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...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.展开更多
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 ...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.展开更多
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 structures.We present some recent devel...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 structures.We 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 equations.In 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.展开更多
This special issue of Acta Mathematica Scientia is dedicated to Professor Banghe Li on the occasion of his 80th birthday.Professor Banghe Li was born in Yueqing City,Zhejiang Province.He graduated from the University ...This special issue of Acta Mathematica Scientia is dedicated to Professor Banghe Li on the occasion of his 80th birthday.Professor Banghe Li was born in Yueqing City,Zhejiang Province.He graduated from the University of Science and Technology of China in 1965 and has been working in the Chinese Academy of Sciences since then.He was elected as an Academician of the Chinese Academy of Sciences in 2001.展开更多
Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are surveyed.The existence and uniqueness of invariant measures are e...Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are surveyed.The existence and uniqueness of invariant measures are established for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white noises.Some further developments,problems,and challenges in this direction are also discussed.展开更多
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 h...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.展开更多
This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a sig...This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. The compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws are first analyzed, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropic flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multi-dimensional steady compressible fluids. Then the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry are revealed. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.展开更多
基金supported in part by the UK Engineering and Physical Sciences Research Council Award EP/E035027/1 and EP/L015811/1
文摘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.
文摘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.
基金The research of Gui-Qiang G.Chen was supported in part by the UK Engineering and Physical Sciences Research Council Awards EP/L015811/1,EP/V008854/1,EP/V051121/1the Royal Society-Wolfson Research Merit Award WM090014.
文摘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 structures.We 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 equations.In 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.
文摘This special issue of Acta Mathematica Scientia is dedicated to Professor Banghe Li on the occasion of his 80th birthday.Professor Banghe Li was born in Yueqing City,Zhejiang Province.He graduated from the University of Science and Technology of China in 1965 and has been working in the Chinese Academy of Sciences since then.He was elected as an Academician of the Chinese Academy of Sciences in 2001.
基金supported by the UK Engineering and Physical Sciences Research Council Award EP/E035027/1,EP/L015811/1the Royal Society-Wolfson Research Merit Award(UK)an Oxford Croucher Scholarship
文摘Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are surveyed.The existence and uniqueness of invariant measures are established for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white noises.Some further developments,problems,and challenges in this direction are also discussed.
文摘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.
基金supported by the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE(No.EP/E035027/1)the UK EPSRC Award to the EPSRC Centre for Doctoral Training in PDEs(No.EP/L015811/1)+1 种基金the National Natural Science Foundation of China(No.10728101)the Royal Society-Wolfson Research Merit Award(UK)
文摘This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. The compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws are first analyzed, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropic flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multi-dimensional steady compressible fluids. Then the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry are revealed. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.
