A mathematically rigorous framework for singular limits of the magnetohydrodynamic rotating shallow water equations with ill-prepared data is developed when the Rossby and Froude numbers tend to zero at different rate...A mathematically rigorous framework for singular limits of the magnetohydrodynamic rotating shallow water equations with ill-prepared data is developed when the Rossby and Froude numbers tend to zero at different rates.The reduced systems are derived,respectively,for the stratification-dominant and the rotation-dominant cases by means of the developed three-scale fast averaging method.展开更多
The existence of singular limit solutions are investigated by establishing a new Liouville type theorem for nonlinear elliptic system with sub-quadratic convection term and by using the nonlinear domain decomposition ...The existence of singular limit solutions are investigated by establishing a new Liouville type theorem for nonlinear elliptic system with sub-quadratic convection term and by using the nonlinear domain decomposition method.展开更多
In this paper, we consider the second-grade fluid equations in a 2D exterior domain satisfying the non-slip boundary conditions. The second-grade fluid model is a wellknown non-Newtonian fluid model, with two paramete...In this paper, we consider the second-grade fluid equations in a 2D exterior domain satisfying the non-slip boundary conditions. The second-grade fluid model is a wellknown non-Newtonian fluid model, with two parameters: α, which represents the length-scale,while ν > 0 corresponds to the viscosity. We prove that, as ν, α tend to zero, the solution of the second-grade fluid equations with suitable initial data converges to the one of Euler equations, provided that ν = o(α^(4/3)). Moreover, the convergent rate is obtained.展开更多
Singular limit is investigated for reaction-diffusion equations with an additive noise in a bounded domain of R^2. The solution converges to one of the two stable phases {+1, -1} determined from the reaction term; acc...Singular limit is investigated for reaction-diffusion equations with an additive noise in a bounded domain of R^2. The solution converges to one of the two stable phases {+1, -1} determined from the reaction term; accordingly a phase separation curve is generated in the limit. We shall derive a randomly perturbed motion by curvature for the dynamics of the phase separation curve.展开更多
The quasi-neutral limit of the Navier-Stokes-Poisson system modeling a viscous plasma with vanishing viscosity coefficients in the half-space■is rigorously proved under a Navier-slip boundary condition for velocity a...The quasi-neutral limit of the Navier-Stokes-Poisson system modeling a viscous plasma with vanishing viscosity coefficients in the half-space■is rigorously proved under a Navier-slip boundary condition for velocity and the Dirichlet boundary condition for electric potential.This is achieved by establishing the nonlinear stability of the approximation solutions involving the strong boundary layer in density and electric potential,which comes from the breakdown of the quasi-neutrality near the boundary,and dealing with the difficulty of the interaction of this strong boundary layer with the weak boundary layer of the velocity field.展开更多
In this paper we consider the asymptotic expression of the solution of the Cauchy’sproblem for a higher order equation when the limit equation has singularity. In orderto construct the asymptotic expression of the so...In this paper we consider the asymptotic expression of the solution of the Cauchy’sproblem for a higher order equation when the limit equation has singularity. In orderto construct the asymptotic expression of the solution, the region is divided into threesub-areas. In every small region, the solution of the differential equation is different.展开更多
This paper considers the singular perturbation of a fourth order elliptic equation when the limit equation is elliptic-parabolic. The equation involves a positive parameter, a positive real number, a Laplacian operato...This paper considers the singular perturbation of a fourth order elliptic equation when the limit equation is elliptic-parabolic. The equation involves a positive parameter, a positive real number, a Laplacian operator, and sufficient smoothness. Under appropriate condition the sufficient condition of solvability is derived, the existence of solution is proved and a uniformly valid asymptotic solution of arbitrary order is given.展开更多
Li has introduced the concepts of inverse system and direct system for fuzzy topological spaces and studied inverse limits and direct limits on such spaces by presenting the explicit constructions of these limits.In t...Li has introduced the concepts of inverse system and direct system for fuzzy topological spaces and studied inverse limits and direct limits on such spaces by presenting the explicit constructions of these limits.In this paper some important concepts of fuzzy topology,such as,product fuzzy topology,quotient fuzzy topology,fuzzy continuity etc.,are used for further study of inverse limits and direct limits for fuzzy topological spaces.展开更多
The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine ...The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.展开更多
The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the econom...The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the economic profit passes through zero, this model exhibits the transcritical bifurcation, the Hopf bifurcation, and the limit cycle. In particular, the system undergoes the singularity induced bifurcation at the positive equilibrium, which can result in impulse. Then, state feedback controllers closer to the actual control strategies are designed to eliminate the unexpected singularity induced bifurcation and stabilize the positive equilibrium under the positive profit. Finally, numerical simulations verify the results and illustrate the effectiveness of the controllers. Also, the model with positive economic profit is shown numerically to have different dynamics.展开更多
This paper is concerned with a singular limit for the one-dimensional compress- ible radiation hydrodynamics model. The singular limit we consider corresponds to the physical problem of letting the Bouguer number infi...This paper is concerned with a singular limit for the one-dimensional compress- ible radiation hydrodynamics model. The singular limit we consider corresponds to the physical problem of letting the Bouguer number infinite while keeping the Boltzmann number constant. In the case when the corresponding Euler system admits a contact discontinuity wave, Wang and Xie （2011） [12] recently verified this singular limit and proved that the solution of the compressible radiation hydrodynamics model converges to the strong contact 1 discontinuity wave in the L∞-norm away from the discontinuity line at a rate of ε1/4, as the reciprocal of the Bouguer number tends to zero. In this paper, Wang and Xie＇s convergence rate is improved to ε7/8 by introducing a new a priori assumption and some refined energy estimates. Moreover, it is shown that the radiation flux q tends to zero in the L∞-norm away from the discontinuity line, at a convergence rate as the reciprocal of the Bouguer number tends to zero.展开更多
基金supported by Natural Science Foundation of China(12001506)the Natural Science Foundation of Shandong Province(ZR2020QA014).
