ARCHIVUM MATHEMATICUM (BRNO)
Tomus 40 (2004), 435 – 456
QUANTUM EULER-POISSON SYSTEMS:
EXISTENCE OF STATIONARY STATES
ANSGAR JÜNGEL† , HAILIANG LI∗
Abstract. A one-dimensional quantum Euler-Poisson system for semiconductors for the electron density and the electrostatic potential in bounded
intervals is considered. The existence and uniqueness of strong solutions with
positive electron density is shown for quite general (possibly non-convex or
non-monotone) pressure-density functions under a “subsonic” condition, i.e.
assuming sufficiently small current densities. The proof is based on a reformulation of the dispersive third-order equation for the electron density as a
nonlinear elliptic fourth-order equation using an exponential transformation
of variables.
1. Introduction
In 1927, Madelung gave a fluiddynamical description of quantum systems governed by the Schrödinger equation for the wave function ψ :
ε2
∆ψ − φψ
2
ψ(·, 0) = ψ0 in Rd ,
iε∂t ψ
= −
in Rd × (0, T ),
where T > 0, d ≥ 1, ε > 0 is the scaled Planck constant, and φ = φ(x, t) is some
(given) potential. Separating the amplitude and phase of ψ = |ψ| exp(iS/ε), the
particle density ρ = |ψ|2 and the particle current density j = ρ∇S for irrotational
flows satisfy the so-called Madelung equations [26]
(1.1)
(1.2)
∂t ρ + div j = 0 ,
∂t j + div
j⊗j
ρ
− ρ∇φ −
ε2
ρ∇
2
√ ∆ ρ
= 0 in
√
ρ
Rd × (0, T ) ,
2000 Mathematics Subject Classification: 35J40, 35J60, 76Y05.
Key words and phrases: quantum hydrodynamics, existence and uniqueness of solutions, non-monotone pressure, semiconductors.
Received December 10, 2002.
436
A. JÜNGEL AND H. LI
where the i-th component of the convective term div (j ⊗ j/ρ) equals
d
X
∂
ji jk
.
∂xk
ρ
k=1
The equations (1.1)–(1.2) can be interpreted as the pressureless Euler equations
including the quantum Bohm potential
√
ε2 ∆ ρ
√ .
2
ρ
They have been used for the modeling of superfluids like Helium II [23, 25].
Recently, Madelung-type equations have been derived for the modeling of quantum semiconductor devices, like resonant tunneling diodes, starting from the
Wigner-Boltzmann equation [11] or from a mixed-state Schrödinger-Poisson system [14, 15]. There are several advantages of the fluiddynamical description of
quantum semiconductors. First, kinetic equations, like the Wigner equation, or
Schrödinger systems are computationally very expensive, whereas for Euler-type
equations efficient numerical algorithms are available [10, 30]. Second, the macroscopic description allows for a coupling of classical and quantum models. Indeed,
setting the Planck constant ε in (1.2) 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 domains, 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 [11] and Gasser et al. [14]
also include a pressure term and a momentum relaxation term taking into account
interactions of the electrons in the semiconductor crystal, and are self-consistently
coupled to the Poisson equation for the electrostatic potential φ:
(1.3)
(1.4)
(1.5)
∂t ρ + div j = 0 ,
∂t j + div
j ⊗j
ρ
+ ∇p(ρ) − ρ∇φ −
λ2 ∆φ = ρ − C(x)
ε2
ρ∇
2
√ ∆ ρ
j
=− ,
√
ρ
τ
in Ω × (0, T ) ,
where Ω ⊂ Rd is a bounded domain, τ > 0 is the (scaled) momentum relaxation
time constant, λ > 0 the (scaled) Debye length, and C(x) is the doping concentration modeling the semiconductor device under consideration [18, 20, 28]. The
pressure is assumed to depend only on the particle density and, like in classical
fluid dynamics, often the expression
(1.6)
p(ρ) =
T γ
ρ ,
γ
ρ ≥ 0,
with the temperature constant T > 0 is employed [11, 19]. Isothermal fluids correspond to γ = 1, isentropic fluids to γ > 1. Notice that the particle temperature
is T (ρ) = T ργ−1 .
QUANTUM EULER-POISSON SYSTEMS
437
The equations (1.3)–(1.5) are referred to as the quantum Euler-Poisson system
or as the quantum hydrodynamic model.
In this paper we study the stationary system (1.3)–(1.5) in one space dimension
with λ = 1:
(1.7)
(1.8)
(1.9)
j0 = const.,
√
( ρ)xx
ε
j0
+ p(ρ) − ρφx − ρ
=− ,
√
ρ
2
ρ
τ
x
x
j02
2
φxx = ρ − C(x) in
Ω = (0, 1)
subject to the boundary conditions
(1.10)
ρ(0) = ρ1 ,
ρ(1) = ρ2 , ρx (0) = ρx (1) = 0 ,
(1.11)
φ(0) = 0 ,
φ(1) = Φ0 ,
where ρ1 , ρ2 > 0 and Φ0 ∈ R. In this formulation, the electron current density is
a given constant. From the equations, the applied voltage U can be computed by
U = φ(1) − φ(0).
