We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions ...We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an n-dimensional incompressible Navier-Stokes equations.We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.展开更多
Liénard’s equation is a kind of important ordinary differential equations frequently appearing in engineering and technology, and hence receives great attention of many mathematicians. In 1949, H. J. Eckweiler c...Liénard’s equation is a kind of important ordinary differential equations frequently appearing in engineering and technology, and hence receives great attention of many mathematicians. In 1949, H. J. Eckweiler conjectured that the equation +μsin+x=0 has infinite number of limit cycles. Then H. S. Hochstadt and B. Stephan, R. N. D’Heedene and others proved that this equation has at least n limit cycles in the interval |x|<(n+1)π for specified parameter μ. In 1980, Professor Zhang Zhifen proved that this equation has exact n limit cycles in the interval |x|<(n+1)π for any nonzero parameter μ, and thus pushed the related work forward greatly. In this paper, we shall prove that the Liénard’s equation has exact n limit cycles in a finite interval under a class of very general condition.展开更多
Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak soluti...Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.展开更多
Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0...Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.展开更多
Define the incremental fractional Brownian field with parameter H ∈ （0, 1） by ZH（τ, s） = BH（s-＋τ） - BH（S）, where BH（s） is a fractional Brownian motion with Hurst parameter H ∈ （0, 1）. We firstly deriv...Define the incremental fractional Brownian field with parameter H ∈ （0, 1） by ZH（τ, s） = BH（s-＋τ） - BH（S）, where BH（s） is a fractional Brownian motion with Hurst parameter H ∈ （0, 1）. We firstly derive the exact tail asymptoties for the maximum MH＊（T） = max（τ,s）∈[a,b]×[0,T] ZH（τ, s）/τH of the standardised fractional Brownian motion field, with any fixed 0 〈 a 〈 b 〈 ∞ and T 〉 0; and we, furthermore, extend the obtained result to the ease that T is a positive random variable independent of {BH（s）, s ≥ 0}. As a by-product, we obtain the Gumbel limit law for MH＊r（T） as T →∞.展开更多
In this paper we consider a class of differential systems with positive definite polynomial having exactly one and two limit cycles. Such a system is more extensive than paper [1,2].
文摘We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an n-dimensional incompressible Navier-Stokes equations.We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.
文摘Liénard’s equation is a kind of important ordinary differential equations frequently appearing in engineering and technology, and hence receives great attention of many mathematicians. In 1949, H. J. Eckweiler conjectured that the equation +μsin+x=0 has infinite number of limit cycles. Then H. S. Hochstadt and B. Stephan, R. N. D’Heedene and others proved that this equation has at least n limit cycles in the interval |x|<(n+1)π for specified parameter μ. In 1980, Professor Zhang Zhifen proved that this equation has exact n limit cycles in the interval |x|<(n+1)π for any nonzero parameter μ, and thus pushed the related work forward greatly. In this paper, we shall prove that the Liénard’s equation has exact n limit cycles in a finite interval under a class of very general condition.
文摘Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.
文摘Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.
基金supported by National Natural Science Foundation of China(Grant Nos.11326175 and 71471090)Natural Science Foundation of Zhejiang Province of China(Grant No.LQ14A010012)+2 种基金Research Start-up Foundation of Jiaxing University(Grant No.70512021)China Postdoctoral Science Foundation(Grant No.2014T70449)Natural Science Foundation of Jiangsu Province of China(Grant No.BK20131339)
文摘Define the incremental fractional Brownian field with parameter H ∈ （0, 1） by ZH（τ, s） = BH（s-＋τ） - BH（S）, where BH（s） is a fractional Brownian motion with Hurst parameter H ∈ （0, 1）. We firstly derive the exact tail asymptoties for the maximum MH＊（T） = max（τ,s）∈[a,b]×[0,T] ZH（τ, s）/τH of the standardised fractional Brownian motion field, with any fixed 0 〈 a 〈 b 〈 ∞ and T 〉 0; and we, furthermore, extend the obtained result to the ease that T is a positive random variable independent of {BH（s）, s ≥ 0}. As a by-product, we obtain the Gumbel limit law for MH＊r（T） as T →∞.
文摘In this paper we consider a class of differential systems with positive definite polynomial having exactly one and two limit cycles. Such a system is more extensive than paper [1,2].
