数理解析研究所講究録
1495 巻 2006 年 71-77

71

A stationary solution for a fluid dynamical
model of semiconductor
東京工業大学 西畑伸也 (Shinya Nishibata)
東京工業大学 鈴木政尋 (Masahiro Suzuki)

Department of Mathematical and Computing Sciences
Tokyo Institute of Technology, Tokyo 152-8552, Japan

Dedicated to Professor $Tai$ -Ping $Liu$ on his 60th birthday

1

Introduction

We study the existence and the asymptotic stability of a stationary solution to
the initial boundary value problem for a one-dimensional hydrodynamic model ofsemiconductors. In the previous researches [1] and [2], this problem is considered
under the assumption that a doping profile is flat. However, this assumption is too
narrow, from the physical point of view. In the present paper we briefly discuss
the asymptotic stability of the stationary solution without this assumption on thedoping profile. For the detailed discussion on this research, please see the paper [9].
The motion of electrons in semiconductors is governed by the system of equations$\rho_{l}+(\rho u)_{x}=0$$(\rho u)_{t}+(\rho u^{2}+p(\rho))_{x}=\rho\phi_{x}-pu$$\phi_{xx}=\rho-D$(l.la)(l.lb)(l.lc),,.

We study this system over the bounded domain $\Omega:=(0,1)$ for the spatial variable .
Here, the unknown functions
and stand for the electron density, the electron
velocity and the electrostatic potential, respectively. Thus, a product $j:=\rho u$ means
the current density. The pressure is assumed to be a function of the electron density
given by$p=p(\rho)=K\rho^{\gamma}$ ,(1.2). In thewhere the constants $K$ and are supposed to satisfy $K>0$ andphysical point of view, the case $\gamma=1$ is important. The doping profile
is a function of the spatial variable and satisfy
(1.3)
.
$x$

$\rho,$$u$$\phi$$p$$\rho$$\gamma\geq 1$$\gamma$$D\in B^{0}(\overline{\Omega})$$x$$\inf_{x\in\Omega}D(x)>0$

We prescribe the initial and the boundary data as$(\rho, u)(0, x)=(\rho_{0}, u_{0})(x)$$\rho(t,\mathrm{O})=\rho_{l}>0$$\phi(t, 0)=0$

,

,

,

$\rho(t, 1)=\rho_{r}>0$$\phi(t, 1)=\phi_{r}>0$

,

,

(1.4)
(1.5)
(1.6)

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and
where
conditions on
Namely,$\rho_{1},$are constants. In addition, we assume that the compatibility$\phi_{r}$$\rho_{r}$$\rho(t, x)$with orders$\rho(0,0)=p_{l}$,$0$and 1 hold at $(t,x)=(0,0)$ and $(t,x)=(0,1)$ .

$\rho(0,1)=\rho_{\mathrm{r}}$,$(\rho u)_{x}(0,0)=0$,$(\rho u)_{x}(0,1)=0$

.

(1.7)

This initial boundary value problem is studied in the region where the subsoniccondition (1.8a) and positivity of the density (1.8b) hold$\inf_{x\in\Omega}(p’(\rho)-u^{2})>0$$\inf_{x\in\Omega}\rho>0$,(1.8a).

$(1.8\mathrm{b})$Thus, the initial data is supposed to satisfy these conditions:$\inf_{x\in\Omega}(p’(\rho_{0}(x))-u_{0}^{2}(x))>0$,$\inf_{x\in\Omega}\rho_{0}(x)>0$

.

(1.9)

Here, note that the subsonic condition is equivalent to that one characteristic speedof the hyperbolic equations (l.lb), (l.lc) is negative and another characteristic
is positive, that is,$\lambda_{1}$$\lambda_{2}$$\lambda_{1}:=u-\sqrt{p’(\rho)}<0$,$\lambda_{2}:=u+\sqrt{p’(\rho)}>0$.

