Abstract
This paper is concerned with the construction of global measure-valued solutions to the extended Riemann problem for a non-strictly hyperbolic system of two conservation laws with delta-type initial data. The wave interaction problems have been extensively studied for all kinds of situations by using the initial condition consisting of constant states in three pieces instead of delta-type initial data under the perturbation method. The measure-valued solutions of the extended Riemann problem are achieved constructively when the perturbed parameter tends to zero. During the process of constructing solutions, a new and interesting nonlinear phenomenon is discovered, in which the initial Dirac delta function travels along the trajectory of either delta shock wave or contact discontinuity (or delta contact discontinuity). Moreover, a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of delta shock wave penetrating a composite wave composed of a rarefaction wave and a contact discontinuity. In addition, we further consider the constructions of global measure-valued solutions when the initial condition contains Dirac delta functions at two different initial points.
Acknowledgements
This work is partially supported by the Shandong Provincial Natural Science Foundation (Funder Id: http://dx.doi.org/10.13039/501100007129, Grant Number: ZR2019MA058) and the National Natural Science Foundation of China (Funder Id: http://dx.doi.org/10.13039/501100001809, Grant Number: 11441002). The authors are very grateful to the two anonymous referees for their valuable suggestions and corrections.
References
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© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Discrete-Time Fractional Order SIR Epidemic Model with Saturated Treatment Function
- Lyapunov Stability of a Fractionally Damped Oscillator with Linear (Anti-)Damping
- Stability and Synchronization of a Fractional Neutral Higher-Order Neural Network System
- Adaptive Synchronization of Time-Delay Chaotic Systems with Intermittent Control
- Rational Type Inequality with Applications to Voltera–Hammerstein Nonlinear Integral Equations
- Numerical Study of the Movement of Water Surface of Dam Break Flow by VOF Methods for Various Obstacles
- Measure-Valued Solutions to a Non-Strictly Hyperbolic System with Delta-Type Riemann Initial Data
- Erratum
- A Class of Exact Solution of (3+1)-Dimensional Generalized Shallow Water Equation System
- Corrigendum
- Almost Periodic Solution in a Lotka–Volterra Recurrent Neural Networks with Time-Varying Delays
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Discrete-Time Fractional Order SIR Epidemic Model with Saturated Treatment Function
- Lyapunov Stability of a Fractionally Damped Oscillator with Linear (Anti-)Damping
- Stability and Synchronization of a Fractional Neutral Higher-Order Neural Network System
- Adaptive Synchronization of Time-Delay Chaotic Systems with Intermittent Control
- Rational Type Inequality with Applications to Voltera–Hammerstein Nonlinear Integral Equations
- Numerical Study of the Movement of Water Surface of Dam Break Flow by VOF Methods for Various Obstacles
- Measure-Valued Solutions to a Non-Strictly Hyperbolic System with Delta-Type Riemann Initial Data
- Erratum
- A Class of Exact Solution of (3+1)-Dimensional Generalized Shallow Water Equation System
- Corrigendum
- Almost Periodic Solution in a Lotka–Volterra Recurrent Neural Networks with Time-Varying Delays