# Existence and Large Time Behavior of Entropy Solutions to One-Dimensional Unipolar Hydrodynamic Model for Semiconductor Devices with Variable Coefficient Damping

Authors: Yan Li; Yanqiu Cheng; Huimin Yu
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we investigate the global existence and large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Possion equations with time and spacedependent damping in a bounded interval. Firstly, we prove the existence of entropy solutions through vanishing viscosity method and compensated compactness framework. Based on the uniform estimates of density, we then prove the entropy solutions converge to the corresponding unique stationary solution exponentially with time. We generalize the existing results to the variable coefficient damping case.


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## Body

## 1. Introduction

The present paper is concerned with the one-dimensional isentropic Euler-Possion model for semiconductor devices with damping:(1)ρt+mx=0,mt+m2ρ+Pρx=ρE+Hx,tm,Ex=ρ−bx,where space variable x∈L1,L2(L1 and L2 are two positive constants) and time variable t∈0,TT>0. Here, ρ≥0, m, Hx,t, Pρ, and E stand for electron density, electron current density, damping coefficient, pressure, and electric filed, respectively. We assume the damping coefficient Hx,t is bounded, and the pressure function is given by Pρ=p0ργ, where p0=θ2/γ and θ=γ−1/2. Here, γ presents the adiabatic coefficient, and γ>1 corresponds to the isentropic case. The doping profile bx≥0 stands for the density of fixed, positively charged background ions. In this paper, we assume
(2)bx∈CL1,L2,0<b∗≤bx≤b∗,where b∗ andb∗ are two positive constants. The initial-boundary value conditions of system (1) are
(3)ρ,mx,0=ρ0x,m0x,L1<x<L2,mL1,t=mL2,t=0,t≥0,EL1,t=E−,t≥0,where ρ0x satisfies
(4)∫L1L2ρ0x−bxdx=0.Firstly, let us survey the related mathematical results. In 1990, Degond and Markowich [1] firstly proved the existence and uniqueness of the steady-state to (1) in subsonic case, which is characterized by a smallness assumption on the current flowing through the device. It was proved that the existence of local smooth solution to the time-dependent problem by using Lagrangian mass coordinates in [2]. However, Chen-Wang in [3] had studied the smooth solution would blow up in finite time; therefore, it is worthwhile considering the existence and other properties of weak solutions. As for weak solutions, Zhang [4] and Marcati-Natalini [5] proved the global existence of entropy solutions to the initial-boundary value and Cauchy problems for γ>1, respectively. Li [6] and Huang et al. [7] proved the existence of L∞ entropy solution of (1) with γ=1 on a bounded interval and the whole space by using a fractional Lax-Friedrichs scheme. It is worth noting that the L∞ estimates of entropy solution, especially the estimate of density, in all of the above works [4–7] depend on time t, which restricted us to consider their large time behavior further. We refer [8–10] for more results on this model and topic. In this paper, for 1<γ≤3 and variable coefficient damping, we shall first verify the assumption in [11], where the density is assumed to be uniformly bounded with respect to space x and time t and then use the entropy inequality to consider the large time behavior of the obtained solutions.Based on the related results in [12–16], we are convinced that the method developed in this paper can be used to bipolar Euler-Poisson system with time depended damping. We will investigate this problem in next papers.To start our main theorem, we define the entropy solution of system (1) as.Definition 1.
For everyT>0, a pair of bounded measurable functions vx,t=ρx,t,mx,t,Ex,t is called a L∞ weak solution of (1) with initial-boundary condition (3) if
(5)∫0T∫L1L2ρφt+mφxdxdt+∫L1L2ρ0φx,0dx=0,∫0T∫L1L2mφt+m2ρ+Pρφxdxdt+∫0T∫L1L2ρE+Hx,tmφdxdt+∫L1L2m0φx,0dx=0,Ex,t=∫L1xρ−bsds+E−,holds for any test function φ∈C0∞L1,L2×0,T, and the boundary condition is satisfied in the sense of divergence-measure field [17]. Furthermore, we call the weak solution ρ,m,Ex,t to be an entropy solution if the entropy inequality
(6)ηt+qx≤ηmρE+Hx,t,satisfies in the sense of distribution for any weak convex entropy pairs ηρ,m,qρ,m.Definition 2.
The stationary solution of problems (1) and (3) is the smooth solution of
(7)Pρ~x=ρ~E~,E~x=ρ~−bx,with the boundary condition
(8)E~L1=E~L2=0.Our main results in this paper are as follows.Theorem 3 (Existence).
Let1<γ≤3, we assume that the initial data and the damping coefficient satisfy
(9)0≤ρ0x≤M0,m0x≤M0ρ0x,Hx,t≤M1,for some positive constants M0 and M1. Then, there exists a global entropy solution ρ,m,Ex,t of the initial-boundary value problems (1) and (3) satisfying
(10)0≤ρx,t≤C,mx,t≤Cρx,t,Ex,t≤C,x,t∈L1,L2×0,T,where C is independent of t.Remark 4.
To get the global existence of theL∞ weak solution, we only need Hx,t is bounded. However, to get the large time behavior of the obtained solution, the uniform negative upper bound is necessary.Theorem 5 (Large time behavior).
Suppose there exists a positive constantδ0>0, such that the damping coefficient
(11)Hx,t<−δ0andHtx,t>−2b∗,for any L1,L2×ℝ+. Denote ρ,m,Ex,t is the global entropy solution of (1) and (3) obtained in Theorem 3, and ρ~,E~ is the stationary solution; then, it holds that
(12)∫L1L2ρ−ρ~2E−E~2+m2x,tdx≤Ce−Ct,for some positive constant C.Remark 6.
Theorems3 and 5 are generalizations of the corresponding theorem of [18], in which the damping coefficient Hx,t=−1. Suppose α,β, and λ are three positive constants, then Hx,t=−α/1+tλ−β satisfies all the assumptions of Hx,t in Theorems 3 and 5.