(1.10)The subsonic condition means that two boundary conditions (1.5), (1.6) are sufficientis negative
for the well-posedness of this initial boundary value problem, since
is positive.
andThe stationary solution is a solution to (1.1) independent of time variable ,satisfying the same boundary conditions (1.5) and (1.6). Hence, the stationary
satisfies the system of equations
solution$\lambda_{1}$$\lambda_{2}$$t$$(\tilde{\rho},\tilde{u},\tilde{\phi})$$(\tilde{p}\tilde{u})_{x}=0$(l.lla),$(\tilde{p}\tilde{u}^{2}+p(\tilde{\rho}))_{x}=\tilde{\rho}\tilde{\phi}_{x}-\tilde{\rho}\tilde{u}$$\tilde{\phi}_{xx}=\tilde{p}-D$,(l.llb),(l.llc)and boundary conditions$\tilde{\rho}(0)=\rho\iota>0$$\tilde{\phi}(0)=0$,,$\tilde{\rho}(1)=\rho_{\mathrm{r}}>0$$\tilde{\phi}(1)=\phi_{r}>0$.

,(1.12)(1.13)The strength of the boundary data$\delta_{b}:=|\rho_{t}-p_{\iota}|+|\phi,|$(1.14)

plays a crucial role in showimg the unique existence and the asymptotic stability ofthe stationary solution. The unique existence of the stationary solution $(p, u, \phi)$ ,satisfying (1.11), (1.12) and (1.13), is summarized in the next lemma.

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Lemma 1.1. Suppose that the doping profile $D(x)$ and the boundary data satisfy(1.3), (1.5) and (1.6). Then, for an arbitrary , there exists a positive constant, the stationary problem (1.11), (1.12) and (1.13) has a uniquesuch that if
satisfying the condition (1.8).
solution$\delta_{1}$$\rho\iota$$\delta_{b}\leq\delta_{1}$$(\tilde{p},\tilde{u},\tilde{\phi})(x)\in B^{2}(\overline{\Omega})$

The proof of the existence of the stationary solution is given by the Schauderfixed-point theorem. On the other hand, the uniqueness is shown by the maximum
principle.
In order to discuss the asymptotic stability of the stationary solution, we introduce function spaces$X_{i}^{j}([0, T]):= \bigcap_{k=0}^{i}C^{k}([0, T];H^{j+i-k}(\Omega))$$X_{i}([0, T]):=X_{i}^{0}([0, T])$

for

$i,$

$j=0,1,2$ ,.

The stability of the stationary solution is given by the following theorem.
Theorem 1.2. Let
be the stationary solution of (1.11), (1.12) and (1.13).
and theSuppose that the initial data
belongs to the function spacesatisfy conditions (1.5), (1.6), (1.7) and (1.9). Thenboundary data
and
, the
that ifthere exists a positive constantuniquesolution(1.6)
aproblem(1.4),
(1.5)
and
has
boundary(1.1),
initialvalue$(\rho, u, \phi)(t, x)$ in the. Moreover, the solution $(\rho, u, \phi)(t, x)$function space
and a decay estimate
verifies the additional regularity$(\tilde{\rho},\tilde{u},\tilde{\phi})$$H^{2}(\Omega)$$(\rho_{0}, u_{0})$$\rho\iota,$$\phi_{f}$$\rho_{\Gamma}$$\delta_{2}su\mathrm{c}h$$\delta_{b}+||(\rho_{0}-\tilde{p}, u_{0}-\tilde{u})||_{2}\leq\delta_{2}$$X_{2}([0, \infty))$$\phi-\tilde{\emptyset}\in \mathfrak{X}_{2}^{2}([0, \infty))$$||(\rho-\tilde{\rho}, u-\tilde{u})(t)||_{2}+||(\phi-\tilde{\phi})(t)||_{4}\leq C||(\rho_{0}-\tilde{\rho}, u_{0}-\tilde{u})||_{2}e^{-\alpha t}$

