QUARTERLY OF APPLIED MATHEMATICS
VOLUME LXII, NUMBER 3
SEPTEMBER 2004, PAGES 569-600
QUANTUM EULER-POISSON SYSTEMS: GLOBAL EXISTENCE AND
EXPONENTIAL DECAY
By
ANSGAR JUNGEL (Fachbereich Mathematik und Statistik, Universitat Konstanz, Fach D193, 78^57
Konstanz, Germany)
HAILIANG LI (Institut fur Mathematik, Universitat Wien, 1090 Vienna, Austria and Department of
Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka
University, Toyonaka, Osaka 560-0043, Japan)
Abstract. A one-dimensional transient quantum Euler-Poisson system for the elec-
tron density, the current density, and the electrostatic potential in bounded intervals is
considered. The equations include the Bohm potential accounting for quantum mechan-
ical effects and are of dispersive type. They are used, for instance, for the modelling of
quantum semiconductor devices.
The existence of local-in-time solutions with small initial velocity is proven for general
pressure-density functions. If a stability condition related to the subsonic condition for
the classical Euler equations is imposed, the local solutions are proven to exist globally in
time and tend to the corresponding steady-state solution exponentially fast as the time
tends to infinity.
1. Introduction.
1.1. The Model Equations. In 1927, Madelung gave a fluid-dynamical description of
quantum systems governed by the linear Schrodinger equation for the wave function tjj-.
£2
iedtip = ——Aip — Vip in x (0, oo),
^(•,0) = tpo in Kd,
Received November 6, 2003.
2000 Mathematics Subject Classification. Primary 35A07, 35B40.
Key words and phrases. Quantum Euler-Poisson system, existence of global-in-time classical solutions,
nonlinear fourth-order wave equation, exponential decay rate, long-time behavior of the solutions.
E-mail address: juengel@mathematik.uni-mainz.de
Current address: Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universitat, 55099
Mainz, Germany.
E-mail address: lihlSmath.sci.osaka-u.ac.jp
Current address: Department of Mathematics, Capital Normal University, Beijing 100037, P. R. China.
©2004 Brown University
569
570 A. JUNGEL and H.-L. LI
where d > 1 is the space dimension, e > 0 denotes the scaled Planck constant, and
V = V(x, t) is some (given) potential. Separating the amplitude and phase of ip =
l^l exp(iS/e), the particle density p = \il>\2, and the particle current density j = pVS for
irrotational flow satisfy the so-called Madelung equations [21]
dtp + divj = 0, (1)
dtj + div - PV4> - jpV = 0 inKdx(0,oo), (2)
where the i-t.li component of the convective term div(j ® j/p) equals
y- 9 f jijk
F)ti. I n
k=1 dxk V p
Equations (l)-(2) can be interpreted as the pressureless Euler equations including the
quantum Bohm potential
<3)
2 \fp
They have been used for the modeling of superfluids like Helium II [16, 20].
Recently, Madelung-type equations have been derived to model quantum phenom-
ena in semiconductor devices, like resonant tunneling diodes, starting from the Wigner-
Boltzmann equation [6] or from a mixed-state Schrodinger-Poisson system [8, 9]. There
are several advantages of the fluid-dynamical description of quantum semiconductors.
First, kinetic equations, like the Wigner equation or Schrodinger systems, are computa-
tionally very expensive, whereas for Euler-type equations, efficient numerical algorithms
are available [5, 25]. Second, the macroscopic description allows for a coupling of classi-
cal and quantum models. Indeed, setting the Planck constant e in (2) equal to zero, we
obtain the classical pressureless equations, so in both pictures, the same (macroscopic)
variables can be used. Finally, as semiconductor devices are modeled in bounded do-
mains, it is easier to find physically relevant boundary conditions for the macroscopic
variables than for the Wigner function or for the wave function.
The Madelung-type equations derived by Gardner [6] and Gasser et al. [8] also include
a pressure term and a momentum relaxation term taking into account interactions of the
electrons with the semiconductor crystal, and are self-consistently coupled to the Poisson
equation for the electrostatic potential <fi:
dtp + divj = 0, (4)
dtj + div + vp(p) -Pv<t>- = -3~, (5)
A2 A4> — p — C(x) in x (0, oo), (6)
where Q C is a bounded domain, r > 0 is the (scaled) momentum relaxation time
constant, A > 0 is the (scaled) Debyc length, and C(x) is the doping concentration
modelling the semiconductor device under consideration [12, 24], The pressure is assumed
QUANTUM EULER-POISSON SYSTEMS 571
to depend only on the particle density and, like in classical fluid dynamics, often the
expression
P(p) = —p1, p> 0, 7 > 1, (7)
7
with the temperature constant To > 0, is employed [6, 11]. Isothermal fluids correspond to
7=1, isentropic fluids to 7 > 1. Notice that the particle temperature is T(p) = Top7-1.
