Zero relaxation time limits to isothermal hydrodynamic model for semiconductor

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Abstract

In this paper, we remove the bounded total variation condition on the initial data and the restriction of the concentration of a fixed background charge being a constant in the paper “Relaxation of the Isothermal Euler–Poisson System to the Drift-Diffusion Equations,” (Quart. Appl. Math., 58 (2000), 511–521), and obtain the zero relaxation time limits to isothermal hydrodynamic model for semiconductor by using the varying viscosity method.

Introduction

In this paper, we study the relaxation limit of the one-dimensional isothermal Euler–Poisson model for semiconductor devices: ρt+(ρu)x=0,(ρu)t+(ρu2+P(ρ))x=ρEρuτ,Ex=ρn(x),in the region (,+)×(0,), with bounded initial data (ρ,u)|t=0=(ρ0(x),u0(x)),lim|x|(ρ0(x),u0(x))=(0,0),ρ0(x)0and a condition at for the electric field limxE(x,t)=E, for a.e.t(0,),where T,E are fixed constants, ρ0 denotes the electron density, the pressure–density relation is P(ρ)=ρ, u the (average) particle velocity and E the electric field, which is generated by the Coulomb force of the particles. The given function n(x) represents the concentration of a fixed background charge [1], [2] and τ>0 is the momentum relaxation time. From the physical and engineering point of view, the isothermal case P(ρ)=ρ is very important. The global existence of entropy solutions of (1.1) with BV(R) initial data was obtained in [2], [3] by using the Glimm method [4], [5], and with bounded L(R) initial data or initial–boundary values was well studied in [6], [7] by using the compensated compactness method.

In this paper, we are concerned with the relaxation limit of the problem (1.1)–(1.3) when τ0+. In the isentropic case P(ρ)=1γργ,γ>1, Marcati and Natalini introduced a “parabolic scaling” sτt,xx, and showed that in the new variables, the solution converges to the solution of the drift-diffusion system [8] (See also [9], [10] for the solutions in Lp,1<p< space). In the isothermal case P(ρ)=ρ, under the bounded total variation condition on the initial data and the restriction n(x)=N, where N0 is a constant, Junca and Rascle [11] proved that the BV(R) solution (ρτ,Eτ), obtained in [2] converges to the solution of the drift–diffusion equations ρs+x(ρEρx)=0Ex=ρNin the sense of distributions, where sτt and (ρ,E) is the relaxation limit of (ρτ,Eτ) as τ0+. In this paper, under the assumptions of the initial data u0(x)L(R),ρ0(x)L(R)L1(R) and n(x)L1(R), we obtain the similar zero-relaxation limit by using the varying viscosity method.

The classical viscosity method is to add the diffusion terms to the right-hand side of system (1.1) and to study the following parabolic system ρt+(ρu)x=ερxx,(ρu)t+(ρu2+P(ρ))x=(ρu)xx+ρEρuτ,Ex=ρn(x).If we consider the momentum m=ρu as an independent variable, we must first obtain the positive, lower estimate of ρε since ρu2=m2ρ in the second equation of (1.5) is not well defined at ρ=0. However, if we apply the third equation in (1.5) to resolve Eε(x,t), the new problem arises of how to control the integral of xρε(x,t)dx.

To overcome the above difficulty, we construct the approximate solutions of (1.1) by adding the classical viscosity coupled with the flux approximation ρt+((ρ2δ)u)x=ερxx,(ρu)t+(ρu2δu2+ρ2δlnρ)x=ε(ρu)xx+(ρ2δ)E1τ(ρ2δ)u,Ex=(ρ2δ)n(x)with the initial data (ρε,δ(x,0),uε,δ(x,0))=(ρ0(x)+2δ,u0(x))Gε,where (ρ0(x),u0(x)) are given in (1.2), δ>0 denotes a regular perturbation constant, Gε is a mollifier such that (ρε,δ(x,0),uε,δ(x,0)) are smooth and lim|x|(ρε,δ(x,0),uε,δ(x,0))=(2δ,0),lim|x|(ρxε,δ(x,0),uxε,δ(x,0))=(0,0).One obvious advantage of the above viscosity-flux approximation is that we may obtain the bound ρε,δ2δ>0 immediately, by applying the maximum principle to the first equation in (1.7), which guarantees that both the term ρu2=m2ρ, and the function Eε,δ(x,t)=xρε,δ(x,t)2δn(x)dx are well defined. More precisely, the following lemma was obtained in [10] by using the compensated compactness method [12]