where

$C$

and

$a$

,

(1.15)are certain positive constants, independent of time .
$t$Related results. The hydrodynamic model of semiconductors was introduced by. Recently, many engineers and mathematicians study this model. For
the derivation of this model, the book [6] is a good reference.
From physical and engineering point of view, it is very interesting to consider
the hydrodynamic model of semiconductors with the Dirichlet boundary condition.
Degond and Markowich [1] investigated the stationary solution to the hydrodynamic
model with the Dirichlet boundary condition. They proved the existence of thestationary solution satisfying the subsonic condition. Li, Markowich and Mei [2]also considered the existence and the asymptotic stability of the stationary solution.
However, they assumed that the doping profile is flat, that is, $|D-p_{l}|\ll 1$ . This
assumption is too narrow to cover actual semiconductor-devices. For instance thetypical example of the doping profile, drawn in [7], does not satisfies this assumption.
Matsumura and Murakami removed this assumption in the paper [3]. Precisely, they
proved the asymptotic stability of the stationary solution without the assumption$\mathrm{B}\iota_{\emptyset}\mathrm{t}\mathrm{e}\mathrm{k}\mathrm{j}\mathrm{a}\mathrm{e}\mathrm{r}[8]$74that the doping profile is flat. However, their boundary condition is periodic in spacevariable. Therefore, the asymptotic stability of the stationary solution under the
Dirichlet boundary condition for the non-flat doping profile had not been proved.
Recently, the problem is solved by the authors in [9]. This short note is devoted to
discussion on the outline of this paper.
denotes the l-th order Sobolev
Notation. For a nonnegative integer $l\geq 0,$. We note $H^{0}=L^{2}$ andsense,
equipped
with
the
norm
space in the$||\cdot||:=||\cdot||_{0}$ .
denotes the space of the -times continuously
differentiable functions on the interval $[0, T]$ with values in (St). For a nonnegative
denotes the space of the functions whose derivatives up to k-th
integer $k\geq 0,$.
order are continuous and bounded over , equipped with the norm
$H^{l}(\Omega)$$L^{2}$$||\cdot||\iota$$k$$C^{k}([0, T];H^{\mathrm{I}}(\Omega))$$H^{\iota}$$B^{k}(\overline{\Omega})$$\overline{\Omega}$$|\cdot|_{i}$Acknowledgments. The authors would like to express their sincere gratitude toProfessor Akitaka Matsumura and Professor Shinji Odanaka for stimulus discussions
and helpful comments.

2

The outline of the proof of Theorem 1.2

This section is devoted to the brief discussion on the proof of Theorem 1.2. The
existence of the time local solution is proved by the similar method as in [4] and[5].
Lemma 2.1. Suppose the initial data $(p_{0}, u_{0})\in H^{2}(\Omega)$ and the boundary datasatisfy (1.9), (1.5),(1.6) and (1.7). Then there exists a constantand$T_{1}(||p_{0}||_{2}, ||u_{0}||_{2})>0$ , such that the initial boundary value problem (1.1),(1.4),. Moreover, the solution(1.5) and (1.6) has a solution$t\in[0,T_{1}]$ .
$(\rho, u)(t, x)$
arbitrary
(1.8)
an
the
conditionforsatisfies$\rho_{l},$$\phi_{r}$$\rho_{\mathrm{r}}$$(\rho, u, \phi)(t, x)\in \mathfrak{X}_{2}([0, T_{1}])$

In order to complete the proof of Theorem 1.2, we have to derive an a-priori
uniform estimate with respect to the time variable . For this purpose, it is conveas a perturbation from the stationary solution
nient to regard the solution. Hence, we introduce new unknown functions as$t$$(\rho, u, \phi)$$(\tilde{p},\tilde{u},\tilde{\phi})$$\psi(t, x):=\rho(t,x)-\tilde{\rho}(x)$,$\eta(t,x):=u(t,x)-\tilde{u}(x)$,$\omega(t, x):=\phi(t,x)-\tilde{\phi}(x)$. $(2.1)$

Multiplying $1/p$ by (l.lb), we have$u_{t}+uu_{x}+(h(\rho))_{x}=\phi_{x}-u$,$h(\xi)$$:= \int_{1}^{\xi}\frac{p’(\zeta)}{\zeta}d\zeta$

.

(2.2)

Similarly, we have from (2.2) that$\tilde{u}\tilde{u}_{x}+(h(\tilde{\rho}))_{x}=\tilde{\phi}_{x}-\tilde{u}$

.

(2.3)

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Subtracting (l.lla) from (l.la), (2.3) from (2.2) and (l.llc) from (l.lc), respectively, we obtain the equations for the perturbation$(\psi, \eta, \omega)$$\psi_{t}+((\tilde{\rho}+\psi)(\tilde{u}+\eta)-\tilde{\rho}\tilde{u})_{x}=0$,(2.4a),(2.4b)$\eta_{t}+\frac{1}{2}((\tilde{u}+\eta)^{2}-\tilde{u}^{2})_{x}+(h(\tilde{\rho}+\psi)-h(\tilde{\rho}))_{x}-\omega_{x}+\eta=0$$\omega_{xx}=\psi$.

