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Two-Dimensional Riemann Problems:Transonic Shock Waves and Free Boundary Problems

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Communications on Applied Mathematics and Computation 2023年第3期5卷 1015-1052页

作者： gui-qiang g.chen Oxford Centre for Nonlinear PDE Mathematical InstituteUniversity of OxfordOxford OX26GGUK

We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and *** present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential *** particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

关键词： Riemann problems Two-dimensional(2-D) Transonic shocks Solution structure Free boundary problems Mixed elliptic-hyperbolic type global configurations Large-time asymptotics global attractors Multidimensional(M-D) Shock capturing methods

STABILITY OF STEADY MULTI-WAVE CONFIgURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

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Acta Mathematica Scientia 2018年第5期38卷 1485-1514页

作者： gui-qiang g.chen Matthew RIgBY Mathematical Institute University of OxfordOxfordOX2 6GGU

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem （initial value problem） in the flow direction, and then develop a modified glimm difference scheme and identify a glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

关键词： stability multi-wave configuration vortex sheet entropy wave shock wave BV perturbation full Euler equations steady wave interactions glimm scheme

Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

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Chinese Annals of Mathematics,Series B 2019年第6期40卷 967-1004页

作者： gui-qiang g.chen Peter H.C.PANg Mathematical Institute University of Oxford

Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are *** existence and uniqueness of invariant measures are established for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white *** further developments,problems,and challenges in this direction are also discussed.

关键词： Stochastic solutions Entropy solutions Invariant measures Existence Uniqueness Stochastic forcing Anisotropic degenerate Parabolichyperbolic equations Long-time behavior

A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUgH COMPLIANT AXISYMMETRIC VESSELS

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Acta Mathematica Scientia 2010年第2期30卷 391-427页

作者： gui-qiang g.chen School of Mathematical Sciences,Fudan University,Shanghai 200433,China Mathematical Institute,University of Oxford,24-29 St giles,Oxford,OX1 3LB,UK Department of Mathematics,Northwestern University,Evanston,IL 60208-2730,USA Weihua Ruan Department of Mathematics,Computer Science and Statistics,Purdue University Calumet,Hammond,IN 46323-2094,USA [a] School of Mathematical Sciences Fudan University Shanghai 200433 China [b] Mathematical Institute University of Oxford 24-29 St Giles Oxford OX1 3LB UK [c] Department of Mathematics Northwestern University Evanston IL 60208-2730 USA [d] Department of Mathematics Computer Science Statistics Purdue University Calumet Hammond IN 46323-2094 USA

We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisyminetric vessels. Early models derived are nonconservative and/or nonho- mogeneous with measure source terms, which are endowed with infinitely many Riemann solutions for some Riemann data. In this paper, we derive a one-dimensional hyperbolic system that is conservative and homogeneous. Moreover, there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data, under a natural stability entropy criterion. The Riemann solutions may consist of four waves for some cases. The system can also be written as a 3 × 3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.

关键词： conservation laws hyperbolic system fluid flow blood flow vessel hyper-bolicity Riemann problem Riemann solution wave curve shock wave rarefaction wave standing wave stability

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Preface

Acta Mathematica Scientia 2015年第4期35卷 761-762页

作者： Stefano Bianchini gui-qiang g.chen Xiaqi Ding Tong Yang Department of Information Engineering and Mathematics University of Siena Siena Italy Department of Computer Science University of Milan Milan Italy Regional Campus Manipur Indira gandhi National Tribal University Manipur Imphal India Bharath Institute of Higher Education and Research Tamil Nadu Chennai India

ISBN: (纸本)9789811929793

This special issue of Acta Mathematica Scientia is dedicated to Tai-Ping Liu on the occasion of his 70th birthday. In the very active and far-reaching field of hyperbolic conservation laws, balance laws, and kinetic equations, it is impossible not to come across the name of Tai-Ping Liu. Certainly, it is not needed to invoke evaluations of experts to realize the impor- tance of the contributions that Tai-Ping Liu has made in the field of partial differential equations.

关键词： Preface very

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Preface

Chinese Annals of Mathematics,Series B 2019年第5期40卷 642-642,641页

作者： gui-qiang g.chen Fanghua LIN Xiao-Ming WANg 不详

Professor Andrew J. Majda is one of the most influential mathematicians of our time. Hiscontributions to theoretical partial differential equations and many applied areas are seminaland fundamental. In the course of his phenomenal scientific career, Professor Majda has writtenmore than 300 papers and 7 books, which have been cited more than 10000 times.

关键词： Preface Professor Hiscontributions

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PREFACE

Acta Mathematica Scientia 2022年第6期42卷 2189-2191页

作者： gui-qiang g.chen Bo LI Zizhou TANg Xiping ZHU Chinese Acad Sci Beijing Peoples R China

This special issue of Acta Mathematica Scientia is dedicated to Professor Banghe Li on the occasion of his 80th *** Banghe Li was born in Yueqing City,Zhejiang *** graduated from the University of Science and Technology of China in 1965 and has been working in the Chinese Academy of Sciences since *** was elected as an Academician of the Chinese Academy of Sciences in 2001.

关键词： Academician Province. birthday

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混合型偏微分方程——分析和应用

数学译林 2023年第4期42卷 289-308页

作者：陈贵强龚雪飞(译) 陈贵强(校) 牛津大学数学研究所牛津大学基布尔学院牛津非线性偏微分方程中心不详

偏微分方程(Partial Differential Equations,PDEs)是众多数学和科学进展的核心.尽管在过去的80年中标准型偏微分方程的理论获得了巨大进展,但是混合型非线性偏微分方程的分析仍处于发展初期.本文的目标是通过若干流体力学,微分几何以... 详细信息

偏微分方程(Partial Differential Equations,PDEs)是众多数学和科学进展的核心.尽管在过去的80年中标准型偏微分方程的理论获得了巨大进展,但是混合型非线性偏微分方程的分析仍处于发展初期.本文的目标是通过若干流体力学,微分几何以及其他领域中的长期基本问题.

关键词：偏微分方程微分几何流体力学分析和应用标准型科学进展混合型

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems Dedicated to Professor LI Ta Tsien on the Occasion of His 80th Birthday

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Science China Mathematics 2017年第8期60卷 1353-1370页

作者： chen gui-qiang School of Mathematical Sciences Fudan University Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Mathematical Institute University of Oxford

When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge,governed by the Euler equations,there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle—the steady weak shock with supersonic or subsonic downstream flow(determined by the wedge angle that is less than or greater than the sonic angle)and the steady strong shock with subsonic downstream flow,both of which satisfy the entropy *** fundamental issue—whether one or both of the steady weak and strong shocks are physically admissible solutions—has been vigorously debated over the past eight *** this paper,we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic *** the static stability,we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small,and we finally present some recent results on the static stability of the steady supersonic and transonic *** the dynamic stability for potential flow,we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem,and we finally survey some recent developments in solving this free boundary problem for the existence of the PrandtlMeyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to *** further developments and mathematical challenges in this direction are also discussed.

关键词： shock wave free boundary wedge problem transonic mixed type Euler equations static stability dynamic stability