TY - JOUR

T1 - Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws

AU - Tan, Dechun

AU - Zhang, Tong

AU - Chang, Tung

AU - Zheng, Yuxi

PY - 1994/8

Y1 - 1994/8

N2 - For simple models of hyperbolic systems of conservation laws, we study a new type of nonlinear hyperbolic wave, a delta-shock wave, which is a Dirac delta function supported on a shock. We prove that delta-shock waves are w*-limits in L1 of solutions to some reasonable viscous perturbations as the viscosity vanishes. Further, we solve the multiplication problem of a delta function with a discontinuous function to show that delta-shock waves satisfy the equations in the sense of distributions. Under suitable generalized Rankine-Hugoniot and entropy conditions, we establish the existence and uniqueness of solutions involving delta-shock waves for the Riemann problems. The existence of solutions to the Cauchy problem is also investigated.

AB - For simple models of hyperbolic systems of conservation laws, we study a new type of nonlinear hyperbolic wave, a delta-shock wave, which is a Dirac delta function supported on a shock. We prove that delta-shock waves are w*-limits in L1 of solutions to some reasonable viscous perturbations as the viscosity vanishes. Further, we solve the multiplication problem of a delta function with a discontinuous function to show that delta-shock waves satisfy the equations in the sense of distributions. Under suitable generalized Rankine-Hugoniot and entropy conditions, we establish the existence and uniqueness of solutions involving delta-shock waves for the Riemann problems. The existence of solutions to the Cauchy problem is also investigated.

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U2 - 10.1006/jdeq.1994.1093

DO - 10.1006/jdeq.1994.1093

M3 - Article

AN - SCOPUS:0002178812

SN - 0022-0396

VL - 112

SP - 1

EP - 32

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 1

ER -