Multidimensional Conservation Laws: Overview, Problems, and Perspective

Chen, Gui-Qiang G.

doi:10.1007/978-1-4419-9554-4_2

Cited by 10 publications

(8 citation statements)

References 192 publications

(236 reference statements)

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“…Shock reflection problem, which has captured the interest of researchers over the years, is one of the most important problems for the mathematical theory of multidimensional conservation laws that is still largely incomplete. The experimental, computational, and asymptotic analyses show that various patterns of reflected shocks may occur including the regular and Mach reflections (see Courant and Friedrichs [1], Glass [2], Zheng [3,4], Ben-Dor [5], Chen [6], Chen and Feldman [7], and Chang and Hsiao [8]). When a weak plane shock hits a wedge head on, two processes take place simultaneously.…”

Section: Introductionmentioning

confidence: 99%

On Shock Reflection–Diffraction in a van der Waals Gas

Gupta

Sharma

2015

Stud Appl Math

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The problem of a weak shock, reflected and diffracted by a wedge, is studied for the two-dimensional compressible Euler system. Some recent developments are overviewed and a perspective is presented within the context of a real gas, modeled by the van der Waals equation of state. The regular reflection configuration and the detachment criterion are studied in the light of real gas effects. Some basic features of the phenomenon and the nature of the self-similar flow pattern are explored using asymptotic expansions. The analysis presented here predicts several inviscid flow properties of the real gases undergoing shock reflection-diffraction phenomenon.

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Section: Introductionmentioning

confidence: 99%

On Shock Reflection–Diffraction in a van der Waals Gas

Gupta

Sharma

2015

Stud Appl Math

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“…Another recent effort has been made on various important physical, mixed elliptic-hyperbolic problems in steady potential flow for which great progress has been made (cf. [28,33,39,117] and the references cited therein).…”

Section: K<0 K>0mentioning

confidence: 99%

On degenerate partial differential equations

Chen

2010

Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena

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Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, fundamental problems in fluid mechanics and differential geometry. The solution to these fundamental problems greatly requires a deep understanding of nonlinear degenerate partial differential equations. Our emphasis is on exploring and/or developing unified mathematical approaches, as well as new ideas and techniques. The potential approaches we have identified and/or developed through these examples include kinetic approaches, free boundary approaches, weak convergence approaches, and related nonlinear ideas and techniques. We remark that most of the important problems for nonlinear degenerate partial differential equations are truly challenging and still widely open, which require further new ideas, techniques, and approaches, and deserve our special attention and further efforts.

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“…Ideas involving entropy have played a crucial role in developing many important mathematical approaches and methods such as variational principles, Lyapunov functionals, relative entropy methods, monotonicity methods, weak convergence methods, divergence-measure fields, and kinetic methods (cf. [2,7,8,9,11,12,16,17,19,20] and [24]- [33]).…”

Section: Entropymentioning

confidence: 99%

Entropy and Convexity for Nonlinear Partial Differential Equations: Introduction

Ball,

Chen

2013

Preprint

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Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological, and social processes. The concept of entropy originated in thermodynamics and statistical physics during the 19 th century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the theme issue is intended to provide an introduction to entropy, convexity, and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this theme issue.Partial differential equations (PDEs) are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological, and social processes. The behaviour of every material object, with length scales ranging from sub-atomic to astronomical and timescales ranging from picoseconds to millennia, can be modelled by PDEs or by equations having similar features.The concept of entropy originated in thermodynamics and statistical physics during the 19 th century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear PDEs in recent decades. For example, for discontinuous or singular solutions to nonlinear conservation laws which may contain shock waves and concentrations, the notion of entropy solutions is based on entropy conditions involving convexity, motivated by and consistent with the Second Law of Thermodynamics. In addition, entropy methods have become one of the most efficient methods in the analysis of physically relevant discontinuous or singular solutions. The notions of convexity appropriate for multi-dimensional problems, such as polyconvexity, quasiconvexity, and rank-one convexity, are responsible for several recent major advances. In the last three decades, various nonlinear methods involving entropy and convexity have been developed to deal with discontinuous and singular solutions in different areas of PDEs, especially in nonlinear conservation laws, the calculus of variations, and gradient flows.

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Multidimensional Conservation Laws: Overview, Problems, and Perspective

Cited by 10 publications

References 192 publications

On Shock Reflection–Diffraction in a van der Waals Gas

On Shock Reflection–Diffraction in a van der Waals Gas

On degenerate partial differential equations

Entropy and Convexity for Nonlinear Partial Differential Equations: Introduction

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