## 2. Preliminary and Formulation

We consider the homogeneous system(13)ρt+mx=0,mt+m2ρ+Pρx=0.Firstly, we user1 and r2 to denote the right eigenvectors corresponding to the eigenvalues λ1 and λ2. After simple calculation, we have
(14)λ1=mρ−θρθ,λ2=mρ+θρθ,θ=γ−12,r1=1λ1,r2=1λ2.The Riemann invariantsw,z are given by
(15)w=mρ+ρθ,z=mρ−ρθ,satisfying ∇w·r1=0 and ∇z·r2=0, where ∇=∂ρ,∂m is the gradient with respect to U=ρ,m.A pair of functionsη,q: ℝ×ℝ+↦ℝ2 is called an entropy-entropy flux of system (13) if it satisfies
(16)∇qU=∇ηU∇mm2ρ+Pρ.Furthermore, if for any fixedm/ρ∈−∞,+∞, η vanishes on the vacuum ρ=0; then, η is called a weak entropy. For example, the mechanical energy-energy flux pair
(17)ηe=m22ρ+p0ργγ−1,qe=m32ρ2+p0ργ−1γ−1m,should be a strictly convex entropy pair. We approximate the equations in (1) by adding artificial viscosity to get the smooth approximate solutions ρε,mε, that is,
(18)ρt+mx=ερxx,mt+m2ρ+Pρx=εmxx+ρE−2Mερx+Hx,tm,Ex,t=∫L1xρ−bsds+E−,with initial-boundary value conditions
(19)ρ,mx,0=ρ0εx,m0εx=ρ0x+ε,m0x∗mε,L1<x<L2,mL1,t=mL2,t=0,ρL1,t=ρ0εL1,ρL2,t=ρ0εL2,t≥0,where M in (18) is a big enough constant to be determined later and mε in (19) is the standard mollifier with small parameter ε.We shall prove that the viscosity solutions of (18) and (19) are uniformly bounded with respect to time t.


