Invariant Manifolds for Steady Boltzmann Flows and Applications

Liu, Tai-Ping; Yu, Shih‐Hsien

doi:10.1007/s00205-013-0640-x

Cited by 43 publications

(70 citation statements)

References 26 publications

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“…For the Boltzmann equation with slab symmetry, i. e., the spatial one-dimensional case, the stability of the these three basic wave patterns are well-understood until now. For instance, the pioneering study on the stability of viscous shock wave was first proved by Liu and Yu [24] with the zero total macroscopic mass condition by the energy method based on the micro-macro decomposition while the existence of viscous shock profile to the Boltzmann equations is given by Caflish and Nicolaenko [3] and Liu and Yu [25]. Then stability of rarefaction wave fan is proved by Liu, Yang, Yu, and Zhao [23] and the stability of viscous contact wave, which is the viscous version of contact discontinuity, by Huang and Yang [17] with the zero mass condition and Huang, Xin, and Yang [16] without the zero mass condition.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation

Wang¹,

Wang²

2019

Kinetic &Amp; Related Models

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We investigate the time-asymptotic stability of planar rarefaction wave for the three-dimensional Boltzmann equation, based on the micro-macro decomposition introduced in [24, 22] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of planar rarefaction wave for 3D Boltzmann equation, while the corresponding results for the shock and contact discontinuities are still completely open.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation

Wang¹,

Wang²

2019

Kinetic &Amp; Related Models

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“…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]. The relation between the gas dynamics and the kinetic theory is a rich field.…”

Section: Discussionmentioning

confidence: 99%

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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“…Now we recall the properties of the shock profile F S i 1 (x − s i t, v) (i = 1, 3) that are given or can be induced by Liu-Yu [22] in Theorem 6.8.…”

Section: Stability Of Boltzmann Shock Profiles For the Bipolar Vpb Symentioning

confidence: 99%

Stability of Nonlinear Wave Patterns to the Bipolar Vlasov–Poisson–Boltzmann System

Wang

Yang

et al. 2017

Arch Rational Mech Anal

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Abstract. The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction wave for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in [21,23], we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as the applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction wave, are proved for the 1D bipolar Vlasov-PoissonBoltzmann system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of rarefaction wave fan to compressible Euler equations is proved to 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in [18] sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.

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Invariant Manifolds for Steady Boltzmann Flows and Applications

Cited by 43 publications

References 26 publications

Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation

Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation

Aspects of dissipation for compressible fluids and kinetic theory

Stability of Nonlinear Wave Patterns to the Bipolar Vlasov–Poisson–Boltzmann System

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