Global BV solutions and relaxation limit for a system of conservation laws

Journal of Differential Equations

Natalini

Yang

2000

In this paper, we study the global existence of BV solutions to a p-system with a relaxation source term using the Glimm's scheme. The pressure p is given by a #-law with #=1. By a suitable choice of the measure for the strength of the shock waves, we show that the total strength of the waves and the total variation of the solutions are uniformly bounded with respect to the relaxation parameter. Furthermore, when the relaxation parameter tends to zero, we show that the sequence of BV solutions eventually converges to a weak solution to the equilibrium equation. Academic Press

Section: Introductionsupporting

confidence: 59%

Global BV Solutions to a p-System with Relaxation

Journal of Differential Equations

Natalini

Yang

2000

“…System (1.2) does not have this property. Another type of inhomogeneous hyperbolic system with source term was studied in [2,14,15] by using the wave tracking method or Glimm scheme. For this type of system, the source term is required in L 1 .…”

Section: Remarkmentioning

confidence: 99%

Global structure and asymptotic behavior of weak solutions to flood wave equations

Journal of Differential Equations

Yang

2004

The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0.

“…A quasilinear model of gas dynamics equations was investigated in [53]. A modified Glimm's scheme ( [25]) and a wave front tracking method ( [1]) are used respectively to prove the existence of global BV solutions of p-system with relaxation for p(v) = kv −1 . Closely related to this paper is a result obtained in [15] for system (1.2) with viscosity in the case that the subcharacteristic condition (1.5) is violated, the weakly nonlinear limit is verified, and the underlying relaxation system reduces to the Burgers equation with a source term (cf [15]).…”

Section: Haitao Fan and Tao Luomentioning

confidence: 99%

Convergence to equilibrium rarefaction waves for discontinuous solutions of shallow water wave equations with relaxation

Fan

2005

Quart. Appl. Math.

Abstract. The purpose of this paper is to study the discontinuous solutions to a shallow water wave equation with relaxation. The typical initial value problem of discontinuous solutions is the Riemann problem. Unlike the homogeneous hyperbolic conservation laws, due to the inhomogeneity of the system studied here, the solutions of the Riemann problem do not have a self-similar structure anymore. This problem can be formulated as a free boundary problem. We show that the Riemann solutions still have a piecewise smooth structure globally and converge to the rarefaction waves of the equilibrium equation as time tends to infinity.