## 3. Viscosity Solutions and A Priori Estimates

For any fixedε>0, we denote the solution of (18) and (19) by ρε,mε,Eε, since Eεx,t is uniquely determined by ρεx,t,bx, and E−; then, the system (18) may be seen as one system with the unknowns ρε and mε. Regarding the proof of local existence of approximate solution, the techniques used in this article are similar to those used in [19]. To extend the local solution to global one, the key point is to obtain the uniform upper bound of ρε,∣mε∣ and the lower bound of density ρε. The following theorem gives the uniform bound of ρε,mε.Lemma 7.
For anyT>0, let ρε,mεx,t∈C10,T,C2L1,L2 to be the smooth solution of (18) and (19). Then
(20)0≤ρεx,t≤C,∣mεx,t∣≤Cρεx,t,where C is a positive constant independent of time t.Proof.
(For simplicity of notation, the superscript ofρε and mε will be omitted as ρ,m.) By the formulas of Riemann invariants (15), we can decouple the viscous perturbation equation (18) as
(21)wt+λ2wx=εwxx+2εwx−Mρxρ−εθθ+1ρθ−2ρx2+E+Hx,tmρ,zt+λ1zx=εzxx+2εzx−Mρxρ+εθθ+1ρθ−2ρx2+E+Hx,tmρ.
We set the control functionsφ,ψ as
(22)φ=MM+x,ψ=MM−x.
A direct calculation tells us(23)φt=0,φx=M,φxx=0,ψt=0,ψx=−M,ψxx=0.
Define the modified Riemann invariantsw¯,z¯ as:
(24)w¯=w−φ,z¯=z+ψ.
Then, inserting the above formulas into (21) yields the decoupled equations for w¯ and z¯:
(25)w¯t+λ2w¯x=εw¯xx+2εw¯xρxρ−εθθ+1ρθ−2ρx2+E−λ2φx+Hx,tmρ,z¯t+λ1z¯x=εz¯xx+2εz¯xρxρ+εθθ+1ρθ−2ρx2+E+λ1ψx+Hx,tmρ.
We rewrite (25) into
(26)w¯t+λ2−2ερxρw¯x=εw¯xx+a11w¯+a12z¯+R1,z¯t+λ1−2ερxρz¯x=εz¯xx+a21w¯+a22z¯+R2,with
(27)a11=−1+θ2M+12Hx,t,a12=θ−12M+12Hx,t,a21=θ−12M+12Hx,t,a22=−1+θ2M+12Hx,t,R1=−εθθ+1ρθ−2ρx2+E−θM3−M2x+MHx,tx,R2=εθθ+1ρθ−2ρx2+E+θM3−M2x+MHx,tx.
In above calculation, we have used the relations:(28)λ1=w+z2−θw−z2,λ2=w+z2+θw−z2,mρ=w+z2.
Noting0<θ≤1, ∣Hx,t∣≤M1, and choosing M≥2/1−θM1, we have
(29)a12≤0,a21≤0.
On the other hand, (27) tells us
(30)R1≤E−θM3−M2x+MHx,tx,R2≥E+θM3−M2x+MHx,tx.
And use the same calculations in [18], we estimate the approximate electric fields and obtain
(31)∣Ex,t∣≤M2,where M2 depends only on initial data. Thus, taking M big enough, we have
(32)R1≤M2−θM3+MM1L2≤0,R2≥−M2+θM3−M2L2−MM1L1≥0,and the initial-boundary value conditions satisfy
(33)w¯x,0=wx,0−φx,0=m0ρ0+ρ0θ−M2−Mx≤0,z¯x,0=zx,0+ψx,0=m0ρ0−ρ0θ+M2−Mx≥0,w¯L1,t≤0,z¯L1,t≥0,w¯L2,t≤0,z¯L2,t≥0.
Basing on the above discussion, using Lemma7 of [18], we have
(34)w¯x,t≤0,z¯x,t≥0,∀x,t∈L1,L2×0,T.
Therefore,(35)wx,t≤φx,t≤M2+Mx≤M2+ML2,zx,t≥−ψx,t≥−M2+Mx≥−M2.
By (35), we have
(36)ρ≤w−z21/θ≤32M21/θ,and Lemma 7 is completed.From (20), the velocity u=m/ρ is uniformly bounded, i.e., ∣u∣<C. Then, following the same way of [20], we could obtain
(37)ρx,t≥δt,ε>0.Based on the local existence of smooth solution, the uniform upper estimates (Lemma7) and the lower bound estimate of density (37), we derive the following lemma.Lemma 8.
For any timeT>0, there exists a unique global classical solution ρε,mεx,t∈C10,T,C2L1,L2 to the initial-boundary value problems (18) and (19) satisfying
(38)0≤δt,ε≤ρεx,t≤C,∣mεx,t∣≤Cρεx,t,where C is independent of ε and T.Through Lemma8 and the compensated compactness framework theory established in [19, 21–23], we can prove that there has a subsequence of ρε,mε (still denoted by ρε,mε), so that
(39)ρE,mE⟶ρ,m,inLlocpL1,L2×0,TFurthermore, it is clear for us thatρ,m is an entropy solution of initial-boundary value problems (1) and (3). We complete the proof of Theorem 3.


