Transonic Steady-States of Euler--Poisson Equations for Semiconductor Models with Sonic Boundary

Liang, Chen; Mei, Ming; Zhang, Guojing; Zhang, Kaijun

doi:10.1137/21m1416369

Cited by 17 publications

(5 citation statements)

References 27 publications

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“…In particular, when the boundary states are separated in the supersonic regime and the subsonic regime, they obtained that the Euler-Poisson system with supersonic doping profile possesses two C ∞ -smooth transonic solutions, where one is from supersonic region to subsonic region and the other is of the inverse direction. Moreover, the existence of 2D and 3D radial subsonic/supersonic/transonic steady-states with the sonic boundary conditions were technically proved by Chen-Mei-Zhang-Zhang in [11] and [12], respectively.…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

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Nonlinear structural stability and linear dynamic instability of transonic steady-states to a hydrodynamic model for semiconductors

Feng¹,

Mei²,

Zhang³

2022

Preprint

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For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, the existence of steady transonic shock solutions and C ∞ -smooth steady transonic solutions for Euler-Poisson Equations were established in [27] and [41], respectively. In this paper we further study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. When the C 1 -smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the proof of structural stability. For any relaxation time: 0 < τ ≤ +∞, by means of elaborate singularity analysis, we first investigate the structural stability of the C 1 -smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. Moreover, when the relaxation time is large enough τ 1, under the condition that the electric field is positive at the shock location, we prove that the transonic shock steady-states are structurally stable with respect to small perturbations of the supersonic doping profile. Furthermore, we show the linearly dynamic instability for these transonic shock steady-states provided that the electric field is suitable negative. The proofs for the structural stability results are based on singularity analysis, a monotonicity argument on the shock position and the downstream density, and the stability analysis of supersonic and subsonic solutions. The linear dynamic instability of the steady transonic shock for Euler-Poisson equations can be transformed to the ill-posedness of a free boundary problem for the Klein-Gordon equation. By using a nontrivial transformation and the shooting method, we prove that the linearized problem has a transonic shock solution with exponential growths. These results enrich and develop the existing studies.

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Section: Introduction and Main Resultsmentioning

confidence: 97%

“…Recently, the study in this topic has made some profound progress [10,11,12,26,27,41]. For Euler-Poisson equations with relaxation effect (1.1), when the boundary is subjected to be sonic (the critical case), Li-Mei-Zhang-Zhang [26,27] first classified the structure of all type of physical solutions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nonlinear structural stability and linear dynamic instability of transonic steady-states to a hydrodynamic model for semiconductors

Feng¹,

Mei²,

Zhang³

2022

Preprint

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“…where J = nu stands for the current density. However, with different settings on the boundary, the doping profile b(x) and the relaxation time τ , the stationary Euler-Poisson system (1.2) may or may not possess the physical subsonic/supersonic/transonic solutions, or may have totally different regularities [1,4,6,8,9,13,14,18,19]. The influence from these physical quantities, in particular, from the doping profile, is essential and important for the structure of solutions.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, in [18], they proved that the critical boundary value problem admits a unique subsonic solution, at least one supersonic solution, infinitely many transonic shocks if α ≪ 1, and infinitely many transonic C 1 -smooth solutions if α ≫ 1; in [19], they showed the nonexistence of all types of physical steady states to the critical boundary-value problem assuming that the doping profile is small enough and α ≫ 1, and they also discussed the existence of supersonic and transonic shock solutions under the hypothesis that the doping profile is close to the sonic state and α ≪ 1. Inspired by the groundbreaking works [18,19], there is a series of interesting generalizations into the transonic doping profile case in [4], the case of transonic C ∞ -smooth steady states in [32], the multi-dimensional cases in [5,6], and even the bipolar case [25].…”

Section: Introductionmentioning

confidence: 99%

Structural stability of subsonic steady states to the hydrodynamic model for semiconductors with sonic boundary

Feng,

Hu,

Mei

2024

Nonlinearity

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The hydrodynamic model for semiconductors with sonic boundary, represented by Euler–Poisson equations, possesses the various physical steady states including interior-subsonic/interior-supersonic/shock-transonic/C 1-smooth-transonic steady states. Since these physical steady states result in some serious singularities at the sonic boundary (their gradients are infinity), this makes that the structural stability for these physical solutions is more difficult and challenging, and has remained open as we know. In this paper, we investigate the structural stability of interior subsonic steady states. Namely, when the doping profiles are as small perturbations, the differences between the corresponding subsonic solutions are also small. To overcome the singularities at the sonic boundary, we propose a novel approach, which combines the weighted multiplier technique, local singularity analysis, monotonicity argument and squeezing skill. Both the result itself and the technique developed here will give us some truly enlightening insights into our follow-up study on the structural stability of the remaining types of solutions. A number of numerical approximations are also carried out, which intuitively confirm our theoretical results.

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“…More precisely, in [14], the authors proved that the critical boundary-value problem admits a unique subsonic solution, at least one supersonic solution, infinitely many transonic shocks if α ≪ 1, and infinitely many transonic C 1 -smooth solutions if α ≫ 1; in [15], the authors showed the nonexistence of all types of physical steady states to the critical boundary-value problem assuming that the doping profile is small enough and α ≫ 1, and they also discussed the existence of supersonic and transonic shock solutions under the hypothesis that the doping profile is close to the sonic state and α ≪ 1. Inspired by the groundbreaking works [14,15], there is a series of interesting generalizations into the transonic doping profile case in [4], the case of transonic C ∞ -smooth steady states in [24], the multi-dimensional cases in [5,6], and even the bipolar case [19].…”

Section: Introductionmentioning

confidence: 99%

Structural stability of interior subsonic steady-states to hydrodynamic model for semiconductors with sonic boundary

Feng¹,

Hu²,

Mei³

2022

Preprint

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For the stationary hydrodynamic model for semiconductors with sonic boundary, represented by Euler-Poisson equations, it possesses the various physical solutions including interior subsonic solutions/interior supersonic solutions/shock transonic solutions/C 1 -smooth transonic solutions. However, the structural stability for these physical solutions is challenging and has remained open as we know. In this paper, we investigate the structural stability of interior subsonic solutions when the doping profiles are restricted in the subsonic region. The main result is proved by using the local (weighted) singularity analysis and the monotonicity argument. Both the result itself and techniques developed here will give us some truly enlightening insights into our follow-up study on the structural stability of the remaining types of solutions.

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Transonic Steady-States of Euler--Poisson Equations for Semiconductor Models with Sonic Boundary

Cited by 17 publications

References 27 publications

Nonlinear structural stability and linear dynamic instability of transonic steady-states to a hydrodynamic model for semiconductors

Nonlinear structural stability and linear dynamic instability of transonic steady-states to a hydrodynamic model for semiconductors

Structural stability of subsonic steady states to the hydrodynamic model for semiconductors with sonic boundary

Structural stability of interior subsonic steady-states to hydrodynamic model for semiconductors with sonic boundary

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