On a Drift–Diffusion System for Semiconductor Devices

Granero-Belinchón, Rafael

doi:10.1007/s00023-016-0493-6

Cited by 26 publications

(22 citation statements)

References 49 publications

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“…Remark 4. The proof of Lemma 2.3 can be referred to [48], and the case of R d can be seen in [28]. Nonlinear maximum principle is an important technique for establishing the L ∞ estimate of solution to Equation (1.1), the more details can be seen in Section 5.…”

Section: Preliminariesmentioning

confidence: 99%

Suppression of blow up by mixing in generalized Keller–Segel system with fractional dissipation

Shi

Wang

2020

Communications in Mathematical Sciences

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In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel equation with a fractional dissipation and advection by a weakly mixing (see Definition 2.4). Here the attractive kernel has strong singularity, namely, the derivative appears in the nonlinear term by singular integral. Without advection, the solution of equation blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of classical solution with large initial data in the case of the derivative of dissipative term is higher than that of nonlinear term. Since the attractive kernel is strong singularity, the weakly mixing has destabilizing effect in addition to the enhanced dissipation effect, which makes the problem more complicated and difficult. In this paper, we establish the L ∞ -criterion and obtain the global L ∞ estimate of the solution through some new ideas and techniques. Combined with [48], we discuss all cases of generalized Keller-Segel system with mixing effect, which was proposed by Kiselev, Xu (see [36]) and Hopf, Rodrigo (see [31]). Based on more precise estimate of solution and the resolvent estimate of semigroup operator, we introduce a new method to study the enhanced dissipation effect of mixing in generalized parabolic-elliptic Keller-Segel equation with a fractional dissipation. And the RAGE theorem is no longer needed in our analysis.

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Section: Preliminariesmentioning

confidence: 99%

Suppression of blow up by mixing in generalized Keller–Segel system with fractional dissipation

Shi

Wang

2020

Communications in Mathematical Sciences

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“…By nonlinear maximum principle (see e.g. [19]), we can assume that there exists a constant C(α) > 0, such that…”

Section: Binbin Shi and Weike Wangmentioning

confidence: 99%

“…There are many different mathematical problems and physical problems relevant to this operator. For example, the fractional Laplacian operator yields the anomalous diffusion, which is related to the dynamics of electrons in a semiconductor [19]. In the stochastic process [3], it is related to random trajectories, generalizing the concept of Brownian motion, which may contain jump discontinuities.…”

Section: Introduction We Consider the 2d Burgers Equation With Fractmentioning

confidence: 99%

Existence and blow up of solutions to the $ 2D $ Burgers equation with supercritical dissipation

Shi¹,

Wang²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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This paper is concerned with the Cauchy problem for a fractal Burgers equation in two dimensions. When α ∈ (0, 1), the same problem has been studied in one dimensions, we can refer to [1, 17, 24]. In this paper, we study well-posedness of solutions to the Burgers equation with supercritical dissipation. We prove the local existence with large initial data and global existence with small initial data in critical Besov space by energy method. Furthermore, we show that solutions can blow up in finite time if initial data is not small by contradiction method.

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“…Denote by Ω + = {x ∈ Ω : h(x) > 0}, Ω − = {x ∈ Ω : h(x) < 0} the (open) subsets of Ω where h is positive or negative, respectively. We apply (27) to h, cf. (10),…”

Section: The Nonlinear Poisson Equationmentioning

confidence: 99%

The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

Disser

Rehberg

2019

Math. Models Methods Appl. Sci.

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We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.

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On a Drift–Diffusion System for Semiconductor Devices

Abstract: Abstract. In this note we study a fractional Poisson-Nernst-Planck equation modeling a semiconductor device. We prove several decay estimates for the Lebesgue and Sobolev norms in one, two and three dimensions. We also provide the first term of the asymptotic expansion as t → ∞.

Cited by 26 publications

References 49 publications

Suppression of blow up by mixing in generalized Keller–Segel system with fractional dissipation

Suppression of blow up by mixing in generalized Keller–Segel system with fractional dissipation

Existence and blow up of solutions to the $ 2D $ Burgers equation with supercritical dissipation

The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

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