In this paper, we prove L^P-boundedness of hyperbolic singular integral operators for kernels satisfying weakened regularity conditions, where 1 〈 p 〈 ∞. This extends previous results of A.R. Nahmod.
In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the b...In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the bicharacteristics bearing weak singularities we proved a theorem on regularity propagation across the shock front.展开更多
In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constru...In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.展开更多
In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the a...In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.展开更多
Let X be a C^(1+)vector field on a compact Riemannian manifold M with dimension d≥3.LetΛbe a transitive singular hypebolic set with positive volume.We show thatΛ=M andΛis a uniformly hyperbolic set without singula...Let X be a C^(1+)vector field on a compact Riemannian manifold M with dimension d≥3.LetΛbe a transitive singular hypebolic set with positive volume.We show thatΛ=M andΛis a uniformly hyperbolic set without singularities.展开更多
In this paper, we prove that every star flow on the closed surface has finitely many chain recurrent classes. Furthermore, it is singular hyperbolic if every non-trivial singular chain component is a graph. As a conse...In this paper, we prove that every star flow on the closed surface has finitely many chain recurrent classes. Furthermore, it is singular hyperbolic if every non-trivial singular chain component is a graph. As a consequence, every star flow on the 2-sphere or the projective plane is singular hyperbolic.展开更多
Deformation theory is an important aspect of the study about isolated singularities. The invariant called irregularity is very useful in the study on the deformation of isolated singu- larities.In this note we give an...Deformation theory is an important aspect of the study about isolated singularities. The invariant called irregularity is very useful in the study on the deformation of isolated singu- larities.In this note we give an optimal upper bound for a class of surface singularities by the computation of cohomology.Moreover a sufficient condition is given for the positivity of irreg- ularity of some simple hyperbolic surface singularities.Therefore a class of surface singularities with non-rigid deformation is constructed.展开更多
Let X be a C1 vector field on a compact boundaryless Riemannian manifold M（dim M≥2）,and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = Es Eu with Es uniformly contracting and E...Let X be a C1 vector field on a compact boundaryless Riemannian manifold M（dim M≥2）,and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = Es Eu with Es uniformly contracting and Eu uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.展开更多
基金The NNSF (10171111) of Chinathe Foundation of Zhongshan University Advanced Research Center
文摘In this paper, we prove L^P-boundedness of hyperbolic singular integral operators for kernels satisfying weakened regularity conditions, where 1 〈 p 〈 ∞. This extends previous results of A.R. Nahmod.
文摘In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the bicharacteristics bearing weak singularities we proved a theorem on regularity propagation across the shock front.
文摘In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.
文摘In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.
基金support of Shanghai Key Laboratory of Pure Mathematics and Mathematical Practicethe Project funded by China Postdoctoral Science Foundation(Grant No.2020TQ0098)。
文摘Let X be a C^(1+)vector field on a compact Riemannian manifold M with dimension d≥3.LetΛbe a transitive singular hypebolic set with positive volume.We show thatΛ=M andΛis a uniformly hyperbolic set without singularities.
基金Supported by National Natural Science Foundation of China(Grant No.11201023)Specialized Research Fund for the Doctoral Program of Higher Education
文摘In this paper, we prove that every star flow on the closed surface has finitely many chain recurrent classes. Furthermore, it is singular hyperbolic if every non-trivial singular chain component is a graph. As a consequence, every star flow on the 2-sphere or the projective plane is singular hyperbolic.
文摘Deformation theory is an important aspect of the study about isolated singularities. The invariant called irregularity is very useful in the study on the deformation of isolated singu- larities.In this note we give an optimal upper bound for a class of surface singularities by the computation of cohomology.Moreover a sufficient condition is given for the positivity of irreg- ularity of some simple hyperbolic surface singularities.Therefore a class of surface singularities with non-rigid deformation is constructed.
基金supported by the State Key Development Program for Basic Research of China(973 Project)(Grant No.2011CB808002)National Natural Science Foundation of China(Grant Nos.11025101 and 11231001)
文摘Let X be a C1 vector field on a compact boundaryless Riemannian manifold M（dim M≥2）,and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = Es Eu with Es uniformly contracting and Eu uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.
