Progress in hydrate thermodynamic study necessitates robust and fast models to be incorporated in reservoir simulation softwares. However, numerous models presented in the literature makes selection of the best,proper...Progress in hydrate thermodynamic study necessitates robust and fast models to be incorporated in reservoir simulation softwares. However, numerous models presented in the literature makes selection of the best,proper predictive model a cumbersome task. It is of industrial interest to make use of cubic equations of state(EOS) for modeling hydrate equilibria. In this regard, this study focuses on evaluation of three common EOSs including Peng–Robinson, Soave–Redlich–Kwong and Valderrama–Patel–Teja coupled with van der Waals and Platteeuw theory to predict hydrate P–T equilibrium of a real natural gas sample. Each EOS was accompanied with three mixing rules, including van der Waals(vd W),Avlonitis non-density dependent(ANDD) and general nonquadratic(GNQ). The prediction of cubic EOSs was in sufficient agreement with experimental data and with overall AARD% of less than unity. In addition, PR plus ANDD proved to be the most accurate model in this study for prediction of hydrate equilibria with AARD% of 0.166.It was observed that the accuracy of cubic EOSs studied in this paper depends on mixing rule coupled with them,especially at high-pressure conditions. Lastly, the present study does not include any adjustable parameter to be correlated with hydrate phase equilibrium data.展开更多
Inspired by Cardano's method for solving cubic scalar equations, the addi- tive decomposition of spherical/deviatoric tensor (DSDT) is revisited from a new view- point. This decomposition simplifies the cubic tenso...Inspired by Cardano's method for solving cubic scalar equations, the addi- tive decomposition of spherical/deviatoric tensor (DSDT) is revisited from a new view- point. This decomposition simplifies the cubic tensor equation, decouples the spher- ical/deviatoric strain energy density, and lays the foundation for the von Mises yield criterion. Besides, it is verified that under the precondition of energy decoupling and the simplest form, the DSDT is the only possible form of the additive decomposition with physical meanings.展开更多
In this paper, the stability of a cubic functional equation in the setting of intuitionistic random normed spaces is proved. We first introduce the notation of intuitionistic random normed spaces. Then, by virtue of t...In this paper, the stability of a cubic functional equation in the setting of intuitionistic random normed spaces is proved. We first introduce the notation of intuitionistic random normed spaces. Then, by virtue of this notation, we study the stability of a cubic functional equation in the setting of these spaces under arbitrary triangle norms. Furthermore, we present the interdisciplinary relation among the theory of random spaces, the theory of intuitionistic spaces, and the theory of functional equations.展开更多
In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double d...In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt + bux4 - duxxt = f(u)xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.展开更多
This paper considers the one-dimensional dissipative cubic nonlinear SchrSdinger equation with zero Dirichlet boundary conditions on a bounded domain. The equation is discretized in time by a linear implicit three-lev...This paper considers the one-dimensional dissipative cubic nonlinear SchrSdinger equation with zero Dirichlet boundary conditions on a bounded domain. The equation is discretized in time by a linear implicit three-level central difference scheme, which has analogous discrete conservation laws of charge and energy. The convergence with two orders and the stability of the scheme are analysed using a priori estimates. Numerical tests show that the three-level scheme is more efficient.展开更多
For further improving the representation of mixture VLE data,the local composition version of CCORequation of state has been developed and tested on 42 sets low-pressure and high-pressure as well as polarand nonpolar ...For further improving the representation of mixture VLE data,the local composition version of CCORequation of state has been developed and tested on 42 sets low-pressure and high-pressure as well as polarand nonpolar VLE data.The data reduction results were compared with conventional quadratic mixing ruleand activity coefficient method.The comparison with quadratic mixing rule showed that the local composition version significantly im-proved the data fitting of polar systems,especially for those highly nonideal mixtures where quadratic mixingrule failed to fit satisfactorily.The comparison with the well-known activity coefficient method——Hayden-O’Connell-Wilson model,indicated that this new version gave,in general,better fit to those low-pressure strongly polar systems,which traditionally has to be treated by activity coefficient approach.展开更多
