Systems of Conservation Laws
Two-Dimensional Riemann Problems
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Chapter and Conference Paper
Enhancing Active Visual Tracking Under Distractor Environments
Active Visual Tracking (AVT) faces significant challenges in distracting environments characterized by occlusions and confusion. Current methodologies address this challenge through the integration of a mixed ...
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Article
Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday
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Article
Causal complexity of environmental pollution in China: a province-level fuzzy-set qualitative comparative analysis
Environmental problems are endowed with the causal complexity of multiple factors. Traditional quantitative research on the influencing mechanism of environmental pollution has tended to focus on the marginal ...
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Chapter and Conference Paper
Performance Simulation and Application Research of Typical Directional Valve Based on Neural Network
The two position four way direction valve is a commonly used hydraulic component, which is widely used in some hydraulic oil circuits. In this paper, a typical cartridge valve consisting of cartridge two posit...
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Article
Semi-hyperbolic Patches of Solutions to the Two-dimensional Euler Equations
We construct semi-hyperbolic patches of solutions, in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves, to the two-dimensional Euler eq...
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Chapter and Conference Paper
Semi-Hyperbolic Waves in Two-Dimensional Compressible Euler Systems
We describe a patch in the self-similar plane as a semi-hyperbolic wave which is locally hyperbolic, but not all characteristics in the patch leads to given boundary data, for the two-dimensional compressible ...
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Article
Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations
The global existence and structures of solutions to multi-dimensional unsteady compressible Euler equations are interesting and important open problems. In this paper, we construct global classical solutions t...
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Article
Conservative Solutions to a System of Variational Wave Equations of Nematic Liquid Crystals
In this paper, we solve a system of hyperbolic equations by modelling nematic liquid crystals.
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Article
Two-dimensional Riemann problems for the compressible Euler system
Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. The author lists merits of one-dimensional Riemann problems a...
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Article
Interaction of Rarefaction Waves of the Two-Dimensional Self-Similar Euler Equations
We construct classical self-similar solutions to the interaction of two arbitrary planar rarefaction waves for the polytropic Euler equations in two space dimensions. The binary interaction represents a major ...
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Article
Asymptotic Variational Wave Equations
We investigate the equation (u t +(f(u)) x ) x =f ′ ′(u) (u ...
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Article
Simple Waves and a Characteristic Decomposition of the Two Dimensional Compressible Euler Equations
We present a characteristic decomposition of the potential flow equation in the self-similar plane. The decomposition allows for a proof that any wave adjacent to a constant state is a simple wave for the adia...
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Article
Conservative Solutions to a Nonlinear Variational Wave Equation
We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation u tt − c(u)(c(u)u ...
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Article
Two-Dimensional Regular Shock Reflection for the Pressure Gradient System of Conservation Laws
We establish the existence of a global solution to a regular reflection of a shock hitting a ramp for the pressure gradient system of equations. The set-up of the reflection is the same as that of Mach’s experime...
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Article
A Global Solution to a Two-dimensional Riemann Problem Involving Shocks as Free Boundaries
We present a global solution to a Riemann problem for the pressure gradient system of equations. The Riemann problem has initially two shock waves and two contact discontinuities. The angle between the two shock ...
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Article
Weak Solutions to a Nonlinear Variational Wave Equation
We establish the global existence of weak solutions to the Cauchy problem for a nonlinear variational wave equation, where the wave speed is a given monotone function of the wave amplitude. The equation arise...
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Book
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Chapter
Problems
In this chapter we outline the problems to be studied, namely, the two-dimensional (2-D) compressible Euler system. When special conditions such as irrotationality are assumed, we reduce the Euler system to si...
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Chapter
Riemann Problems
In this chapter we solve one-dimensional Riemann problems for the following systems: isentropic and adiabatic Euler systems for polytropic gases, Lax’s strictly hyperbolic and genuinely nonlinear systems, nonc...
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Chapter
A 2-D Scalar Riemann Problem
Our study of the Burgers equation has helped us in studying the 1-D Euler systems. We hope that the study of 2-D scalar Riemann problems will be similarly useful. As we will see, 2-D Riemann problems are a lot...
