xh, defined respectively on one coarse grid with grid ...A global finite element nonlinear Galerkin method for the penalized Navier-Stokes equations is presented. This method is based on two finite element spaces XH and Xh, defined respectively on one coarse grid with grid size H and one fine grid with grid size h << H. Comparison is also made with the finite element Galerkin method. If we choose H = O(), E > 0 being the penalty parameter, then two methods are of the same order of approximation. However, the global finite element nonlinear Galerkin method is much cheaper than the standard finite element Galerkin method. In fact, in the finite element Galerkin method the nonlinearity is treated on the fine grid finite element space Xh and while in the global finite element nonlinear Galerkin method the similar nonlinearity is treated on the coarse grid finite element space XH and only the linearity needs to be treated on the fine grid incremeat finite element space Wh. Finally, we provide numerical test which shows above results stated.展开更多
In this paper, problems of the flow over a fat plate in the large Reynolds numbercase are studied by using the method of multiple scales￣[1,2].We have obtained N-orderuniformly valid asymptotic solutions of the Naver...In this paper, problems of the flow over a fat plate in the large Reynolds numbercase are studied by using the method of multiple scales￣[1,2].We have obtained N-orderuniformly valid asymptotic solutions of the Naver-Stodes equations.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
Biomass is among the most important state variables used to characterize ecosystems. Estimation of tree biomass involves the development of species-specific “allometric equations” that describe the relationship betw...Biomass is among the most important state variables used to characterize ecosystems. Estimation of tree biomass involves the development of species-specific “allometric equations” that describe the relationship between tree biomass and tree diameter and/or height. While many allometric equations were developed for northern hemisphere and tropical species, rarely have they been developed for trees in arid ecosystems, limiting, amongst other things, our ability to estimate carbon stocks in arid regions. Acacia raddiana and A. tortilis are major components of savannas and arid regions in the Middle East and Africa, where they are considered keystone species. Using the opportunity that trees were being uprooted for land development, we measured height (H), north-south (C1) and east-west (C2) canopy diameters, stem diameter at 1.3 meters of the largest stem (D1.3 or DBH), and aboveground fresh and dry weight (FW and DW, respectively) of nine trees (n = 9) from each species. For A. tortilis only, we recorded the number of trunks, and measured the diameter of the largest trunk at ground level (D0). While the average crown (canopy) size (C1 + C2) was very similar among the two species, Acacia raddiana trees were found to be significantly taller than their Acacia tortilis counterparts. Results show that in the arid Arava (southern Israel), an average adult acacia tree has ~200 kg of aboveground dry biomass and that a typical healthy acacia ecosystem in this region, may include ~41 tons of tree biomass per km2. The coefficients of DBH (tree diameter at breast height) to biomass and wood volume, could be used by researchers studying acacia trees throughout the Middle East and Africa, enabling them to estimate biomass of acacia trees and to evaluate their importance for carbon stocks in their arid regions. Highlights: 1) Estimations of tree biomass in arid regions are rare. 2) Biomass allometric equations were developed for A. raddiana and A. tortilis trees. 3) Equations contribute to the estimation of carbon stocks in arid regions.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
In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use ...In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use the Z1transformation to get the general solutions of some nonlinear partial differential equations for the first time, and use the general solutions to obtain the exact solutions of some typical definite solution problems.展开更多
ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lew...ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed CCFL=0.95for their experiments. Nonetheless, we noted that the experiments conducted in this study with CCFL=0.95produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.展开更多
In this paper, we get the N-fold Darboux transformation with multi-parameters for the coupled mKdV equations with the help of a guage transformation of the spectral problem. As an application, some new multi-soliton s...In this paper, we get the N-fold Darboux transformation with multi-parameters for the coupled mKdV equations with the help of a guage transformation of the spectral problem. As an application, some new multi-soliton solutions and complexiton solutions are obtained from choosing the appropriate seed solution. All obtained solutions and N-fold Darboux transformations are expressed using the Vandermonde-like determinants.展开更多
本文证明带有临界型阻尼项的Navier-Stokes方程在Lei-Lin-Gevrey空间Xa,σ0(ℝ3)中存在唯一的局部解。文章利用不动点定理和热方程解的有关性质来证明这一主要结论。In this paper, it is proved that the Navier-Stokes equation with c...本文证明带有临界型阻尼项的Navier-Stokes方程在Lei-Lin-Gevrey空间Xa,σ0(ℝ3)中存在唯一的局部解。文章利用不动点定理和热方程解的有关性质来证明这一主要结论。In this paper, it is proved that the Navier-Stokes equation with critical damping terms has a unique local solution in the Lei-Lin-Gevrey space Xa,σ0(ℝ3). In this paper, the main conclusion is proved by using the fixed point theorem and the related properties of the solution of the heat equation.展开更多
We examine a quasilinear system of viscoelastic equations in this study that have fractional boundary conditions,dispersion,source,and variable-exponents.We discovered that the solution of the system is global and con...We examine a quasilinear system of viscoelastic equations in this study that have fractional boundary conditions,dispersion,source,and variable-exponents.We discovered that the solution of the system is global and constrained under the right assumptions about the relaxation functions and initial conditions.After that,it is demonstrated that the blow-up has negative initial energy.Subsequently,the growth of solutions is demonstrated with positive initial energy,and the general decay result in the absence of the source term is achieved by using an integral inequality due to Komornik.展开更多
In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with ...In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with N≥1,m>0,p>1,such that m(p-1)>1.We give a clear classification of the self-similar solutions of the form u(x,t)=(βt)^(-α/β)((βt)^(-1/β)|x|)withα∈R andβ=α[m(p-1)-1]+p,regular or singular at the origin point.The existence and uniqueness of some solutions are established by the phase plane analysis method,and the asymptotic properties of the solutions near the origin and the infinity are also described.This paper extends the classical results of self-similar solutions for degeneratep-Laplace heat equation by Bidaut-Véron[Proc Royal Soc Edinburgh,2009,139:1-43]to the doubly nonlinear degenerate diffusion equations.展开更多
文摘This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.
