Gui-Qiang G. Chen
University of Oxford, Mathematical Institute, Faculty Member
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We present our recent results on the mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the potential flow equation. The shock diffraction problem can be formulated... more
We present our recent results on the mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the potential flow equation. The shock diffraction problem can be formulated as an initial-boundary value problem, which is invariant under self-similar scaling. Then, by employing its self-similar invariance, the problem is reduced to a boundary value problem for a first-order nonlinear system of partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It is further reformulated as a free boundary problem for a nonlinear degenerate elliptic system of first-order in a bounded domain with a boundary corner whose angle is bigger than π. A first global theory of existence and regularity has been established for this shock diffraction problem for the potential flow equation.
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Abstract In this paper, the one-dimensional motion for compressible viscous and heat conductive fluids is considered.
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This article is a survey of Cathleen Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz’s fundamental... more
This article is a survey of Cathleen Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz’s fundamental work on the nonexistence of continuous transonic flows past profiles, Morawetz’s program regarding the construction of global steady weak transonic flow solutions past profiles via compensated compactness, and a potential theory for regular and Mach reflection of a shock at a wedge. The profound impact of Morawetz’s work on recent developments and breakthroughs in these research directions and related areas in pure and applied mathematics are also discussed.
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Abstract Lax, Gel'fand, Oleinik and others have established some fundamental concepts about system of one spatial dimensional conservation laws, such as hyperbolicity, genuine nonlinearity (convexity), linear degenerate,... more
Abstract Lax, Gel'fand, Oleinik and others have established some fundamental concepts about system of one spatial dimensional conservation laws, such as hyperbolicity, genuine nonlinearity (convexity), linear degenerate, Rankine–Hugoniot condition, stability condition (entropy condition), stability condition (E) and so on. We extend these concepts to system of two spatial dimension case in this note. It is a necessary preliminary for advance study.
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A hierarchy of centered (non-upwind) difference schemes is identified for solving hyperbolic equations. The bottom of the hierarchy is the classical Lax–Friedrichs scheme, which is the least accurate in computation, and the top of the... more
A hierarchy of centered (non-upwind) difference schemes is identified for solving hyperbolic equations. The bottom of the hierarchy is the classical Lax–Friedrichs scheme, which is the least accurate in computation, and the top of the hierarchy is the FORCE scheme, which is the optimal scheme in the family. The FORCE scheme is optimal in the sense that it is monotone, has the optimal stability condition for explicit methods, and has the smallest numerical viscosity. It is shown that the FORCE scheme is consistent with the Lax entropy inequality, that is, the limit functions of the FORCE approximate solutions are entropy solutions. The convergence of the FORCE scheme is also established for the isentropic Euler equations and the shallow water equations. Some related centered difference schemes are also surveyed and discussed.
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ABSTRACT
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ABSTRACT The authors study the Cauchy problem for a scalar conservation law ρ t +F(x,ρ) x =0 with generally discontinuous flux F(x,ρ) satisfying the assumptions of E. Audusse and B. Perthame [Proc. R. Soc. Edinb., Sect. A, Math. 135, No.... more
ABSTRACT The authors study the Cauchy problem for a scalar conservation law ρ t +F(x,ρ) x =0 with generally discontinuous flux F(x,ρ) satisfying the assumptions of E. Audusse and B. Perthame [Proc. R. Soc. Edinb., Sect. A, Math. 135, No. 2, 253–265 (2005; Zbl 1071.35079)]. They generalize the uniqueness result of Audusse-Perthame to the case of measure-valued entropy solutions. Using this result, the authors justify the hydrodynamic limit of large particle systems.
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We develop a general L --framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach [9]. We apply our L --framework to... more
We develop a general L --framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach [9]. We apply our L --framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without the requirement of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an L error estimate for an upwind-central finite difference scheme.
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Page 1. Conditional Gain is Necessary and Sufficient for the Robust Stabilization of Nonlinear Systems Anna L. Chen, Gui-Qiang Chen, and Randy A. Freeman Abstract For the feedback interconnection of general non-linear ...
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ABSTRACT The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous... more
ABSTRACT The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous initial value function ϕ(x) is continuous outside a set Γ of measure zero and satisfies (∗)ϕ(x)⩾ϕ∗∗(x):=liminfy→x,y∈Rd⧹Γϕ(y). We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying (∗) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L1-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L∞-solutions is clarified. To cite this article: G.-Q. Chen, B. Su, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 113–118
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Hyperbolic systems of conservaiton laws with a symmetry are studied. Some peculiar phenomena for such systems are shown. Admissibility criteria for solutions to such systems are discussed. propagation and cancellation of initial... more
Hyperbolic systems of conservaiton laws with a symmetry are studied. Some peculiar phenomena for such systems are shown. Admissibility criteria for solutions to such systems are discussed. propagation and cancellation of initial oscillations for the systems are classified. As a byproduct of this study, an existence theorem of global solutions for the Cauchy of the systems is established.
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Some evidence indicates that spherically symmetric solutions of the compressible Euler equations blow up near the origin at some time under certain circumstances (cf. [4,19]). In this paper, we observe a criterion forL∞Cauchy data of... more
Some evidence indicates that spherically symmetric solutions of the compressible Euler equations blow up near the origin at some time under certain circumstances (cf. [4,19]). In this paper, we observe a criterion forL∞Cauchy data of arbitrarily large amplitude to ensure the existence ofL∞spherically symmetric solutions in the large, which model outgoing blast waves and large-time asymptotic solutions. The equilibrium states of the solutions and their asymptotic decay to such states are analysed. Some remarks on global spherically symmetric solutions are discussed.