In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is...In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.展开更多
Tidal flow is a periodic movement of unsteady and non-uniform, which has acceleration and deceleration process obviously, especially in coastal shallow waters. Many researches show that vertical distribution of tidal ...Tidal flow is a periodic movement of unsteady and non-uniform, which has acceleration and deceleration process obviously, especially in coastal shallow waters. Many researches show that vertical distribution of tidal flow Reynolds stress deviated from linear distribution. The parabolic distribution of the tidal flow Reynolds stress was proposed by Song et al. （2009）. Although the model fills better with field observations and indoor experimental data, it has the lower truncated series expansion of tidal flow Reynolds stress, and the description of the distribution is not very comprehensive By introducing the motion equation of tidal flow and improving the parabolic distribution established by Song et al. （2009）, the cubic distribution of the tidal flow Reynolds stress is proposed. The cubic distribution is verified well by field data （Bowden and Fairbairn, 1952; Bowden et al., 1959; Rippeth et al., 2002） and experimental data （Anwar and Atkins, 1980）, is consistent with the numerical model results of Kuo et al. （1996）, and is compared with the parabolic distribution of the tidal flow Reynolds stress. It is shown that this cubic distribution is not only better than the parabolic distribution, but also can better reflect the basic features of Reynolds stress deviating from linear distribution downward with the tidal flow acceleration and upward with the tidal flow deceleration, for the foundation of further study on the velocity profile of tidal flow.展开更多
This article studies the nonlinear evolution of disturbance waves in supersonic nonparallel boundary layer flows by using nonlinear parabolic stability equations （NPSE）. An accurate numerical method is developed to ...This article studies the nonlinear evolution of disturbance waves in supersonic nonparallel boundary layer flows by using nonlinear parabolic stability equations （NPSE）. An accurate numerical method is developed to solve the equations and march the NPSE in a stable manner. Through computation,are obtained the curves of amplitude and disturbance shape function of harmonic waves. Especially are demonstrated the physical characteristics of nonlinear stability of various harmonic waves,including instantaneous stream wise vortices,spanwise vortices and Λ structure etc,and are used to study and analyze the mechanism of the transition process. The calculated results have evidenced the effectiveness of the proposed NPSE method to research the nonlinear stability of the supersonic boundary layers.展开更多
The nonlinear stability problem in nonparallel boundary layer flow fortwo-dimensional disturbances was studied by using a newly presented method called ParabolicStability Equations (PSE). A series of new modes generat...The nonlinear stability problem in nonparallel boundary layer flow fortwo-dimensional disturbances was studied by using a newly presented method called ParabolicStability Equations (PSE). A series of new modes generated by the nonlinear interaction ofdisturbance waves were tabu-lately analyzed, and the Mean Flow Distortion (MFD) was numericallygiven. The computational techniques developed, including the higher-order spectral method and themore effective algebraic mapping, increased greatly the numerical accuracy and the rate ofconvergence. With the predictor-corrector approach in the marching procedure, the normalizationcondition was satisfied, and the stability of numerical calculation could be ensured. With differentinitial amplitudes, the nonlinear stability of disturbance wave was studied. The results ofexamples show good agreement with the data given by the DNS using the full Navier-Stokes equations.展开更多
Based on the results of the tidal flow Reynolds stresses of the field observations, indoor experiments, and numerical models, the parabolic distribution of the tidal flow Reynolds stress is proposed and its coefficien...Based on the results of the tidal flow Reynolds stresses of the field observations, indoor experiments, and numerical models, the parabolic distribution of the tidal flow Reynolds stress is proposed and its coefficients are determined theoretically in this paper. Having been well verified with the field data and experimental data, the proposed distribution of Reynolds stress is also compared with numerical model results, and a good agreement is obtained, showing that this distribution can well reflect the basic features of Reynolds stress deviating from the linear distribution that is downward when the tidal flow is of acceleration, upward when the tidal flow is of deceleration. Its dynamics cause is also discussed preliminarily and the influence of the water depth is pointed out from the definition of Reynolds stress, turbulent generation, transmission, and so on. The established expression for the vertical distribution of the tidal flow Reynolds stress is not only simple and explicit, but can also well reflect the features of the tidal flow acceleration and deceleration for further study on the velocity profile of tidal flow.展开更多
Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） e...Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） evolving by the Ricci flow gij/ t=-2Rij.In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u~ = /△u - aulogu - bu on M x （0,T], where a 〉 0 and b ∈ R.展开更多
In this paper,we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation:u_(t)=u_(xx)+f(t,u,u_(x)),t>0,0<x<L...In this paper,we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation:u_(t)=u_(xx)+f(t,u,u_(x)),t>0,0<x<L(0.1)with the periodic boundary condition u(t,0)=u(t,L),u_(x)(t,0)=u_(x)(t,L),(0.2)where f is uniformly almost periodic in t.In particular,we study the topological structure of the limit sets of the skew-product semiflow.It is proved that any compact minimal invariant set(throughout this paper,we refer to it as a minimal set)can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle S^(1)=R/LZ(see Definition 2.4 for“residually embedded”).Particularly,if f(t,u,p)=f(t,u,-p),then the flow on a minimal set can be embedded into an almost periodically-forced minimal flow on R(see Definition 2.4 for“embedded”).Moreover,it is proved that the ω-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase translation.In addition,we further consider the asymptotic dynamics of the skew-product semiflow generated by(0.1)with the Neumann boundary condition u_(x)(t,0)=u_(x)(t,L)=0 or the Dirichlet boundary condition u(t,0)=u(t,L)=0.For such a system,it has been known that theω-limit set of any bounded orbit contains at most two minimal sets.By applying the new results for(0.1)+(0.2),under certain direct assumptions on f,we prove in this paper that the flow on any minimal set of(0.1)with the Neumann boundary condition or the Dirichlet boundary condition can be embedded into an almost periodically-forced minimal flow on R.Finally,a counterexample is given to show that even for quasi-periodically-forced equations,the results we obtain here cannot be further improved in general.展开更多
We study the evolution of convex hypersurfaces H（., t） with initial H（., 0） = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H（., t）, H0 is a smooth, cl...We study the evolution of convex hypersurfaces H（., t） with initial H（., 0） = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H（., t）, H0 is a smooth, closed, and uniformly convex hypersurface. We find a θ^＊ 〉 0 and a sufficient condition about the anisotropic function f, such that if θ 〉 θ^＊, then H（.,t） remains uniformly convex and expands to infinity as t →∞ and its scaling, H（-, t）e^-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H - log f instead of H - f.展开更多
We study the long time behavior of J-flows on toric manifolds. By introducing the tran- sition maps between moment maps, we get a quasilinear parabolic system for J-flows. Some basic estimates for transition maps are ...We study the long time behavior of J-flows on toric manifolds. By introducing the tran- sition maps between moment maps, we get a quasilinear parabolic system for J-flows. Some basic estimates for transition maps are obtained.展开更多
In this paper an elliptic-parabolic coupled system arising from the fluid-solute-heat flowthrough a saturated porous medium is considerd.The uniqueness and the existence of classicalsolutions are proved.The asymptoti... In this paper an elliptic-parabolic coupled system arising from the fluid-solute-heat flowthrough a saturated porous medium is considerd.The uniqueness and the existence of classicalsolutions are proved.The asymptotic behavior of solutions for large time is shown,too.展开更多
In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (△-■t)u=pu+qu^(a+1) under the Yamabe flow. Here p,q∈C...In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (△-■t)u=pu+qu^(a+1) under the Yamabe flow. Here p,q∈C^(2,1)(M^(n)×[0,T]) and a is a positive constant.展开更多
基金This work was partially supported by the National Natural Science Foundation of China (Grant No. 10631020)Basic Research Grant of Tsinghua University (Grant No. JCJC2005071).
