Zero-relaxation-time limits in the hydrodynamic equations for plasmas revisited

Jüngel, Ansgar; Peng, Yue‐Jun

doi:10.1007/s000330050004

Cited by 52 publications

(23 citation statements)

References 12 publications

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“…It can be derived from the Euler-Poisson equations when the relaxation time goes to 0. The mathematical justification of this zero-relaxation-time limit has been rigorously performed in [16,17]. In the numerical context, it is much simpler to deal with the elliptic-parabolic coupled system of drift-diffusion equations than the Euler-Poisson hyperbolic system.…”

Section: Introductionmentioning

confidence: 99%

Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis

2003

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Abstract. We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal with coupled terms in the continuity equations. Finally, a numerical example is given to show the efficiency of the scheme.Mathematics Subject Classification. 65M60, 76X05.

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Section: Introductionmentioning

confidence: 99%

Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis

2003

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“…In [17], Marcati and Natalini first established the relation between the simplified (isentropic or isothermal) hydrodynamic model and drift-diffusion model rigorously, via the above zero-relaxation-time limits. Since then, this kind of limit problem has been investigated by various authors for entropy weak solutions [10][11][12][13][14][15], and for smooth solutions [5,21,23]. However, these results are all restricted in the simplified models and the momentum relaxation-time τ is a unique small parameter.…”

Section: Introductionmentioning

confidence: 99%

Relaxation-time limits of non-isentropic hydrodynamic models for semiconductors

Yong

2009

Journal of Differential Equations

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This work deals with non-isentropic hydrodynamic models for semiconductors with short momentum and energy relaxationtimes. The high-and low-frequency decomposition methods are used to construct uniform (global) classical solutions to Cauchy problems of a scaled hydrodynamic model in the framework of critical Besov spaces. Furthermore, it is rigorously justified that the classical solutions strongly converge to that of a drift-diffusion model, as two relaxation times both tend to zero. As a by-product, global existence of weak solutions to the drift-diffusion model is also obtained.

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“…Without this assumption, the limit has been proved in [4] for smooth solutions which are small perturbations of a steady state and then for weak solutions in [26,27] (for the isentropic equations) and in [20] (for the isothermal model). The multidimensional equations are considered in [30].…”

Section: Introductionmentioning

confidence: 99%

“…In [1,5] the relaxation-time limit in the hydrodynamic model including an energy equation has been shown. The idea of [26,27] was to derive estimates uniform in the relaxation time by employing so-called higher-order entropies which allow to obtain L p bounds for any p < ∞. Unfortunately, this idea cannot be used here since we are not able to control the dispersive quantum term.…”

Section: Introductionmentioning

confidence: 99%

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

Jüngel

Matsumura

2006

Journal of Differential Equations

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The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R 3 is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R 3 close to the steady-state is obtained.

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Zero-relaxation-time limits in the hydrodynamic equations for plasmas revisited

Cited by 52 publications

References 12 publications

Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis

Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis

Relaxation-time limits of non-isentropic hydrodynamic models for semiconductors

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

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