In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of cl...In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.展开更多
In this paper, using the characteristic analysis method, we study the relativistic Euler equations of conservation laws in energy and momentum in special relativity. The interactions of elementary waves for the relati...In this paper, using the characteristic analysis method, we study the relativistic Euler equations of conservation laws in energy and momentum in special relativity. The interactions of elementary waves for the relativistic Euler equations are shown. The collision of two shocks, two centered rarefaction waves, a shock and a rarefaction wave yield corresponding ransmitted waves. The overtaking of two shocks appears a transmitted shock wave, together with a reflected centered rarefaction wave.展开更多
In this paper we consider the Riemann problem for the nonlinear degenerate wave equations. This problem has been studied by Sun and Sheng, however the so-called degenerate shock solutions did not satisfy the R-H condi...In this paper we consider the Riemann problem for the nonlinear degenerate wave equations. This problem has been studied by Sun and Sheng, however the so-called degenerate shock solutions did not satisfy the R-H condition. In the present paper, the Riemann solutions of twelve regions in the v - u plane are completely constructed by the Liu-entropy condition. Our Riemann solutions are very different to that one obtained by Sun and Sheng in some regions.展开更多
The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By con...The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By constructing an example, first it is illustrated that the classical solution to this kind of Cauchy problem may blow up in a finite time, even if the system is weakly linearly degenerate. Then some lower bounds of the life-span of classical solutions are given in the case that the system is weakly linearly degenerate. These estimates imply that, when the system is weakly linearly degenerate, the classical solution exists almost globally in time. Finally, it is proved that Theorems 1.1-1.3 in [2] are still valid for this kind of initial data.展开更多
文摘In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.
基金Project supported by the National Natural Science Foundation of China (Grant No.10671120)
文摘In this paper, using the characteristic analysis method, we study the relativistic Euler equations of conservation laws in energy and momentum in special relativity. The interactions of elementary waves for the relativistic Euler equations are shown. The collision of two shocks, two centered rarefaction waves, a shock and a rarefaction wave yield corresponding ransmitted waves. The overtaking of two shocks appears a transmitted shock wave, together with a reflected centered rarefaction wave.
基金Supported by the National Natural Science Foundation of China (11171340)
文摘In this paper we consider the Riemann problem for the nonlinear degenerate wave equations. This problem has been studied by Sun and Sheng, however the so-called degenerate shock solutions did not satisfy the R-H condition. In the present paper, the Riemann solutions of twelve regions in the v - u plane are completely constructed by the Liu-entropy condition. Our Riemann solutions are very different to that one obtained by Sun and Sheng in some regions.
基金Project supported by the National Natural Science Foundation of China
文摘The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By constructing an example, first it is illustrated that the classical solution to this kind of Cauchy problem may blow up in a finite time, even if the system is weakly linearly degenerate. Then some lower bounds of the life-span of classical solutions are given in the case that the system is weakly linearly degenerate. These estimates imply that, when the system is weakly linearly degenerate, the classical solution exists almost globally in time. Finally, it is proved that Theorems 1.1-1.3 in [2] are still valid for this kind of initial data.