## 4. Large Time Behavior of Weak Solutions

This section is devoted to the proof of Theorem5. Firstly, for stationary solution, from the result in [24], we have the following argument:Lemma 9.
Under the assumption (2) of bx, there exists a unique solution ρ~,E~ to problems (7) and (8) satisfying
(40)0<b∗≤ρ~x≤b∗,ρ~′x≤C,ρ~′′x≤C,x∈L1,L2,where C only depends on γ,b∗ and b∗.Now, we shall derive that the entropy solutionρ,m,E acquired in Theorem 3 converges strongly to the corresponding stationary solution ρ~,E~ in the norm of L2 with exponential decay rate. From (7) and (8), we see that
(41)∫L1L2ρ~x−bxdx=∫L1L2E~xdx=E~L2−E~L1=0.Give the definition of the new function as follows(42)yx,t=−∫L1xρs,t−ρ~sds=−E−E~,x,t∈L1,L2×0,∞.Obviously, we observe that(43)yx=−ρ−ρ~,yt=m,yL1=yL2.From (1) and (7), we have
(44)ytt+m2ρx+pρ−pρ~x−Hx,tyt=−ρ~y−E~yx+yyx.Multiplyingy with (44) and integrating from L1 to L2, we have
(45)ddt∫L1L2yyt−12y2Hdx+∫L1L212y2Htdx+∫L1L2Pρ−Pρ~ρ−ρ~+ρ~−E~x2y2dx≤∫L1L2yt2dx+∫L1L2yt2ρyxdx=∫L1L2ρ~ρyt2dx.Lemma7 of [25] tells us there exist two nonnegative constants C~1 and C~2 such that
(46)C~2ρ−ρ~2≥Pρ−Pρ~ρ−ρ~≥C~1ρ−ρ~2=C~1yx2.Putting (46) into (45), we have
(47)ddt∫L1L2yyt−12y2Hdx+∫L1L212y2Htdx+C~1∫L1L2yx2dx+∫L1L2b∗y2dx≤∫L1L2ρ~ρyt2dx.Additionally, denote the relative entropy-entropy flux by(48)η∗=ηe−p0ρ~γγ−1−p0γγ−1ρ~γ−1ρ−ρ~,q∗=qe−p0γγ−1ρ~γ−1m.From the entropy inequality (16), we have the following inequality holds in the sense of distribution:
(49)ηt∗+qx∗=ηet+qex−p0γγ−1ρ~γ−1ρ−ρ~t−p0γγ−1ρ~γ−1mx≤mE+Hm2ρ−p0γγ−1ρ~γ−1ρ−ρ~t−p0γγ−1ρ~γ−1mx=mE+Hm2ρ−p0γρ~γ−2ytρ~x.We notice that(50)mE=mE~+mE−E~=ytE~−yyt=Pρ~xρ~yt−yyt=p0γρ~γ−2ytρ~x−yyt,and use the theory of divergence-measure fields [17] to arrive at
(51)ddt∫L1L2η∗+12y2dx−∫L1L2Hyt2ρdx≤0.LetΛ sufficiently big so that Λ>b∗/δ0+∥ρ∥L∞+1. Multiply (51) by Λ and add the result to (47), we have
(52)ddt∫L1L2Λη∗+Λy22+yyt−y22Hdx+∫L1L212Ht+b∗y2+C~1yx2+−ΛH−ρ~ρyt2dx≤0.Since(53)−ΛH−ρ~>b∗δ0+∥ρ∥L∞δ0−b∗>∥ρ∥L∞δ0,then there exists C~3>0 such that
(54)ddt∫L1L2Λη∗+Λy22+yyt−y22Hdx+C~3∫L1L2y2+yx2+yt2ρdx≤0.Since∥ρx,t∥L∞≤C and
(55)η∗~yx2+yt2ρ,we can directly conclude that
(56)Λη∗+Λy22+yyt−y22H~y2+yx2+yt2ρ.Now from (54), the Gronwall inequality implies Theorem 5.



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