Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

Abstract

MSC: 35L60 35L65 35L67Keywords: Quasi-linear wave equations Hyperbolic integro-differential systems Nonlinear balance laws Entropy solutions Cauchy problem Riemann problem Perturbed Riemann problem Generalized Glimm scheme This research explores the Cauchy problem for a class of quasilinear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.

Global existence and strong trace property of entropy solutions by the source-concentration Glimm scheme for nonlinear hyperbolic balance laws

“…However, we observe that, since a(x, t) is not continuous, the source terms a x G(U ) and a t H(U ) are not defined in the sense of distributions. The technique of asymptotic expansion to a in [18] also fails in our case. To overcome this difficulty, Chou-Lin [2] choose 0 < ε 1, decompose D κδ into the following six sub-regions:…”

Global entropy solutions of the general nonlinear hyperbolic balance laws with time-evolution flux and source

“…Here we point out that condition (A 2 ) gives the existence and generic structure of the standing wave discontinuities for the Riemann and boundary Riemann problems [12]. Comparing to the results in [5,12,29], the contribution of this paper is that the global existence result can be extended to more general flux and source without the dissipative condition.…”

“…The method in [18] showed that incorporating the source term as a wave gave sharp time independent bounds for solutions of the initial value problem, while the operator-splitting method gave only time dependent bounds in this nonstrictly hyperbolic setting. Recently, this framework was extended to quasi-linear wave equations [4,14,29],…”

“…In [29], ρ is considered as a more general form ρ = ρ(x, t), and the result of global existence was obtained under a dissipative condition. The initial-boundary value problem for (1.10) was also studied in [15].…”

Global solutions to a class of nonlinear damped wave operator equations

“…Equation (1.1) appears naturally in the study for liquid crystals [1][2][3][4]. In addition, Chang et al [5], Su [6] and Kian [7] where (x,t) R × R + , u 0 (x),ω 0 (x) R. Furthermore, Nishihara [8] and Hayashi [9] obtained the global solution to one dimension semilinear damped wave equation…”