文摘In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.
基金supported by the National Natural Science Foundation of China (Grant No. 41076008)the Science and Technology Project of Chongqing Education Committee (Grant No. KJ110409 and No. KJ111501)+1 种基金the National Engineering Research Center for Inland Waterway Regulation Program (Grant No. SLK2012A02)the National Key Technology R&D Program (Grant No. 2012BAB05B03)
文摘Tidal flow is a periodic movement of unsteady and non-uniform, which has acceleration and deceleration process obviously, especially in coastal shallow waters. Many researches show that vertical distribution of tidal flow Reynolds stress deviated from linear distribution. The parabolic distribution of the tidal flow Reynolds stress was proposed by Song et al. （2009）. Although the model fills better with field observations and indoor experimental data, it has the lower truncated series expansion of tidal flow Reynolds stress, and the description of the distribution is not very comprehensive By introducing the motion equation of tidal flow and improving the parabolic distribution established by Song et al. （2009）, the cubic distribution of the tidal flow Reynolds stress is proposed. The cubic distribution is verified well by field data （Bowden and Fairbairn, 1952; Bowden et al., 1959; Rippeth et al., 2002） and experimental data （Anwar and Atkins, 1980）, is consistent with the numerical model results of Kuo et al. （1996）, and is compared with the parabolic distribution of the tidal flow Reynolds stress. It is shown that this cubic distribution is not only better than the parabolic distribution, but also can better reflect the basic features of Reynolds stress deviating from linear distribution downward with the tidal flow acceleration and upward with the tidal flow deceleration, for the foundation of further study on the velocity profile of tidal flow.
基金National Natural Science Foundation of China (10772082)Doctoral Foundation of Ministry of Education of China (20070287005)
文摘This article studies the nonlinear evolution of disturbance waves in supersonic nonparallel boundary layer flows by using nonlinear parabolic stability equations （NPSE）. An accurate numerical method is developed to solve the equations and march the NPSE in a stable manner. Through computation,are obtained the curves of amplitude and disturbance shape function of harmonic waves. Especially are demonstrated the physical characteristics of nonlinear stability of various harmonic waves,including instantaneous stream wise vortices,spanwise vortices and Λ structure etc,and are used to study and analyze the mechanism of the transition process. The calculated results have evidenced the effectiveness of the proposed NPSE method to research the nonlinear stability of the supersonic boundary layers.
文摘The nonlinear stability problem in nonparallel boundary layer flow fortwo-dimensional disturbances was studied by using a newly presented method called ParabolicStability Equations (PSE). A series of new modes generated by the nonlinear interaction ofdisturbance waves were tabu-lately analyzed, and the Mean Flow Distortion (MFD) was numericallygiven. The computational techniques developed, including the higher-order spectral method and themore effective algebraic mapping, increased greatly the numerical accuracy and the rate ofconvergence. With the predictor-corrector approach in the marching procedure, the normalizationcondition was satisfied, and the stability of numerical calculation could be ensured. With differentinitial amplitudes, the nonlinear stability of disturbance wave was studied. The results ofexamples show good agreement with the data given by the DNS using the full Navier-Stokes equations.
基金supported by the National Natural Science Foundation of China(Grant No.50339010)the Public Fund Project of Ministry of Water Resource of China(Grant No.200701026)
文摘Based on the results of the tidal flow Reynolds stresses of the field observations, indoor experiments, and numerical models, the parabolic distribution of the tidal flow Reynolds stress is proposed and its coefficients are determined theoretically in this paper. Having been well verified with the field data and experimental data, the proposed distribution of Reynolds stress is also compared with numerical model results, and a good agreement is obtained, showing that this distribution can well reflect the basic features of Reynolds stress deviating from the linear distribution that is downward when the tidal flow is of acceleration, upward when the tidal flow is of deceleration. Its dynamics cause is also discussed preliminarily and the influence of the water depth is pointed out from the definition of Reynolds stress, turbulent generation, transmission, and so on. The established expression for the vertical distribution of the tidal flow Reynolds stress is not only simple and explicit, but can also well reflect the features of the tidal flow acceleration and deceleration for further study on the velocity profile of tidal flow.
