Lifespan  of classical  discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem   for hyperbolic conservation laws with small BV   data: shocks and contact discontinuities

Zhi-Qiang Shao

doi:10.3934/cpaa.2015.14.759

Article Contents

2015, Volume 14, Issue 3: 759-792. Doi: 10.3934/cpaa.2015.14.759

This issue Previous Article General types of spherical mean operators and $K$-functionals of fractional orders Next Article Differential Harnack estimates for backward heat equations with potentials under geometric flows

Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities

Zhi-Qiang Shao^1,

1.
Department of Mathematics, Fuzhou University, Fuzhou 350002

Received: July 31, 2012

Revised: December 31, 2014

Published: April 2015

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Abstract

In the present paper the author investigates the generalized nonlinear initial-boundary Riemann problem with small BV data for general $n\times n$ quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space $\{(t,x)|t\geq 0,x\geq 0\}$, where the Riemann solution only contains shocks and contact discontinuities. Combining the techniques employed by Li-Kong with the modified Glimm's functional, the author obtains the almost global existence and lifespan of classical discontinuous solutions to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem. This result is also applied to the system of traffic flow on a road network using the Aw-Rascle model.

Keywords:

Generalized nonlinear initial-boundary Riemann problem,
quasilinear hyperbolic system of conservation laws,
piecewise $c^1$ solution,
shock wave,
contact discontinuity,
lifespan.

Mathematics Subject Classification: Primary: 35L65, 35L45; Secondary: 35L67.

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