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
We investigate a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids with the goal to prove for it the existence of weak solutions for arbitrary large initial data o...We investigate a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids with the goal to prove for it the existence of weak solutions for arbitrary large initial data on a large time interval.We transform the one velocity Baer-Nunziato system to another"more academic"system which possesses the clear"Navier-Stokes structure".We solve the new system by adapting to its structure the Lions approach for solving the(mono-fluid)compressible Navier-Stokes equations.An extension of the theory of renormalized solutions to the transport equation to more continuity equations with renormalizing functions of several variables is essential in this process.We derive a criterion of almost uniqueness for the renormalized solutions to the pure transport equation without the classical assumption on the boundedness of the divergence of the transporting velocity.This result does not follow from the DiPerna-Lions transport theory and it is of independent interest.This criterion plays the crucial role in the identification of the weak solutions to the original one velocity Baer-Nunziato problem starting from the weak solutions of the academic problem.As far as we know,this is the first result on the existence of weak solutions for a version of the one velocity bi-fluid system of the Baer-Nunziato type in the mathematical literature.展开更多
A complex network consisting of chaotic systems is considered and the existence of the HSlder continuous gen- eralized synchronization in the network is studied. First, we divide nodes of the network into two parts ac...A complex network consisting of chaotic systems is considered and the existence of the HSlder continuous gen- eralized synchronization in the network is studied. First, we divide nodes of the network into two parts according to their dynamical behaviour. Then, based on the Schauder fixed point theorem, sufficient conditions for the existence of the generalized synchronization between them are derived. Moreover, the results are theoretically proved. Numerical simulations validate the theory.展开更多
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.
文摘We investigate a version of one velocity Baer-Nunziato model with dissipation for the mixture of two compressible fluids with the goal to prove for it the existence of weak solutions for arbitrary large initial data on a large time interval.We transform the one velocity Baer-Nunziato system to another"more academic"system which possesses the clear"Navier-Stokes structure".We solve the new system by adapting to its structure the Lions approach for solving the(mono-fluid)compressible Navier-Stokes equations.An extension of the theory of renormalized solutions to the transport equation to more continuity equations with renormalizing functions of several variables is essential in this process.We derive a criterion of almost uniqueness for the renormalized solutions to the pure transport equation without the classical assumption on the boundedness of the divergence of the transporting velocity.This result does not follow from the DiPerna-Lions transport theory and it is of independent interest.This criterion plays the crucial role in the identification of the weak solutions to the original one velocity Baer-Nunziato problem starting from the weak solutions of the academic problem.As far as we know,this is the first result on the existence of weak solutions for a version of the one velocity bi-fluid system of the Baer-Nunziato type in the mathematical literature.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.11002061 and 10901073)the Fundamental Research Funds for the Central Universities,China (Grant No.JUSRP11117)
文摘A complex network consisting of chaotic systems is considered and the existence of the HSlder continuous gen- eralized synchronization in the network is studied. First, we divide nodes of the network into two parts according to their dynamical behaviour. Then, based on the Schauder fixed point theorem, sufficient conditions for the existence of the generalized synchronization between them are derived. Moreover, the results are theoretically proved. Numerical simulations validate the theory.
