Shock waves in conservation laws with physical viscosity

Liu, Tai-Ping; Zeng, Yanni

doi:10.1090/memo/1105

Cited by 37 publications

(31 citation statements)

References 30 publications

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“…There is another more quantitative approach based on concrete constructions [14,18,21,[42][43][44] on hyperbolic conservation laws (see [17] and references therein). Green's function approach has been useful for the study of nonlinear waves for viscous conservation laws [4,22,23,45,46]. Green's function approach [32,34,47,48] for the Boltzmann equation has yielded quantitative understanding of nonlinear waves and the boundary behaviour [41,[49][50][51].…”

Section: Discussionmentioning

confidence: 99%

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Aspects of dissipation for compressible fluids and kinetic theory

Liu

2013

Phil. Trans. R. Soc. A.

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There are several types of dissipation—the viscosity and heat conductivity, the nonlinearity and the coupling of distinct characteristics—that can occur in the system of conservation laws. In addition to these corresponding types of dissipation, there are other consequences of the H-theorem and the boundary dissipative effects for the kinetic theory. We discuss these issues and raise questions.

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Section: Discussionmentioning

confidence: 99%

“…Even for such a system, the study of the wave propagation over shock waves was understood only recently, for artificial viscosity [22] and for physical viscosity [23]. The study of rarefaction waves was also initiated recently [24].…”

Section: System Of Conservation Lawsmentioning

confidence: 99%

Aspects of dissipation for compressible fluids and kinetic theory

Liu

2013

Phil. Trans. R. Soc. A.

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“…It is well-known (eg. see Chapter 2 in [25]) that the propagation of shock profiles for viscous conservation laws such as the classical Navier-Stokes equations can be approximately described by the rather simple viscous Burgers equation in the weak shock regime. However, when the dispersive effect is involved, the generation of dispersive plasma shock waves has been observed in physical experiments and investigated by numerical simulations, for instance, [11,30].…”

Section: Formal Kdv-burgers Approximationmentioning

confidence: 99%

Ion-acoustic shock in a collisional plasma

Duan

Liu

Zhang

2020

Journal of Differential Equations

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The paper is concerned with the propagation of ion-acoustic shock waves in a collision dominated plasma. We firstly establish the existence and uniqueness of a small-amplitude smooth travelling wave, then justify its approximation to the shock profile of the KdV-Burgers equations in a suitable asymptotic regime where dissipation in terms of viscosity coefficient is much stronger than dispersion by the Debye length, and prove in the end the large time asymptotic stability of travelling waves under suitably small smooth perturbations.

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“…For the study of shock waves, the Burgers Green's function is used in the construction of the Green's function, [28]. The main motivation of J. M. Burgers for the study of the equation u t + uu x = µu xx was to understand certain key elements of turbulence for compressible flows, [3,4,5].…”

Section: ] Hopf-cole Transformation 91mentioning

confidence: 99%