文摘This study examines the mathematical foundations of the Euler and Navier-Stokes equations of fluid dynamics, identifying some inconsistencies in the mathematical definitions of flow velocity and the material derivative. We show that the flow velocity of a fluid parcel, which in the Lagrangian description is traditionally modeled as a bivariate function of the presumed independent variables of initial parcel position and time, is more accurately defined as a parametric function of time, with the initial parcel position treated as a time-dependent parameter. This finding leads to the result that the standard form of the material derivative in the Lagrangian description is mathematically inconsistent. We also show that if the fluid flow is non-unidirectional, then the map from parcel position to flow velocity becomes a one-to-many map, leading to the conclusion that the flow velocity is not a valid mathematical function of position in both the Lagrangian and Eulerian descriptions under such conditions. Therefore, if flow velocity is not a valid mathematical function of position, we conclude that the inability to integrate the Euler and Navier-Stokes differential equations in the spatial domain implies the nonexistence of a mathematical solution of these equations under these conditions. Additionally, through mathematical and theoretical analysis, supported by experimental and numerical simulations, we uncover challenges in the material consistency of the definition of the material derivative in the Eulerian description. This inconsistency leads to a decoupling between the Lagrangian and Eulerian descriptions, especially under complex non-unidirectional flow conditions and multi-directional flows with intersecting pathlines. We also show that the Eulerian description is a quasi-continuum mechanics model that, when applied to certain fluids, especially gases and low-viscosity liquids where intermolecular forces are weak or intermediate, limits the ability to accurately model the bi-directional transmission of deformation and force continuously between neighboring parcels. While the Euler and Navier-Stokes equations remain largely valid and effective for modeling unidirectional flows in viscous fluids, our findings suggest the need to refocus on developing fluid dynamics solutions rooted in the Lagrangian model to more accurately capture complex flow behaviors and improve applicability across fields such as atmospheric sciences, oceanography, and plasma physics. These insights aim to advance our understanding of the limits of existing fluid dynamics models by addressing foundational inconsistencies, the understanding of which can contribute to refining these mathematical models.
文摘In this paper,we prove that there exists a unique local solution for the Cauchy problem of a system of the incompressible Navier-Stokes-Landau-Lifshitz equations with the Dzyaloshinskii-Moriya interaction and V-flow term inR^(2) and R^(3).Our methods rely upon approximating the system with a perturbed parabolic system and parallel transport.
文摘The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA0012 airfoil is used to verify the efficiency of the proposed method. Two cases of the low Mach number flows around the multi-element airfoil and the circular cylinder are also used to test the proposed method. Numerical results show that the methods combined the preconditioning method and compressible Navier-Stokes equations are efficient to solve low Mach number flows.
基金The project supported by National Natural Science Foundation of China.
文摘In this paper we propose a new method for obtaining the exact solutions of the Navier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author. Using the present method we can find some new exact solu- tions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a va- riety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates.
文摘This paper deals with the inertial manifold and the approximate inertialmanifold concepts of the Navier-Stokes equations with nonhomogeneous boundary conditions and inertial algorithm. Furtheremore,we provide the error estimates of the approximate solutions of the Navier-Stokes Equations.
基金Subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China19971067, NSF of Shaanxi P
文摘A global finite element nonlinear Galerkin method for the penalized Navier-Stokes equations is presented. This method is based on two finite element spaces XH and Xh, defined respectively on one coarse grid with grid size H and one fine grid with grid size h << H. Comparison is also made with the finite element Galerkin method. If we choose H = O(), E > 0 being the penalty parameter, then two methods are of the same order of approximation. However, the global finite element nonlinear Galerkin method is much cheaper than the standard finite element Galerkin method. In fact, in the finite element Galerkin method the nonlinearity is treated on the fine grid finite element space Xh and while in the global finite element nonlinear Galerkin method the similar nonlinearity is treated on the coarse grid finite element space XH and only the linearity needs to be treated on the fine grid incremeat finite element space Wh. Finally, we provide numerical test which shows above results stated.
