Basic Aspects of Hyperbolic Relaxation Systems

Yong, Wen‐An

doi:10.1007/978-1-4612-0193-9_4

Cited by 90 publications

(120 citation statements)

References 47 publications

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“…This approach is based on the convergence-stability lemma (Lemma 9.1) in [9] -a continuation principle first formulated in [9] for general (hyperbolic) singular limt problems.…”

Section: Introductionmentioning

confidence: 99%

A note on the zero Mach number limit of compressible Euler equations

Yong

2005

Proc. Amer. Math. Soc.

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“…This approach is based on the convergence-stability lemma (Lemma 9.1) in [9] -a continuation principle first formulated in [9] for general (hyperbolic) singular limt problems.…”

Section: Introductionmentioning

confidence: 99%

A note on the zero Mach number limit of compressible Euler equations

Yong

2005

Proc. Amer. Math. Soc.

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“…To prove this result, we shall adopt and modify the arguments in [25,26]. More precisely, we first construct the approximate solutions, based on the existence of smooth solutions for the corresponding classical drift-diffusion model in [13,23].…”

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confidence: 99%

“…More precisely, we first construct the approximate solutions, based on the existence of smooth solutions for the corresponding classical drift-diffusion model in [13,23]. Next, we establish the continuation principle (convergence-stability lemma) of the Cauchy problem, under the assumption of the important a priori estimate (which is indeed referred to as a convergence assumption in [25,26]). Finally, we should show the a priori convergence assumption; namely, from the energy methods and the special nonlinear structure of the quantum hydrodynamic models, we can obtain the a priori estimate.…”

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confidence: 99%

“…Finally, we should show the a priori convergence assumption; namely, from the energy methods and the special nonlinear structure of the quantum hydrodynamic models, we can obtain the a priori estimate. However, we have to face up to two difficulties, contrasted with [25,26]. The first one is from the Bohm potential, which is a third-order dispersive term.…”

mentioning

confidence: 99%

“…The second difficulty is that we cannot directly make use of the previous convergence-stability lemma, which was first formulated by W.A. Yong in [25], since it only fits into the symmetrizable hyperbolic system with relaxation term. So here we have to establish the corresponding convergence-stability lemma for our quantum hydrodynamic model with the third-order dispersive term.…”

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confidence: 99%

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From a multidimensional quantum hydrodynamic model to the classical drift-diffusion equation

2009

Quart. Appl. Math.

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Abstract. In the paper, we discuss the combined semiclassical and relaxation-time limits of a multidimensional isentropic quantum hydrodynamical model for semiconductors with small momentum relaxation time and Planck constant. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohn potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with thirdorder derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. With the help of the Maxwell-type iteration, we prove that, as the Planck constant and the relaxation time tend to zero, periodic initial-value problems of a scaled isentropic quantum hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion models have smooth solutions. Introduction.In the present paper, we consider the initial value problem of the quantum hydrodynamic model for semiconductors where an additional relaxation term is involved in the linear momentum equation to model the interaction between the electron and crystal lattice. The rescaled multidimensional quantum hydrodynamic models for semiconductors (see [5,6]) are then given by ⎧ ⎪ ⎨

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Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

Bianchini

Hanouzet

Natalini

2007

Comm Pure Appl Math

160

248

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We study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach a constant equilibrium state in the L p -norm at a rate O(t −(m/2)(1−1/ p) ) as t → ∞ for p ∈ [min{m, 2}, ∞]. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m ≥ 2, and by a parabolic equation, in the spirit of Chapman-Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem.

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Basic Aspects of Hyperbolic Relaxation Systems

Cited by 90 publications

References 47 publications

A note on the zero Mach number limit of compressible Euler equations

A note on the zero Mach number limit of compressible Euler equations

From a multidimensional quantum hydrodynamic model to the classical drift-diffusion equation

Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

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