This paper considers the pointwise estimate of the solutions to Cauchy problem for quasilin-ear hyperbolic systems, which bases on the existence of the solutions by using the fundamental solutions. It gives a sharp po...This paper considers the pointwise estimate of the solutions to Cauchy problem for quasilin-ear hyperbolic systems, which bases on the existence of the solutions by using the fundamental solutions. It gives a sharp pointwise estimates of the solutions on domam under consideration. Specially, the estimate is precise near each characteristic direction.展开更多
Abstract In this paper, the author studies the global existence, singularities and life span of smooth solutions of the Cauchy problem for a class of quasilinear hyperbolic systems with higher order dissipative terms ...Abstract In this paper, the author studies the global existence, singularities and life span of smooth solutions of the Cauchy problem for a class of quasilinear hyperbolic systems with higher order dissipative terms and gives their applications to nonlinear wave equations with higher order dissipative terms.展开更多
Suppose that the two eigenvalues of system （0.1） are λ<sub>1</sub>（u, v）, λ<sub>2</sub>（u, v）, the corres-ponding Riemann invariants are w=w（u, v）, z=z（u, v）, and w=w（u, v）, z=z（...Suppose that the two eigenvalues of system （0.1） are λ<sub>1</sub>（u, v）, λ<sub>2</sub>（u, v）, the corres-ponding Riemann invariants are w=w（u, v）, z=z（u, v）, and w=w（u, v）, z=z（u, v） give a bijective smooth mapping from （u, v） plane onto （w, z） plane. Throughout this note, we always suppose that A<sub>1</sub> u<sub>0</sub>（x）, v<sub>0</sub>（x） are bounded measurable functions. A<sub>2</sub> λ<sub>1</sub>（u, v）, λ<sub>2</sub>（u, v）∈C<sup>1</sup> and system （0.1） are strictly hyperbolic, i.e. λ<sub>1</sub>（u, v）【λ<sub>2</sub>（u, v）.展开更多
In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space ...In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space Lloc1. The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system.The method used is Lagrangian representation, the essence of which is characteristic analysis.The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables.We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.展开更多
For an inhomogeneous quasilinear hyperbolic system of diagonal form, under the assumptions that the system is linearly degenerate and the C^1 norm of the boundary data is bounded, we show that the mechanism of the for...For an inhomogeneous quasilinear hyperbolic system of diagonal form, under the assumptions that the system is linearly degenerate and the C^1 norm of the boundary data is bounded, we show that the mechanism of the formation of singularities of C^1 classical solution to the Goursat problem with C^1 compatibility conditions at the origin must be an ODE type. The similar result is also obtained for the weakly discontinuous solution with C^0 compatibility conditions at the origin.展开更多
基金the National Natural Science Foundation of China(No.10131050).
文摘This paper considers the pointwise estimate of the solutions to Cauchy problem for quasilin-ear hyperbolic systems, which bases on the existence of the solutions by using the fundamental solutions. It gives a sharp pointwise estimates of the solutions on domam under consideration. Specially, the estimate is precise near each characteristic direction.
基金Supported by the Scientific and Technical Foundation for the Education Commission of Shanghai.
文摘Abstract In this paper, the author studies the global existence, singularities and life span of smooth solutions of the Cauchy problem for a class of quasilinear hyperbolic systems with higher order dissipative terms and gives their applications to nonlinear wave equations with higher order dissipative terms.
基金Project supported by the National Natural Science Foundation of China.
文摘Suppose that the two eigenvalues of system （0.1） are λ<sub>1</sub>（u, v）, λ<sub>2</sub>（u, v）, the corres-ponding Riemann invariants are w=w（u, v）, z=z（u, v）, and w=w（u, v）, z=z（u, v） give a bijective smooth mapping from （u, v） plane onto （w, z） plane. Throughout this note, we always suppose that A<sub>1</sub> u<sub>0</sub>（x）, v<sub>0</sub>（x） are bounded measurable functions. A<sub>2</sub> λ<sub>1</sub>（u, v）, λ<sub>2</sub>（u, v）∈C<sup>1</sup> and system （0.1） are strictly hyperbolic, i.e. λ<sub>1</sub>（u, v）【λ<sub>2</sub>（u, v）.
基金supported by the Central UniversitiesChina University of Geosciences (Wuhan)(CUGL180827)+1 种基金supported by the National Natural Science Foundation of China (11871218, 12071298)supported by the National Natural Science Foundation of China (11771442)。
文摘In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space Lloc1. The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system.The method used is Lagrangian representation, the essence of which is characteristic analysis.The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables.We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.
基金Supported by the National Natural Science Foundation of China(Grant No.10926162)the Fundamental Research Funds for the Central Universities(Grant No.2009B01314)the Natural Science Foundation of HohaiUniversity(Grant No.2009428011)
文摘For an inhomogeneous quasilinear hyperbolic system of diagonal form, under the assumptions that the system is linearly degenerate and the C^1 norm of the boundary data is bounded, we show that the mechanism of the formation of singularities of C^1 classical solution to the Goursat problem with C^1 compatibility conditions at the origin must be an ODE type. The similar result is also obtained for the weakly discontinuous solution with C^0 compatibility conditions at the origin.