文摘In this paper, problems of the flow over a fat plate in the large Reynolds numbercase are studied by using the method of multiple scales￣[1,2].We have obtained N-orderuniformly valid asymptotic solutions of the Naver-Stodes equations.
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Biomass is among the most important state variables used to characterize ecosystems. Estimation of tree biomass involves the development of species-specific “allometric equations” that describe the relationship between tree biomass and tree diameter and/or height. While many allometric equations were developed for northern hemisphere and tropical species, rarely have they been developed for trees in arid ecosystems, limiting, amongst other things, our ability to estimate carbon stocks in arid regions. Acacia raddiana and A. tortilis are major components of savannas and arid regions in the Middle East and Africa, where they are considered keystone species. Using the opportunity that trees were being uprooted for land development, we measured height (H), north-south (C1) and east-west (C2) canopy diameters, stem diameter at 1.3 meters of the largest stem (D1.3 or DBH), and aboveground fresh and dry weight (FW and DW, respectively) of nine trees (n = 9) from each species. For A. tortilis only, we recorded the number of trunks, and measured the diameter of the largest trunk at ground level (D0). While the average crown (canopy) size (C1 + C2) was very similar among the two species, Acacia raddiana trees were found to be significantly taller than their Acacia tortilis counterparts. Results show that in the arid Arava (southern Israel), an average adult acacia tree has ~200 kg of aboveground dry biomass and that a typical healthy acacia ecosystem in this region, may include ~41 tons of tree biomass per km2. The coefficients of DBH (tree diameter at breast height) to biomass and wood volume, could be used by researchers studying acacia trees throughout the Middle East and Africa, enabling them to estimate biomass of acacia trees and to evaluate their importance for carbon stocks in their arid regions. Highlights: 1) Estimations of tree biomass in arid regions are rare. 2) Biomass allometric equations were developed for A. raddiana and A. tortilis trees. 3) Equations contribute to the estimation of carbon stocks in arid regions.
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.
文摘In previous papers, we proposed the important Ztransformations and obtained general solutions to a large number of linear and quasi-linear partial differential equations for the first time. In this paper, we will use the Z1transformation to get the general solutions of some nonlinear partial differential equations for the first time, and use the general solutions to obtain the exact solutions of some typical definite solution problems.
文摘ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed CCFL=0.95for their experiments. Nonetheless, we noted that the experiments conducted in this study with CCFL=0.95produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.
文摘In this paper, we get the N-fold Darboux transformation with multi-parameters for the coupled mKdV equations with the help of a guage transformation of the spectral problem. As an application, some new multi-soliton solutions and complexiton solutions are obtained from choosing the appropriate seed solution. All obtained solutions and N-fold Darboux transformations are expressed using the Vandermonde-like determinants.
文摘本文证明带有临界型阻尼项的Navier-Stokes方程在Lei-Lin-Gevrey空间Xa,σ0(ℝ3)中存在唯一的局部解。文章利用不动点定理和热方程解的有关性质来证明这一主要结论。In this paper, it is proved that the Navier-Stokes equation with critical damping terms has a unique local solution in the Lei-Lin-Gevrey space Xa,σ0(ℝ3). In this paper, the main conclusion is proved by using the fixed point theorem and the related properties of the solution of the heat equation.
文摘We examine a quasilinear system of viscoelastic equations in this study that have fractional boundary conditions,dispersion,source,and variable-exponents.We discovered that the solution of the system is global and constrained under the right assumptions about the relaxation functions and initial conditions.After that,it is demonstrated that the blow-up has negative initial energy.Subsequently,the growth of solutions is demonstrated with positive initial energy,and the general decay result in the absence of the source term is achieved by using an integral inequality due to Komornik.
基金supported by the NSFC(12271178,12171166)the Guangzhou Basic and Applied Basic Research Foundation(2024A04J2022)the TCL Young Scholar(2024-2027).
文摘In this paper,we study the self-similar solutions of the degenerate diffusion equation ut-div(|▽u^(m)|^(p-2)▽u^(m))=0 of polytropic filtration diffusion in R^(N)×(0,±∞)or(R^(N)/{0})×(0,±∞)with N≥1,m>0,p>1,such that m(p-1)>1.We give a clear classification of the self-similar solutions of the form u(x,t)=(βt)^(-α/β)((βt)^(-1/β)|x|)withα∈R andβ=α[m(p-1)-1]+p,regular or singular at the origin point.The existence and uniqueness of some solutions are established by the phase plane analysis method,and the asymptotic properties of the solutions near the origin and the infinity are also described.This paper extends the classical results of self-similar solutions for degeneratep-Laplace heat equation by Bidaut-Véron[Proc Royal Soc Edinburgh,2009,139:1-43]to the doubly nonlinear degenerate diffusion equations.
