Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ...Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ?t ?x t = 0 : u = f(x). We let M = sup |f (x)| < +∞. x∈R The main result of this paper is that under the assumption that the system is weakly linearly degenerated, there exists a positive constant ε independent of M, such that the above Cauchy problem admits a unique global C1 solution u = u(t,x) for all t ∈ R, provided that +∞ |f (x)|dx ≤ ε, ?∞ +∞ ε |f(x)|dx ≤ . M∞展开更多
This paper deals with the blow-up phenomenon, particularly, the geometric blow-up mechanism, of classical solutions to the Cauchy problem for quasilinear hyperbolic systems in the critical case. We prove that it is st...This paper deals with the blow-up phenomenon, particularly, the geometric blow-up mechanism, of classical solutions to the Cauchy problem for quasilinear hyperbolic systems in the critical case. We prove that it is still the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem in the critical case.展开更多
By means of the continuous Glimm functional, a proof is given on the global existence of classical solutions to Cauchy problem for general first order quasilinear hyperbolic systems with small initial total variation.
Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local bound...Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local boundary(weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.展开更多
The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solu...The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in Arch.Rational Mech.Anal.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.展开更多
基金Project supported by the National Natural Science Foundation of China (No.10225102) the 973 Project of the Ministry of Science and Technology of China and the Doctoral Programme Foundation of the Ministry of Education of China.
文摘Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ?t ?x t = 0 : u = f(x). We let M = sup |f (x)| < +∞. x∈R The main result of this paper is that under the assumption that the system is weakly linearly degenerated, there exists a positive constant ε independent of M, such that the above Cauchy problem admits a unique global C1 solution u = u(t,x) for all t ∈ R, provided that +∞ |f (x)|dx ≤ ε, ?∞ +∞ ε |f(x)|dx ≤ . M∞
基金Project supported by the Special Funds for Major State Basic Research Project of ChinaSpecialized Research Fund for the Doctoral Program of Higher Education (No. 20020246002).
文摘This paper deals with the blow-up phenomenon, particularly, the geometric blow-up mechanism, of classical solutions to the Cauchy problem for quasilinear hyperbolic systems in the critical case. We prove that it is still the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem in the critical case.
文摘By means of the continuous Glimm functional, a proof is given on the global existence of classical solutions to Cauchy problem for general first order quasilinear hyperbolic systems with small initial total variation.
文摘Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local boundary(weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.
基金Project supported by the National Natural Science Foundation of China (No. 10728101)the Basic Research Program of China (No. 2007CB814800)+1 种基金the Doctoral Program Foundation of the Ministry of Education of Chinathe "111" Project (No. B08018) and SGST (No. 09DZ2272900)
文摘The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in Arch.Rational Mech.Anal.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.