Let G be an Abelian group and letρ:G×G→[0,∞) be a metric on G. Let E be a normed space. We prove that under some conditions if f:G→E is an odd function and Cx:G→E defined by Cx(y):=2 f (x+y)+2 f ...Let G be an Abelian group and letρ:G×G→[0,∞) be a metric on G. Let E be a normed space. We prove that under some conditions if f:G→E is an odd function and Cx:G→E defined by Cx(y):=2 f (x+y)+2 f (x-y)+12 f (x)-f (2x+y)-f (2x-y) is a cubic function for all x∈G, then there exists a cubic function C:G→E such that f?C is Lipschitz. Moreover, we investigate the stability of cubic functional equation 2 f (x+y)+2 f (x-y)+12 f (x)-f (2x+y)-f (2x-y)=0 on Lipschitz spaces.展开更多
An extended subequation rational expansion method is presented and used to construct some exact,analyt-ical solutions of the (2+1)-dimensional cubic nonlinear Schrdinger equation.From our results,many known solutionso...An extended subequation rational expansion method is presented and used to construct some exact,analyt-ical solutions of the (2+1)-dimensional cubic nonlinear Schrdinger equation.From our results,many known solutionsof the (2+1)-dimensional cubic nonlinear Schrdinger equation can be recovered by means of some suitable selections ofthe arbitrary functions and arbitrary constants.With computer simulation,the properties of new non-travelling waveand coefficient function's soliton-like solutions,and elliptic solutions are demonstrated by some plots.展开更多
A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarit...A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain (loss) coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.展开更多
In this paper, we determine the general solution of the functional equation f1 (2x + y) + f2(2x - y) = f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1...In this paper, we determine the general solution of the functional equation f1 (2x + y) + f2(2x - y) = f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 : R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x - y) = g(x + y) + g(x - y) + h(x) and f(2x + y) - f(2x - y) = g(x + y) - g(x - y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.展开更多
In this paper, we will find out the general solution and investigate the generalized Hyers-Ulam-Rassias stability problem for the following cubic functional equation2f(x + 2y) + f(2x - y) = 5f(x + y) + 5f(x...In this paper, we will find out the general solution and investigate the generalized Hyers-Ulam-Rassias stability problem for the following cubic functional equation2f(x + 2y) + f(2x - y) = 5f(x + y) + 5f(x - y)+ 15f(y)in the spirit of Hyers, Ulam, Rassias and Gavruta.展开更多
According to the inverse solution of elasticity mechanics, a stress function is constructed which meets the space biharmonic equation, this stress functions is about cubic function pressure on the inner and outer surf...According to the inverse solution of elasticity mechanics, a stress function is constructed which meets the space biharmonic equation, this stress functions is about cubic function pressure on the inner and outer surfaces of cylinder. When borderline condition that is predigested according to the Saint-Venant's theory is joined, an equation suit is constructed which meets both the biharmonic equations and the boundary conditions. Furthermore, its analytic solution is deduced with Matlab. When this theory is applied to hydraulic bulging rollers, the experimental results inosculate with the theoretic calculation. Simultaneously, the limit along the axis invariable direction is given and the famous Lame solution can be induced from this limit. The above work paves the way for mathematic model building of hollow cylinder and for the analytic solution of hollow cvlinder with randomly uneven pressure.展开更多
The nonlinear stability of thermal convection in a layer of an Oldroyd-B fluid-saturated Darcy porous medium with anisotropic permeability and thermal diffu- sivity is investigated with the perturbation method. A modi...The nonlinear stability of thermal convection in a layer of an Oldroyd-B fluid-saturated Darcy porous medium with anisotropic permeability and thermal diffu- sivity is investigated with the perturbation method. A modified Darcy-Oldroyd model is used to describe the flow in a layer of an anisotropic porous medium. The results of the linear instability theory are delineated. The thresholds for the stationary and oscillatory convection boundaries are established, and the crossover boundary between them is de- marcated by identifying a codimension-two point in the viscoelastic parameter plane. The stability of the stationary and oscillatory bifurcating solutions is analyzed by deriving the cubic Landau equations. It shows that these solutions always bifurcate supercritically. The heat transfer is estimated in terms of the Nusselt number for the stationary and oscillatory modes. The result shows that, when the ratio of the thermal to mechanical anisotropy parameters increases, the heat transfer decreases.展开更多
