Initial-boundary value problem of nonlinear hyperbolic system for conservation laws with delta-shock waves

For a nonlinear hyperbolic system of conservation laws, the initial-boundary value problem is concerned with the boundary conditions. A boundary entropy condition is derived based on Dubois F and Le Floch P＇s results by taking a suitable entropy-flux pair （Journal of Differential Equations, 1988, 71（1）： 93-122）. The solutions of the initial-boundary value problem for the system are constructively obtained, in which initial-boundary data are in piecewise constant states. The delta-shock waves appear in their solutions.

Mechanism of the Formation of Singularities to the Goursat Problem for Diagonal Systems with Linearly Degenerate Characteristic Fields

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.

EXISTENCE AND UNIQUENESS OF THE GLOBAL L^(1) SOLUTION OF THE EULER EQUATIONS FOR CHAPLYGIN GAS

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.

Cauchy Problem for 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.