Lemma 1.1

Let (ρ0(x),u0(x)) be bounded in L(R) and (ρ0(x),n(x)) be bounded in L1(R). Then, for any fixed ε>0,δ>0,τ>0, the problem (1.6)(1.8) has a unique global smooth solution (ρε,δ(x,t),uε,δ(x,t),Eε,δ(x,t)) in R×(0,T], satisfying lim|x|(ρε,δ(x,t),uε,δ(x,t))=(2δ,0),lim|x|(ρxε,δ(x,t),uxε,δ(x,t))=(0,0),lnρε,δ(x,t)uε,δ(x,t)M1+M2t,lnρε,δ(x,t)+uε,δ(x,t)M1+M2t,0<2δρε,δ,|ρε,δ(,t)2δ|L1(R)M3,|Eε,δ|M3,where the constants Mi,i=1,2,3 depend only on the bounds of the initial data, but are independent of ε,δ,τ.

In this paper, we are concerned with the zero-relaxation-time-limit of above viscosity solutions as ε,δ,τ go to zero, without the uniformly time-independent estimates on (ρε,δ,uε,δ).

Theorem 1.1

Let the conditions in Lemma 1.1 and u02(x)L1(R) be satisfied; let s=τt,vτ(x,s)=vε,δ(x,sτ)=vε,δ(x,t) for any function v. Then, there exists a subsequence (still labelled) ({ρτ},{Eτ}) such that ρτ(x,s)ρ(x,s) weakly in Lloc1(R×R+),Eτ(x,s)E(x,s) strongly in Llocp(R×R+),p1 when ε,δ,τ go to zero, and the limit (ρ,E) is a solution of the drift–diffusion equations ρs+x(ρEρx)=0,Ex=ρn(x)in the sense of distributions.

Remark 1.1

It is worthwhile to point out that the results in Theorem 1.1 can be easily extended to the following Euler–Poisson equations of two-carrier types in one dimension ρit+(ρiui)x=0,(ρiui)t+(ρi(ui)2+ρi)x=ρiEρiuiτi,i=1,2,Ex=ρ1+ρ2n(x),in the region (,+)×[0,T], with suitable bounded initial data and the condition (1.3) at for the electric potential E, where (ρ1,u1) and (ρ2,u2) are the (density, velocity) pairs for electrons (i=1) and holes (i=2) respectively, and the given function n(x) represents the impurity doping profile.

In the above case, the drift–diffusion equations (1.10) are replaced by ρis+x(ρiEρix)=0,Ex=ρ1+ρ2n(x),i=1,2.

Section snippets

The proof of Theorem 1.1

Let s=τt and vε,δ(x,t)=vε,δ(x,sτ)=vτ(x,s). Then vε,δt=vτsst=τvτs,vε,δx=vτxand τρτs+x((ρτ2δ)uτ)=ε2ρτx2,τ(ρτuτ)s+x((ρτδ)(uτ)2+ρτ2δlnρτ)=(ρτ2δ)Eτ1τ(ρτ2δ)uτ+ε2(ρτuτ)x2,Eτx=ρτb(x)due to (1.6).

We shall prove Theorem 1.1 by the following several lemmas.

Lemma 2.1

We have the estimates 0LR1τ2(ρτ2δ)(uτ)2dxdsM(L) and 0LRετρτ(ρxτ)2M(L).

Proof of Lemma 2.1

Multiplying the first equation in (2.2) by ηρ, and the second by ηm, then adding the result, we have τηs(ρτ,mτ)+qx(ρτ,mτ)=εηxx(ρτ,mτ)+(

Acknowledgements

This paper is partially supported by the NSFC Grant Nos. LY20A010023 and LY17A010025 of Zhejiang Province of China, the NSFC Grant No. 11471281 of China and a Humboldt renewed research fellowship of Germany.

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