Standing soliton was studied by numerical simulation of ifs governing equation, a cubic Schrodiger equation with a complex conjugate term, which was derived by Miles and was accepted. The value of linear damping in Mi...Standing soliton was studied by numerical simulation of ifs governing equation, a cubic Schrodiger equation with a complex conjugate term, which was derived by Miles and was accepted. The value of linear damping in Miles equation was studied. Calculations showed that linear damping effects strongly on the formation of a standing soliton and Laedke and Spatschek stable condition is only a necessary condition, but not a sufficient one. The interaction of two standing solitons was simulated. Simulations showed that the interaction pattern depends on system parameters. Calculations for the different initial condition and its development indicated that a stable standing soliton can be fanned only for proper initial disturbance, otherwise the disturbance will disappear or develop into several solitons.展开更多
In this paper,the governing equation for the non-propagating solitary waves,similar to the cubicSchr(?)dinger equation,is derived by the multiple scales with the consideration of surface tension.The non-propagatingsol...In this paper,the governing equation for the non-propagating solitary waves,similar to the cubicSchr(?)dinger equation,is derived by the multiple scales with the consideration of surface tension.The non-propagatingsolitary wave solution is given.It is explained by the capillary-gravity wave theory that the crests are sharpened and thetroughs are flattened in the transversal harmonic of the non-propagating solitary waves.On σ~kh plane,twoparameter regions are obtained in which the non-propagating solitary wave can occur,but all existing experimentalparameters are in region 1(Fig.1).展开更多
In this paper, we investigate the general solution and the stability of a cubic functional equation f(x + ny) + f(x - ny) + f(nx) = n^2 f(x + y) + n^2 f(x - y)+ (n^3 - 2n^2 + 2)f(x),where n ≥ 2 i...In this paper, we investigate the general solution and the stability of a cubic functional equation f(x + ny) + f(x - ny) + f(nx) = n^2 f(x + y) + n^2 f(x - y)+ (n^3 - 2n^2 + 2)f(x),where n ≥ 2 is an integer. Furthermore, we prove the stability by the fixed point method.展开更多
Applying Hopf bifurcation theory and qualitative theory, we give the conditions of the existence and uniqueness of one limit cycle and the existence of two limit cycles for the general cubic Lienard equation. Numerica...Applying Hopf bifurcation theory and qualitative theory, we give the conditions of the existence and uniqueness of one limit cycle and the existence of two limit cycles for the general cubic Lienard equation. Numerical simulation results with one and two limit cycles are given to demonstrate the theoretical results.展开更多
Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at ...Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation.展开更多
Let λ<sub>1</sub>, λ<sub>2</sub>,...,λ<sub>7</sub> be real numbers satisfying λ<sub>i</sub>≥1. In this paper, we prove there are integers x<sub>1</sub>,...Let λ<sub>1</sub>, λ<sub>2</sub>,...,λ<sub>7</sub> be real numbers satisfying λ<sub>i</sub>≥1. In this paper, we prove there are integers x<sub>1</sub>,...,x<sub>7</sub> such that the inequalities |λ<sub>1</sub>x<sub>1</sub><sup>3</sup>+λ<sub>2</sub>x<sub>2</sub><sup>3</sup>+...+λ<sub>7</sub>x<sub>7</sub><sup>3</sup>|【1 and 0【sum from i=1 to7(λ<sub>i</sub>|x<sub>i</sub>]<sup>3</sup> (λ<sub>1</sub>λ<sub>2</sub>…λ<sub>7</sub>)<sup>89814</sup>) hold simultaneously.展开更多
We survey the main properties of the cubic Szeg? equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of high Sobolev norms and the effects of weak damping.
文摘Progress in hydrate thermodynamic study necessitates robust and fast models to be incorporated in reservoir simulation softwares. However, numerous models presented in the literature makes selection of the best,proper predictive model a cumbersome task. It is of industrial interest to make use of cubic equations of state(EOS) for modeling hydrate equilibria. In this regard, this study focuses on evaluation of three common EOSs including Peng–Robinson, Soave–Redlich–Kwong and Valderrama–Patel–Teja coupled with van der Waals and Platteeuw theory to predict hydrate P–T equilibrium of a real natural gas sample. Each EOS was accompanied with three mixing rules, including van der Waals(vd W),Avlonitis non-density dependent(ANDD) and general nonquadratic(GNQ). The prediction of cubic EOSs was in sufficient agreement with experimental data and with overall AARD% of less than unity. In addition, PR plus ANDD proved to be the most accurate model in this study for prediction of hydrate equilibria with AARD% of 0.166.It was observed that the accuracy of cubic EOSs studied in this paper depends on mixing rule coupled with them,especially at high-pressure conditions. Lastly, the present study does not include any adjustable parameter to be correlated with hydrate phase equilibrium data.