文摘A mathematically rigorous framework for singular limits of the magnetohydrodynamic rotating shallow water equations with ill-prepared data is developed when the Rossby and Froude numbers tend to zero at different rates.The reduced systems are derived,respectively,for the stratification-dominant and the rotation-dominant cases by means of the developed three-scale fast averaging method.
文摘The existence of singular limit solutions are investigated by establishing a new Liouville type theorem for nonlinear elliptic system with sub-quadratic convection term and by using the nonlinear domain decomposition method.
基金Aibin Zang was supported partially by the National Natural Science Foundation of China (11771382, 12061080, 12261093)the Jiangxi Provincial Natural Science Foundation (20224ACB201004)。
文摘In this paper, we consider the second-grade fluid equations in a 2D exterior domain satisfying the non-slip boundary conditions. The second-grade fluid model is a wellknown non-Newtonian fluid model, with two parameters: α, which represents the length-scale,while ν > 0 corresponds to the viscosity. We prove that, as ν, α tend to zero, the solution of the second-grade fluid equations with suitable initial data converges to the one of Euler equations, provided that ν = o(α^(4/3)). Moreover, the convergent rate is obtained.
文摘Singular limit is investigated for reaction-diffusion equations with an additive noise in a bounded domain of R^2. The solution converges to one of the two stable phases {+1, -1} determined from the reaction term; accordingly a phase separation curve is generated in the limit. We shall derive a randomly perturbed motion by curvature for the dynamics of the phase separation curve.
基金supported by National Natural Science Foundation of China(Grant Nos.12131007 and 12070144).supported by National Natural Science Foundation of China(Grant No.12001506)supported by a General Research Fund of Research Grants Council(Hong Kong)(Grant No.11306117)+1 种基金Natural Science Foundation of Shandong Province(Grant No.ZR2020QA014)supported by the Israel Science Foundation-National Natural Science Foundation of China Joint Research Program(Grant No.11761141008)。
文摘The quasi-neutral limit of the Navier-Stokes-Poisson system modeling a viscous plasma with vanishing viscosity coefficients in the half-space■is rigorously proved under a Navier-slip boundary condition for velocity and the Dirichlet boundary condition for electric potential.This is achieved by establishing the nonlinear stability of the approximation solutions involving the strong boundary layer in density and electric potential,which comes from the breakdown of the quasi-neutrality near the boundary,and dealing with the difficulty of the interaction of this strong boundary layer with the weak boundary layer of the velocity field.
文摘In this paper we consider the asymptotic expression of the solution of the Cauchy’sproblem for a higher order equation when the limit equation has singularity. In orderto construct the asymptotic expression of the solution, the region is divided into threesub-areas. In every small region, the solution of the differential equation is different.
文摘This paper considers the singular perturbation of a fourth order elliptic equation when the limit equation is elliptic-parabolic. The equation involves a positive parameter, a positive real number, a Laplacian operator, and sufficient smoothness. Under appropriate condition the sufficient condition of solvability is derived, the existence of solution is proved and a uniformly valid asymptotic solution of arbitrary order is given.
文摘Li has introduced the concepts of inverse system and direct system for fuzzy topological spaces and studied inverse limits and direct limits on such spaces by presenting the explicit constructions of these limits.In this paper some important concepts of fuzzy topology,such as,product fuzzy topology,quotient fuzzy topology,fuzzy continuity etc.,are used for further study of inverse limits and direct limits for fuzzy topological spaces.
基金Supported by Science Fund of the Education Departmentof Guangxi province( 2 0 0 3) and the NationalNatural Science Foundation of China( 1 0 361 0 0 3)
文摘The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.
基金supported by National Natural Science Foundation of China (No.60974004)Science Foundation of Ministry of Housing and Urban-Rural Development (No.2011-K5-31)
文摘The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the economic profit passes through zero, this model exhibits the transcritical bifurcation, the Hopf bifurcation, and the limit cycle. In particular, the system undergoes the singularity induced bifurcation at the positive equilibrium, which can result in impulse. Then, state feedback controllers closer to the actual control strategies are designed to eliminate the unexpected singularity induced bifurcation and stabilize the positive equilibrium under the positive profit. Finally, numerical simulations verify the results and illustrate the effectiveness of the controllers. Also, the model with positive economic profit is shown numerically to have different dynamics.
基金supported by the Doctoral Scientific Research Funds of Anhui University(J10113190005)the Tian Yuan Foundation of China(11426031)
文摘This paper is concerned with a singular limit for the one-dimensional compress- ible radiation hydrodynamics model. The singular limit we consider corresponds to the physical problem of letting the Bouguer number infinite while keeping the Boltzmann number constant. In the case when the corresponding Euler system admits a contact discontinuity wave, Wang and Xie （2011） [12] recently verified this singular limit and proved that the solution of the compressible radiation hydrodynamics model converges to the strong contact 1 discontinuity wave in the L∞-norm away from the discontinuity line at a rate of ε1/4, as the reciprocal of the Bouguer number tends to zero. In this paper, Wang and Xie＇s convergence rate is improved to ε7/8 by introducing a new a priori assumption and some refined energy estimates. Moreover, it is shown that the radiation flux q tends to zero in the L∞-norm away from the discontinuity line, at a convergence rate as the reciprocal of the Bouguer number tends to zero.
