In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus $ \mathbb{T}^2 $ perturbed by a Lévy process. The existence of invariant me...In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus $ \mathbb{T}^2 $ perturbed by a Lévy process. The existence of invariant measure of the solutions are proved also.展开更多
In this paper we consider the Feller property and the exponential ergodicity for general diffusion processes with state-dependent switching. We prove their Feller continuity by means of intro- ducing some auxiliary pr...In this paper we consider the Feller property and the exponential ergodicity for general diffusion processes with state-dependent switching. We prove their Feller continuity by means of intro- ducing some auxiliary processes and by making use of the Radon-Nikodym derivatives. Furthermore, we also prove their strong Feller continuity and their exponential ergodicity under some reasonable conditions.展开更多
We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordi...We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill’s equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.展开更多
In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean s...In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space ? d , not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.展开更多
基金supported by National Basic Research Program of China (Grant No. 2006CB8059000)Science Fund for Creative Research Groups (Grant No.10721101)+2 种基金National Natural Science Foundation of China (GrantNos.10671197,10671168)Science Foundation of Jiangsu Province (Grant Nos.BK2006032,06-A-038,07-333)Key Lab of Random Complex Structures and Data Science,Chinese Academy of Sciences
文摘In this paper, we prove the global existence and uniqueness of the strong and weak solutions for 2D Navier-Stokes equations on the torus $ \mathbb{T}^2 $ perturbed by a Lévy process. The existence of invariant measure of the solutions are proved also.
基金the National Natural Science Foundation of China (Grant No. 10671037)the Basic Research Foundation of Beijing Institute of Technology (Grant No. 200507A4203)
文摘In this paper we consider the Feller property and the exponential ergodicity for general diffusion processes with state-dependent switching. We prove their Feller continuity by means of intro- ducing some auxiliary processes and by making use of the Radon-Nikodym derivatives. Furthermore, we also prove their strong Feller continuity and their exponential ergodicity under some reasonable conditions.
基金the National Natural Science Foundation of China(Grant Nos.10325102,10531010)the National Basic Research Program of China(Grant No.2006CB805903)Teaching and Research Award Program for Outstanding Young Teachers,Ministry of Education of China(2001)
文摘We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill’s equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.
基金supported by the National Natural Science Foundation of China (Grant No. 10671088)the Major State Basic Research Development Program of China (Grant No. 2006CB805903)
文摘In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space ? d , not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.