基金Supported by National Natural Science Foundation of China （Grant N0s. 10926109 and 11001268） and Chinese Universities Scientific Fund （2009JS32 and 2009-2-05）
文摘Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） evolving by the Ricci flow gij/ t=-2Rij.In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u~ = /△u - aulogu - bu on M x （0,T], where a 〉 0 and b ∈ R.
基金supported by National Science Foundation of USA(Grant No.DMS1645673)supported by National Natural Science Foundation of China(Grant Nos.11825106,11771414 and 12090012)+2 种基金Wu Wen-Tsun Key Laboratory of Mathematics,Chinese Academy of Sciences and University of Science and Technology of Chinasupported by National Natural Science Foundation of China(Grant Nos.11971232,12071217 and 11601498)the Chinese Scholarship Council(Grant No.201906845011)for its financial support。
文摘In this paper,we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation:u_(t)=u_(xx)+f(t,u,u_(x)),t>0,0<x<L(0.1)with the periodic boundary condition u(t,0)=u(t,L),u_(x)(t,0)=u_(x)(t,L),(0.2)where f is uniformly almost periodic in t.In particular,we study the topological structure of the limit sets of the skew-product semiflow.It is proved that any compact minimal invariant set(throughout this paper,we refer to it as a minimal set)can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle S^(1)=R/LZ(see Definition 2.4 for“residually embedded”).Particularly,if f(t,u,p)=f(t,u,-p),then the flow on a minimal set can be embedded into an almost periodically-forced minimal flow on R(see Definition 2.4 for“embedded”).Moreover,it is proved that the ω-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase translation.In addition,we further consider the asymptotic dynamics of the skew-product semiflow generated by(0.1)with the Neumann boundary condition u_(x)(t,0)=u_(x)(t,L)=0 or the Dirichlet boundary condition u(t,0)=u(t,L)=0.For such a system,it has been known that theω-limit set of any bounded orbit contains at most two minimal sets.By applying the new results for(0.1)+(0.2),under certain direct assumptions on f,we prove in this paper that the flow on any minimal set of(0.1)with the Neumann boundary condition or the Dirichlet boundary condition can be embedded into an almost periodically-forced minimal flow on R.Finally,a counterexample is given to show that even for quasi-periodically-forced equations,the results we obtain here cannot be further improved in general.
基金Acknowledgements The author would like to thank professor Huaiyu Jian for his comments and suggestions about this work. This work was supported by the National Natural Science Foundation of China （Grant Nos. 11131005, 11271118, 11301034） and the Doctoral Programme Foundation of Institution of Higher Education of China.
文摘We study the evolution of convex hypersurfaces H（., t） with initial H（., 0） = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H（., t）, H0 is a smooth, closed, and uniformly convex hypersurface. We find a θ^＊ 〉 0 and a sufficient condition about the anisotropic function f, such that if θ 〉 θ^＊, then H（.,t） remains uniformly convex and expands to infinity as t →∞ and its scaling, H（-, t）e^-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H - log f instead of H - f.
基金Supported by National Natural Science Foundation of China(Grant No.11171143)
文摘We study the long time behavior of J-flows on toric manifolds. By introducing the tran- sition maps between moment maps, we get a quasilinear parabolic system for J-flows. Some basic estimates for transition maps are obtained.
文摘 In this paper an elliptic-parabolic coupled system arising from the fluid-solute-heat flowthrough a saturated porous medium is considerd.The uniqueness and the existence of classicalsolutions are proved.The asymptotic behavior of solutions for large time is shown,too.
文摘In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (△-■t)u=pu+qu^(a+1) under the Yamabe flow. Here p,q∈C^(2,1)(M^(n)×[0,T]) and a is a positive constant.