In this paper we consider general (smooth) pressure functions. Equations (4)-(6) are
referred to as the quantum Euler-Poisson system or as the quantum hydrodynamic model.
In this paper, we investigate the (local and global) existence and long-time behavior
of solutions of the following one-dimensional quantum Euler-Poisson system:
Pt + ix = 0, (8)
3, + (Jj tfw)_ =?*. + -/*> <9>
<t*xx = P - C(x), (10)
with the following initial and boundary conditions:
P(x,0) = gi(x) > 0, j(x,0) = ji(x) =: ei(x)vi(x), (11)
p{0,t)=p1, p(l,t)=p2, px(0,t) = px(l,t) = 0, (12)
cj)(0,t) = 0, 0(l,f) = $o, (13)
for (x,t) G (0,1) x (0,oo), where pi, p2, $0 > 0, and vi is the initial velocity.
The existence and uniqueness of steady-state (classical) solutions to the quantum
Euler-Poisson system for current density jo = 0 (thermal equilibrium) has been stud-
ied in [1, 7]. The stationary equations for jo > 0 have been considered in [4, 11, 27] for
general monotone pressure functions, however, with different boundary conditions, as-
suming Dirichlet data for the velocity potential S [11] or employing nonlinear boundary
conditions [4, 27], Existence of steady-state solutions to (8)-(10) subject to the boundary
conditions (12)-(13) is proven in [10] for the linear pressure function P{p) = p and in
[14] for general pressure functions P(p) also allowing for non-convex or non-monotone
pressure-density relations. So far, to our knowledge, the only known results 011 the
existence of the time-dependent system (4)-(6) have been obtained in [13] for smooth
local-in-time solutions on bounded domains and in [17] for "small" irrotational global-
in-time solutions in the whole space assuming strictly convex pressure functions and a
constant doping profile.
In the present paper, we consider the initial-boundary-value problem (IBVP) 8-13
for general pressure and non-constant doping profile, and we focus on the local and
global existence of classical solutions {p,j, <fi) of the IBVP 8-13 and their time-asymptotic
convergence to the stationary solutions (poijo^o) obtained in [14]. First, we show that
there exists a classical local-in-time solution for regular initial data. Second, we prove
that if a certain "subsonic" condition (see 25) holds and if the initial data is a perturbation
of a stationary solution (po,jo, <fio), a classical solution (p, j, (f>) exists globally in time and
tends to {po,jo,<Po) exponentially fast as time tends to infinity.
572 A. JUNGEL AND H.-L. LI
In dealing with the IBVP 8-13 we have to overcome the following difficulties. First,
since the general pressure P(p) can be non-convex (even zero or "negative"; see Re-
mark 1.6), the left part of Eqs. 8-10 may be not hyperbolic any more. Unlike [17],
we cannot apply the local existence theory of quasilinear symmetric hyperbolic sys-
tems [3, 15, 22, 23]. We have to establish a new local existence theory. Second, the
appearance of the nonlinear quantum Bohm potential in 9 requires that the density is
strictly positive for regular solutions. This together with the structure of the quantum
term causes problems in the local and global existence proofs.
1.2. Main results. Before stating our main results we introduce some notation. We
denote by L2 = L2(0,1) and Hk = Hk(0,1) the Lebesgue space of square integrable
functions and the Sobolev space of functions with square integrable weak derivatives of
order /c, respectively. The norm of L2 is denoted by ||-||o = ||-||, and the norm of Hk is ||-||fc.
The space Hq = Hq(0, 1) is the closure of Co°(0,1) in the norm of Hk. Let T > 0 and
let B be a Banach space. Then Ck(0, T; B) (Cfc([0, T];B), respectively) denotes the space
of ^-valued fc-times continuously differentiable functions on (0, T) ([0, T], respectively),
L2{0, T; B) is the space of B-valued L2-functions on (0, T), and Wfc'p(0, T; B) the space of
B-valued PF^-functions on (0, T). Finally, C always denotes a generic positive constant.