基金supported by the National Natural Science Foundation of China(Nos.11072125 and11272175)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20130002110044)the China Postdoctoral Science Foundation(No.2015M570035)
文摘Inspired by Cardano's method for solving cubic scalar equations, the addi- tive decomposition of spherical/deviatoric tensor (DSDT) is revisited from a new view- point. This decomposition simplifies the cubic tensor equation, decouples the spher- ical/deviatoric strain energy density, and lays the foundation for the von Mises yield criterion. Besides, it is verified that under the precondition of energy decoupling and the simplest form, the DSDT is the only possible form of the additive decomposition with physical meanings.
基金supported by the Natural Science Foundation of Yibin University (No. 2009Z003)
文摘In this paper, the stability of a cubic functional equation in the setting of intuitionistic random normed spaces is proved. We first introduce the notation of intuitionistic random normed spaces. Then, by virtue of this notation, we study the stability of a cubic functional equation in the setting of these spaces under arbitrary triangle norms. Furthermore, we present the interdisciplinary relation among the theory of random spaces, the theory of intuitionistic spaces, and the theory of functional equations.
文摘In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt + bux4 - duxxt = f(u)xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
文摘This paper considers the one-dimensional dissipative cubic nonlinear SchrSdinger equation with zero Dirichlet boundary conditions on a bounded domain. The equation is discretized in time by a linear implicit three-level central difference scheme, which has analogous discrete conservation laws of charge and energy. The convergence with two orders and the stability of the scheme are analysed using a priori estimates. Numerical tests show that the three-level scheme is more efficient.
文摘For further improving the representation of mixture VLE data,the local composition version of CCORequation of state has been developed and tested on 42 sets low-pressure and high-pressure as well as polarand nonpolar VLE data.The data reduction results were compared with conventional quadratic mixing ruleand activity coefficient method.The comparison with quadratic mixing rule showed that the local composition version significantly im-proved the data fitting of polar systems,especially for those highly nonideal mixtures where quadratic mixingrule failed to fit satisfactorily.The comparison with the well-known activity coefficient method——Hayden-O’Connell-Wilson model,indicated that this new version gave,in general,better fit to those low-pressure strongly polar systems,which traditionally has to be treated by activity coefficient approach.
文摘Let G be an Abelian group and letρ:G×G→[0,∞) be a metric on G. Let E be a normed space. We prove that under some conditions if f:G→E is an odd function and Cx:G→E defined by Cx(y):=2 f (x+y)+2 f (x-y)+12 f (x)-f (2x+y)-f (2x-y) is a cubic function for all x∈G, then there exists a cubic function C:G→E such that f?C is Lipschitz. Moreover, we investigate the stability of cubic functional equation 2 f (x+y)+2 f (x-y)+12 f (x)-f (2x+y)-f (2x-y)=0 on Lipschitz spaces.
基金The project supported by Natural Science Foundation of Zhejiang Province of China under Grant Nos.Y604056 and 605408the Doctoral Foundation of Ningbo City under Grant No.2005A61030Ningbo Natural Science Foundation under Grant No.2007A610049
文摘An extended subequation rational expansion method is presented and used to construct some exact,analyt-ical solutions of the (2+1)-dimensional cubic nonlinear Schrdinger equation.From our results,many known solutionsof the (2+1)-dimensional cubic nonlinear Schrdinger equation can be recovered by means of some suitable selections ofthe arbitrary functions and arbitrary constants.With computer simulation,the properties of new non-travelling waveand coefficient function's soliton-like solutions,and elliptic solutions are demonstrated by some plots.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175158)the Natural Science Foundation of Zhejiang Province of China(Grant No.LY12A04001)
文摘A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain (loss) coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.
文摘In this paper, we determine the general solution of the functional equation f1 (2x + y) + f2(2x - y) = f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 : R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x - y) = g(x + y) + g(x - y) + h(x) and f(2x + y) - f(2x - y) = g(x + y) - g(x - y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.
基金Korea Research Foundation Grant KRF-2007-313-C00033
文摘In this paper, we will find out the general solution and investigate the generalized Hyers-Ulam-Rassias stability problem for the following cubic functional equation2f(x + 2y) + f(2x - y) = 5f(x + y) + 5f(x - y)+ 15f(y)in the spirit of Hyers, Ulam, Rassias and Gavruta.
