For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H r...For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H rmander. As an application of our result, we show that the solution of three- dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.展开更多
For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state, we prove that the first-order derivatives of ρ, υ blow-up at the blow-up time...For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state, we prove that the first-order derivatives of ρ, υ blow-up at the blow-up time, while ρ, υ remain continuous. In particular, in the irrotational case we prove S. Alinhac’s statement.展开更多
Commutator free method is an effective method for solving rotating integration.Numerical examples show that the use of the proposed combining method can achieve the same order accuracy with less computation than other...Commutator free method is an effective method for solving rotating integration.Numerical examples show that the use of the proposed combining method can achieve the same order accuracy with less computation than other geometry integration method.However,it is difficult to be directly applied to mechanic dynamics solutions.In this paper,commutator free method which is often applied to rotation integration and classical Runge–Kutta(RK)method which is usually operated in Linear space are combined to solve the multi-body dynamic equations.The explicit Runge–Kutta coefficients are reconstructed to meet different order accuracy integration methods.The reconstruction method is discussed and coefficients are given.With this method,the dynamic equations can be solved accurately and economically without much modification on the classical numerical integration.Moreover,CG method and CF method can also be combined with adaptive RK method without many changes.Finally,the results of the examples show that with less computation,fourth-order combining method is as accurate as fourth-order Crouch–Grossman algorithm.展开更多
基金Project supported by the Zheng Ge Ru FoundationTianyuan Foundation of China
文摘For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and H rmander. As an application of our result, we show that the solution of three- dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.
基金Project supported by the Tianyuan Foundation of ChinaLab. of Math, for Nonlinear Problems. Fudan. Univ.
文摘For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state, we prove that the first-order derivatives of ρ, υ blow-up at the blow-up time, while ρ, υ remain continuous. In particular, in the irrotational case we prove S. Alinhac’s statement.
文摘Commutator free method is an effective method for solving rotating integration.Numerical examples show that the use of the proposed combining method can achieve the same order accuracy with less computation than other geometry integration method.However,it is difficult to be directly applied to mechanic dynamics solutions.In this paper,commutator free method which is often applied to rotation integration and classical Runge–Kutta(RK)method which is usually operated in Linear space are combined to solve the multi-body dynamic equations.The explicit Runge–Kutta coefficients are reconstructed to meet different order accuracy integration methods.The reconstruction method is discussed and coefficients are given.With this method,the dynamic equations can be solved accurately and economically without much modification on the classical numerical integration.Moreover,CG method and CF method can also be combined with adaptive RK method without many changes.Finally,the results of the examples show that with less computation,fourth-order combining method is as accurate as fourth-order Crouch–Grossman algorithm.