It is convenient to make use of the variable transformation p = ui2 in 8-10, which
yields the following IBVP for (u>,j,4>):
2u)u}t + jx = 0, (14)
jt+(% + P^2)) =uj24>x+1-sW(^) -i (15)
\(jJz J x 2 V UJ J x T
<\>xx = W2 -C(x), (16)
with the initial and boundary conditions
(w,j)(x, 0) = (wi,ji)(x) = (V?T>eiVi)(a:), (17)
u>(0, t) = VpT, w(l, t) = V/P2, wx(0, t) = wx{l, t) = 0, (18)
<f>(0,t) = 0, <j>(l,t) = $0, (19)
for x € (0,1), t > 0. This problem is equivalent to 8-13 for classical solutions with
positive particle density.
We will assume throughout this paper compatibility conditions for the IBVP 14-19
in the sense that the time derivatives of the boundary values and the spatial derivatives
of the initial data are compatible at (x,t) = (0,0) and {x,t) = (1,0) in 14-16. We will
prove the following local existence result for the IBVP 14 19:
Theorem 1.1. Assume that
PeC\ 0,+oo), CeH2, (20)
(wiiii) £ H6 x H5 such that uji(x) > 0 for x £ [0,1], and for some a G [(1 + 2\Z2e)~1,1)
(1 - ct)w» , .
ll»il|CM[o,u» < <21>
QUANTUM EULER-POISSON SYSTEMS 573
where
= min loAx) > 0.
*e[o,i]
Then, there is a number (determined by 128), such that there exists a unique classical
solution (u>, j,(f>) of the IB VP (14)-(19) in the time interval [0, T], with 0 < T < T**,
satisfying u> > (1 — > 0 in [0,1] x [0, T\ and
Mt)\\i+ymi+\m)\\i<™ for t <t.
Remark 1.2. (1) It is well known that for classical hydrodynamic equations, monotone
pressure-density relations are required to guarantee short-time existence of classical so-
lutions [2, 18]. The condition 20 means that this condition is not necessary (to a certain
extent) when the quantum effects are taken into account.
(2) Condition 21 is needed to prove the positivity of the particle density. A similar
condition has been used to prove the existence of stationary solutions [11]. This condition
allows for arbitrarily large current densities j\ = wfvi, for instance, if Wi is a sufficiently
large constant.
(3) We are able to show the statements of Theorem 1.1 under the slightly more general
condition
Ihllc'ao.i]) < min jae, ' "€(0,1). (22)
Then 21 is a special case for a > (1 + 2\[2e)~l which is equivalent to as > (1 — a)/2\/2.
(4) The local existence of the Cauchy problem in or Td can be shown in the same
framework; see [19].
Theorem 1.1 is proven by an iteration method and compactness arguments. More
precisely, we construct a sequence of approximate solutions that is uniformly bounded in
a certain Sobolev space in a fixed (maybe small) time interval. Compactness arguments
then imply that there is a limiting solution which proves to be a local-in-time solution
of 14-19. Unlike [17], we cannot apply the theory of quasilinear symmetric hyperbolic
systems [3, 15, 22, 23] to construct (local) approximate solutions and obtain uniform
bounds in Sobolev spaces because the pressure can be non-convex, causing the loss of
entropy and hyperbolicity of 14-15.
The idea of the local existence result is first to linearize the system 14-16 around its
initial state {wi,ji,4>i), where <j>\ solves the Dirichlet problem 16 and 19 with o> replaced
by u)\, and to consider the equations for the perturbation (ip, 77, e) =
The main idea is to write the evolution equation for the perturbed particle density
as a semilinear fourth-order wave equation. Then, we construct approximate solutions
(■ipi,r)i,ei) (i > 1) from a fixed-point procedure, which are expected to converge to a
solution (1/), 77, e) of the perturbed problem as i —> 00. For this, we derive uniform
bounds in Sobolev spaces on a uniform time interval and apply standard compactness
arguments,(see Sec. 3). A further analysis shows that (u>,j,(j)) = (a>i +ip,ji +f),<j> 1 + e)
with ui > 0 is the expected local (in time) solution of the original problem 14-19.
To extend the local classical solution globally in time, we need to establish uniform
estimates. We consider the situation when the initial data is close to the stationary