Global classical solutions of a kind of boundary value problem for quasilinear hyperbolic systems

In this paper, we consider a stability problem of a kind of boundary value problem for quasilinear hyperbolic systems. For small boundary data, we prove that the C1$$ {C} circumflex 1 $$ solution exists globally in time when the system is weakly linearly degenerate. In the special case of linear degeneracy, the smallness assumption on the boundary data is weakened. The error estimate in LT infinity L1$$ {L}_T circumflex {\infty }{L} circumflex 1 $$ space between two different solutions with different boundary data is also obtained. In our proof, an important estimate which describes the interaction of different waves is established by constructing a continuous Glimm function. This estimate together with uniform a prior estimates with respect to the time leads to the global-in-time existence of smooth solutions. Finally, we apply the stability results to the isentropic Euler equations and the system of the motion of an elastic string.

Automated summary

This paper discusses the stability of a certain type of boundary value problem for quasilinear hyperbolic systems, proving global existence of solutions under certain conditions and obtaining error estimates between solutions with different boundary data, which has theoretical and practical significance.