Convergence of the viscosity method for a nonstrictly hyperbolic conservation law

Lu, Yun

doi:10.1007/bf02096565

Cited by 16 publications

(11 citation statements)

References 9 publications

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“…For instance, when 1 < γ < 3, Diperna established, by using the Glimms scheme method, the existence of global weak solutions for the Cauchy problem, whereas Cheng obtained the same result by applying the compensated compactness method and the kinetic formulation. For the case γ > 3, Lu established a convergence theorem for the method of artificial viscosity applied to system and obtained the existence of global weak solutions for the Cauchy problem. In addition, he obtained an existence theorem for global entropy solutions to the Cauchy problem by the theory of compensated compactness coupled with some basic ideas of the kinetic formulation in another work …”

Section: Introductionmentioning

confidence: 99%

Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions

Zhang

Wang

2018

Math Methods in App Sciences

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We investigate the interaction of delta shock waves and contact discontinuities for the Chaplygin Euler equations of compressible fluid flow with split delta functions. The perturbed Riemann problem when initial data are three piecewise constant states is constructively solved, and the global structure and large time‐asymptotic behaviors of solutions are discussed case by case via deriving how the solution continues beyond points of interaction. It is shown that the Riemann solutions are stable for such small perturbations with initial data by letting perturbed parameter ε tend to zero. Moreover, some numerical simulations completely coinciding with the theoretical analysis are also exhibited.

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Section: Introductionmentioning

confidence: 99%

Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions

Zhang

Wang

2018

Math Methods in App Sciences

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“…Entropy-entropy flux pairs to more general strictly hyperbolic systems or systems in the strictly hyperbolic domains were well analyzed by Lax [5]. However, to apply the compensated compactness method to some nonstrictly hyperbolic systems just as given in the form of system (1.23), some new techniques to construct entropy-entropy flux pairs were investigated in [6,7].…”

Section: System Of Extended Traffic Flowsmentioning

confidence: 99%

Nonstrictly hyperbolic systems with stiff relaxation terms

2006

Journal of Mathematical Analysis and Applications

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In this paper, a zero factor idea is introduced to extend the convergence framework in [G.-Q. Chen, C.D. Levermore, T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994) 787-830] for the singular limits of stiff relaxation from general 2 × 2 hyperbolic conservation systems to nonstrictly hyperbolic systems and an application of this framework on the socalled system of extended traffic flow is obtained.

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“…Reference [30] gives the existence of global weak solution for the special case of g(r)=> r s(s+c) c − 3 ds, c > 3, c > 0, with bounded measurable initial data. A global existence theorem for a general function g(r) was established in [33], provided that system (3.15) has a strictly convex entropy by applying the compensated compactness.…”

Section: ) Singular Limits Of Stiff Relaxationmentioning

confidence: 99%

Singular Limits of Stiff Relaxation and Dominant Diffusion for Nonlinear Systems

Lu¹

2002

Journal of Differential Equations

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We are concerned with singular limits of stiff relaxation and dominant diffusion for general 2 × 2 nonlinear systems of conservation laws, that is, the relaxation time y tends to zero faster than the diffusion parameter e, y=o(e), e Q 0. We establish the following general framework: If there exists an a priori L . bound that is uniformly with respect to e for the solutions of a system, then the solution sequence converges to the corresponding equilibrium solution of this system. Our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems, which is not the case for other type of limits. This framework applies to some important nonlinear systems with relaxation terms, such as the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates, and the extended models of traffic flows. The singular limits are also considered for some physical models, without L . bounded estimates, including the system of isentropic fluid dynamics in Lagrangian coordinates and the models of traffic flows with stiff relaxation terms. The convergence of solutions in L p to the equilibrium solutions of these systems is established, provided that the relaxation time y tends to zero faster than e.

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Convergence of the viscosity method for a nonstrictly hyperbolic conservation law

Cited by 16 publications

References 9 publications

Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions

Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions

Nonstrictly hyperbolic systems with stiff relaxation terms

Singular Limits of Stiff Relaxation and Dominant Diffusion for Nonlinear Systems

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