(2.4c)The initial and the boundary data to the system (2.4) are derived from (1.4), (1.5),(1.6), (1.12) and (1.13) as$\psi(x, \mathrm{O})=\psi_{0}(x):=\rho_{0}(x)-\tilde{\rho}(x)$,$\eta(x, \mathrm{O})=\eta_{0}(x):=u_{0}(x)-\tilde{u}(x)$$\psi(t, \mathrm{O})=\psi(t, 1)=0$$\omega(t, \mathrm{O})=\omega(t, 1)=0$

Since

,
,

$t\geq 0$,$t\geq 0$

.

,

(2.5)
(2.6)
(2.7)

and cv satisfies (2.4c), the local existence of the solution
to the initial boundary value problem (2.4), (2.5), (2.6) and (2.7) follows
from Lemmas 1.1 and 2.1.
$(\tilde{\rho},\tilde{u},\tilde{\phi})\in X_{2}([0, T])$$(\psi, \eta,\omega)$

Corollary 2.2. Suppose that the initial data
belongs toandsatisfy (1.9). Then there exists a constantsuch thatthe initial boundary value problem (2.4), (2.5), (2.6) and (2.7) has a unique solution(th,).
$(\psi_{0}, \eta_{0})$$\eta,\omega$$H^{2}(\Omega)$$(\tilde{\rho}+$$T_{2}(||\psi_{0}||_{2}, ||\eta_{0}||_{2})>0$$\psi_{0},\tilde{u}+\eta 0)$$\in \mathfrak{X}_{2}([0, T_{2}])\mathrm{x}\mathfrak{X}_{2}([0, T_{2}])\mathrm{x}\mathfrak{X}_{2}^{2}([0, T_{2}])$

Owing to the above Corollary 2.2, it suffices to derive the a-priori uniform estimate (2.8) in order to complete the proof of the existence of the solution globally intime.
Proposition 2.3. Let $(\psi, \eta, \omega)(t, x)\in X_{2}([0, T])\cross X_{2}([0, T])\cross X_{2}^{2}([0, T])$ be a solution to (2.4), (2.5), (2.6) and (2.7). Then there exists a positive constant suchthat if, the following estimate holds for an arbitrary $t\in[0, T]$ .
$\epsilon_{0}$$N(T)+\delta_{b}\leq\epsilon_{0}$$||$$( \psi, \eta)(t)||_{2}^{2}+||\omega(t)||_{4}^{2}+\int_{0}^{t}||(\psi, \eta)(\tau)||_{2}^{2}+||\omega(\tau)||_{4}^{2}d\tau\leq C||(\psi, \eta)(0)||_{2}^{2}$

where

$C$

,

(2.8)

is a positive constant independent of .
$T$

To show the estimate (2.8), it is convenient to use notations$N(t):= \sup_{0\leq\tau\leq t}||(\psi, \eta)(\tau)||_{2}$,.

$M^{2}(t):= \int_{0}^{t}||\psi_{x}(\tau)||^{2}+||\eta_{x}(\tau)||^{2}d\tau$First, we derive the basic estimate to obtain the a-priori estimate (2.8). To thisend, we define the energy form as$\mathcal{E}$,$\mathcal{E}:=\frac{1}{2}p(u-\tilde{u})^{2}+\Psi(\rho,\tilde{\rho})+\frac{1}{2}(\phi-\tilde{\phi})_{x}^{2}$$\Psi(\rho,\tilde{\rho}):=\int_{\rho}^{\rho}h(\xi)-h(\tilde{p})d\xi$

.

(2.9)

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Notice that 8 is equivalent to. Namely, there exist positive constantsto, since$|(\psi, \eta, \omega_{x})|^{2}\mathrm{i}\mathrm{f}|(\psi, \eta, \omega_{x})|<c$$|\psi|^{2}$and$c_{1}$$C_{1}$$G(\rho,\tilde{p})$is equivalentsuch that(2.10)$c_{1}|(\psi, \eta, \omega)|^{2}\leq \mathcal{E}\leq C_{1}|(\psi, \eta,\omega)|^{2}$and then apply the. Multiply the equation (2.4b) byforequation
following
satisfies
theenergyform
integration by parts, to see the(2.11a),, (2.11b)$(pu-\tilde{\rho}\tilde{u})$$|(\psi, \eta, \omega)|\leq c$$\mathcal{E}$$\mathcal{E}_{t}+\tilde{\rho}\eta^{2}=R_{1x}+R_{2}$$R_{1}:=\omega\omega_{xt}+\omega(p\phi-\tilde{p}\tilde{\phi})-(h(p)-h(\tilde{\rho}))(pu-\tilde{p}\tilde{u})+(h(\rho)-h(\tilde{\rho}))(\psi\tilde{u})$$R_{2}:=- \{\frac{1}{2}(u^{2}-\tilde{u}^{2})(\rho u-\tilde{\rho}\tilde{u})\}_{x}-u\psi\eta+(pu-\tilde{p}\tilde{u})_{x}\eta\tilde{u}$$+ \{\frac{1}{2}(u^{2}-\tilde{u}^{2})_{x}-\omega_{x}+\eta\}\psi\tilde{u}-(h(\rho)-h(\tilde{\rho}))(\psi\tilde{u})_{x}$.