文摘According to the inverse solution of elasticity mechanics, a stress function is constructed which meets the space biharmonic equation, this stress functions is about cubic function pressure on the inner and outer surfaces of cylinder. When borderline condition that is predigested according to the Saint-Venant's theory is joined, an equation suit is constructed which meets both the biharmonic equations and the boundary conditions. Furthermore, its analytic solution is deduced with Matlab. When this theory is applied to hydraulic bulging rollers, the experimental results inosculate with the theoretic calculation. Simultaneously, the limit along the axis invariable direction is given and the famous Lame solution can be induced from this limit. The above work paves the way for mathematic model building of hollow cylinder and for the analytic solution of hollow cvlinder with randomly uneven pressure.
基金Project supported by the Innovation in Science Pursuit for the Inspired Research(INSPIRE)Program(No.DST/INSPIRE Fellowship/[IF 150253])
文摘The nonlinear stability of thermal convection in a layer of an Oldroyd-B fluid-saturated Darcy porous medium with anisotropic permeability and thermal diffu- sivity is investigated with the perturbation method. A modified Darcy-Oldroyd model is used to describe the flow in a layer of an anisotropic porous medium. The results of the linear instability theory are delineated. The thresholds for the stationary and oscillatory convection boundaries are established, and the crossover boundary between them is de- marcated by identifying a codimension-two point in the viscoelastic parameter plane. The stability of the stationary and oscillatory bifurcating solutions is analyzed by deriving the cubic Landau equations. It shows that these solutions always bifurcate supercritically. The heat transfer is estimated in terms of the Nusselt number for the stationary and oscillatory modes. The result shows that, when the ratio of the thermal to mechanical anisotropy parameters increases, the heat transfer decreases.
文摘Standing soliton was studied by numerical simulation of ifs governing equation, a cubic Schrodiger equation with a complex conjugate term, which was derived by Miles and was accepted. The value of linear damping in Miles equation was studied. Calculations showed that linear damping effects strongly on the formation of a standing soliton and Laedke and Spatschek stable condition is only a necessary condition, but not a sufficient one. The interaction of two standing solitons was simulated. Simulations showed that the interaction pattern depends on system parameters. Calculations for the different initial condition and its development indicated that a stable standing soliton can be fanned only for proper initial disturbance, otherwise the disturbance will disappear or develop into several solitons.
文摘In this paper,the governing equation for the non-propagating solitary waves,similar to the cubicSchr(?)dinger equation,is derived by the multiple scales with the consideration of surface tension.The non-propagatingsolitary wave solution is given.It is explained by the capillary-gravity wave theory that the crests are sharpened and thetroughs are flattened in the transversal harmonic of the non-propagating solitary waves.On σ~kh plane,twoparameter regions are obtained in which the non-propagating solitary wave can occur,but all existing experimentalparameters are in region 1(Fig.1).
文摘In this paper, we investigate the general solution and the stability of a cubic functional equation f(x + ny) + f(x - ny) + f(nx) = n^2 f(x + y) + n^2 f(x - y)+ (n^3 - 2n^2 + 2)f(x),where n ≥ 2 is an integer. Furthermore, we prove the stability by the fixed point method.
基金supported by the National Natural Sciences Foundation of China.
文摘Applying Hopf bifurcation theory and qualitative theory, we give the conditions of the existence and uniqueness of one limit cycle and the existence of two limit cycles for the general cubic Lienard equation. Numerical simulation results with one and two limit cycles are given to demonstrate the theoretical results.
基金Supported by the National Natural Science Foundation of ChinaNational Key Basic Research Special Found (No. G1998020307).
文摘Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19671051)
文摘Let λ<sub>1</sub>, λ<sub>2</sub>,...,λ<sub>7</sub> be real numbers satisfying λ<sub>i</sub>≥1. In this paper, we prove there are integers x<sub>1</sub>,...,x<sub>7</sub> such that the inequalities |λ<sub>1</sub>x<sub>1</sub><sup>3</sup>+λ<sub>2</sub>x<sub>2</sub><sup>3</sup>+...+λ<sub>7</sub>x<sub>7</sub><sup>3</sup>|【1 and 0【sum from i=1 to7(λ<sub>i</sub>|x<sub>i</sub>]<sup>3</sup> (λ<sub>1</sub>λ<sub>2</sub>…λ<sub>7</sub>)<sup>89814</sup>) hold simultaneously.
文摘We survey the main properties of the cubic Szeg? equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of high Sobolev norms and the effects of weak damping.