By exploiting the contact Hamiltonian dynamics(T*M×R,Φ_(t))around the Aubry set of contact Hamiltonian systems,we provide a relation among the Mather set,theΦ_(t)-recurrent set,the strongly static set,the Aubry...By exploiting the contact Hamiltonian dynamics(T*M×R,Φ_(t))around the Aubry set of contact Hamiltonian systems,we provide a relation among the Mather set,theΦ_(t)-recurrent set,the strongly static set,the Aubry set,the Ma?éset,and theΦ_(t)-non-wandering set.Moreover,we consider the strongly static set,as a new flow-invariant set between the Mather set and the Aubry set in the strictly increasing case.We show that this set plays an essential role in the representation of certain minimal forward weak Kolmogorov-Arnold-Moser(KAM)solutions and the existence of transitive orbits around the Aubry set.展开更多
Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold,where the Hamiltonian satisfies the condition:The Aubry set of the corresponding Hamiltonian system consists of one hyperbol...Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold,where the Hamiltonian satisfies the condition:The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit.It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.展开更多
In the recent works, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental...In the recent works, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental solutions of the associated Hamilton-Jacobi equations. In the present paper, we exploit the relations among Lasry-Lions regularization, Lax-Oleinik operators(or inf/sup-convolution) and generalized characteristics, which are discussed in the context of the variational setting of Tonelli Hamiltonian dynamics, such as Mather theory and weak KAM(Kolmogorov-Arnold-Moser) theory.展开更多
We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general conv...We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12122109)。
文摘By exploiting the contact Hamiltonian dynamics(T*M×R,Φ_(t))around the Aubry set of contact Hamiltonian systems,we provide a relation among the Mather set,theΦ_(t)-recurrent set,the strongly static set,the Aubry set,the Ma?éset,and theΦ_(t)-non-wandering set.Moreover,we consider the strongly static set,as a new flow-invariant set between the Mather set and the Aubry set in the strictly increasing case.We show that this set plays an essential role in the representation of certain minimal forward weak Kolmogorov-Arnold-Moser(KAM)solutions and the existence of transitive orbits around the Aubry set.
基金supported by the National Natural Science Foundation of China(No.11371167)
文摘Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold,where the Hamiltonian satisfies the condition:The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit.It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.
基金supported by National Natural Science Foundation of China (Grant Nos. 11271182 and 11471238)the National Basic Research Program of China (Grant No. 2013CB834100)
文摘In the recent works, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental solutions of the associated Hamilton-Jacobi equations. In the present paper, we exploit the relations among Lasry-Lions regularization, Lax-Oleinik operators(or inf/sup-convolution) and generalized characteristics, which are discussed in the context of the variational setting of Tonelli Hamiltonian dynamics, such as Mather theory and weak KAM(Kolmogorov-Arnold-Moser) theory.
基金supported by National Natural Science Foundation of China(Grant Nos.1132510311301106 and 11201288)+1 种基金China Postdoctoral Science Foundation(Grant No.2014M550210)Guangxi Experiment Center of Information Science(Grant No.YB1410)
文摘We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.