(2.11c)Applying the Sobolev inequality and using the equation (2.4c), we have the following
estimate(2.12)$|R_{2}|\leq C(N(T)+\delta_{b})|(\psi, \psi_{x}, \eta, \eta_{x})|^{2}$

On the other hand, we have$\int_{0}^{1}R_{x}dx=0$

thanks to the boundary conditions (2.6) and (2.7). Therefore, the integration of theshows the next lemma.
equation (2.9) over the domain
Lemma 2.4. Assume the same conditions as in Proposition 2.3 hold. Then there, the following estimate holds
that ifexists a positive constant$\Omega \mathrm{x}[0, t]$$\epsilon_{0}su\mathrm{c}h$$N(T)+\delta_{b}\leq\epsilon_{0}$for an arbitrary $t\in[0, T]$$||( \psi, \eta,\omega_{x})(t)||^{2}+\int_{0}^{t}||(\psi, \eta, \omega_{x})(\tau)||^{2}d\tau\leq C||(\psi, \eta,\omega_{x})(0)||^{2}+C(N(T)+\delta_{b})M^{2}(t)$,(2.13)

where is a positive constant independent of .
, stated in the next
The estimates of the higher order derivatives oflemma, are obtained by an energy method. For details, see [9].
Lemma 2.5. Assume the same conditions as in Proposition 2.3 hold. Then there, the following estimate holds
exists a positive constant such that if$t\in[0,T]$$i=1,2$and an integer
for an arbitrary
$T$

$C$

$(\psi, \eta,\omega)$$\epsilon_{1}$$N(T)+\delta_{b}\leq\epsilon_{1}$$|1( \partial_{t}^{i}\psi, \partial_{t}^{i}\eta_{t}, \partial_{t}^{i}\omega_{x})(t)||^{2}+\int_{0}^{t}||(\partial_{t}^{i}\psi, \partial_{t}^{i}\eta, \partial_{t}^{i}\omega_{x})(\tau)||^{2}d\tau$$\leq C(A_{i}^{2}(0)+A_{i-1}^{2}(t)+\int_{0}^{t}A_{i-1}^{2}(\tau)d\tau)$$A_{i}^{2}(t):= \sum_{j=0}^{:}||(\mathrm{f}\dot{\mathrm{f}}_{t}\psi,\dot{\theta}_{t}\eta)(t)||^{2}$,for$i=1,2$ ,,(2.14)(2.15)

77

where

$C$

is a positive constant independent of .
$T$

Notice that $c||(\psi, \eta)(t)||_{i}\leq A_{i}^{2}(t)\leq C||(\psi, \eta)(t)||_{i}(i=1,2)$ , which holds fromthe equations (2.4a) and (2.4b) under the condition that $N(T)+\delta_{b}$ is sufficiently
small. In addition, the estimate
follows from (2.4c). Hence we obtain
the a-priori estimate (2.8) by using Lemma 2.4 and Lemma 2.5 for the small quantity$N(T)+\delta_{b}$ . Consequently, we complete
the proof of Proposition 2.3 and thus we seethat Theorem 1.2 holds.
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fluid dynamical model of semiconductor equation, to appear.
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[6] P. A. MARKOWICH, C. A. RINGHOFER AND C. SCHMEISER, Semiconductor equations,
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[7] C. L. GARDNER, The quantum hydrodynamic model for semiconductor devices, SIAM J.
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[8] K.
, Transport equations for electrons in two-valley semiconductors, IEEE
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$\mathrm{B}\iota\emptyset \mathrm{T}\mathrm{E}\mathrm{K}\mathrm{J}5\mathrm{E}\mathrm{R}$$I\mathrm{b}n\epsilon$.

[9] S. NISHIBATA AND M. SUZUKI, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, to appear.