As the momentum equation (1.8) is of third order, the mathematical analysis
of the above system of equations is quite difficult. In fact, without the thirdorder quantum term, the above equations represent the Euler-Poisson system of
gas dynamics for which only partial existence results (in several space dimensions)
are available (see, e.g., [5, 9] for several space dimensions and [6] for one space
dimension).
Therefore, we can only expect partial results for the hydrodynamic equations
including the third-order quantum term which makes the problem even more difficult. In the following, we describe some mathematical techniques which have
been successfully applied to the system (1.7)–(1.9) to prove the existence (and
uniqueness) of solutions.
In the literature, there exist essentially two ideas in dealing with the nonlinear
third-order equation (see also [14]). One idea consists in reducing the momentum
equation (1.8) to a second-order equation. The second idea is to differentiate (1.8)
once and to obtain a fourth-order equation.
The first idea has been used in [8, 7, 19, 31]. The existence of solutions to
(1.7)–(1.9) has been shown for sufficiently small j0 > 0, using nonlinear boundary
√
conditions for ρxx or Dirichlet data for the velocity potential. The pressure
function is assumed to be a monotone function of the density.
The second idea has been employed in [16] in order to prove the existence
of solutions to (1.7)–(1.9), again for sufficiently small j0 > 0. In that work,
the boundary conditions (1.10)–(1.11) have been used, but the pressure has been
assumed to be linear: p(ρ) = ρ. The main idea in [16] was to write the density
in exponential form: n = eu and to derive uniform H 1 bounds for u which, by
Sobolev embedding, yields L∞ bounds for u and hence a positive lower bound for
n = eu .
The main aim of this paper is to generalize the results of [16] to general pressure
functions. Compared to the results in [8], we use different boundary conditions
438
A. JÜNGEL AND H. LI
and more general pressure functions. Moreover, the technique of proof is different.
Compared to [16], we allow for more general pressure functions, in particular also
non-convex or non-monotone pressure-density relations.
We mention some related results on the stationary quantum Euler-Poisson system. The semi-classical limit ε → 0 in the case of thermal equilibrium j0 = 0 and
in the case j0 > 0 has been studied in [12, 29] and [16], respectively (also see [13]).
For results on the limit problem ε = 0 (Euler-Poisson system) we refer to the
review paper [24]. The local existence of strong solutions to the transient quantum Euler-Poisson model has been shown in [21]. The global existence of “small”
solutions to the transient model and its asymptotic behavior for large times will
be studied in our forthcoming work [22] based on the results of this paper for the
steady state.
In all cited papers, the existence of (strong) steady-state solutions to the quantum hydrodynamic equations is shown for sufficiently small current densities j0 >
0. In fact, in the case of the nonlinear boundary conditions assumed in [8], the
non-existence of weak solutions to the quantum Euler-Poisson system for sufficiently large j0 > 0 has been proved. We also need the smallness condition on |j0 |
to prove the existence of solutions to (1.7)–(1.11).
In order to explain our main results in detail, we rewrite the equation for the
electron density (1.8) as a nonlinear elliptic fourth-order equation and write the
density in exponential form. Writing
√
( ρ)xx
ε2
ε2
ρ
= (ρ(ln ρ)xx )x ,
√
2
ρ
4
x
dividing (1.8) by ρ > 0, differentiating with respect to x and using (1.7) and (1.9)
to remove the electrostatic potential from the equation, we obtain
0
p (ρ)ρx
j02
ε2 −1
j0
− 3 ρx
− (ρ − C(x)) − (ρ (ρ(ln ρ)xx )x )x = −
.
ρ
ρ
4
τρ x
x
It is convenient to introduce the new variable u = ln ρ. Then the above equation
can be written as
ε2 1 j0
uxx + u2x
− ((p0 (eu ) − j02 e−2u )ux )x + eu − C(x) = (e−u )x .
(1.12)
4
2
τ
xx
The boundary conditions (1.10) transform to
(1.13)
u(0) = u1 ,
u(1) = u2 ,
ux (0) = ux (1) = 0 ,
where u1 = ln ρ1 , u2 = ln ρ2 .
The electrostatic potential can be computed from the formulae
Z 1
(1.14)
φ(x) = Φ0 +
G(x, y)(eu(y) − C(y)) dy ,
0
where the Green’s function G(x, y) is defined by
(
x(1 − y) ,
G(x, y) =
y(1 − x) ,
x < y,
x > y.
QUANTUM EULER-POISSON SYSTEMS
439
The advantage of the above formulation is that bounded solutions u ∈ L∞ (0, 1)
define positive densities ρ = eu and in this case, both formulations (1.8)–(1.9) and
(1.12)–(1.14) are equivalent. Notice that for third-order or fourth-order equations,
no maximum principle is available such that other methods for proving the positivity of the variables have to be devised. Here we use the exponential transformation
of variables combined with Sobolev embeddings as in [8, 16].
Assume that
(1.15)
ε, τ, ρ1 , ρ2 > 0 ,
Φ 0 , j0 ∈ R ,
C ∈ L2 (0, 1) .
Then our main results are as follows:
1. Suppose that the pressure function is given by (1.6) for γ > 0. Then there
exist constants J0 , γ1 > 0 such that if |j0 | ≤ J0 and |γ − 1| ≤ γ1 , there exists
a unique strong solution u, φ ∈ H 4 (0, 1) to (1.12)–(1.14). Since u ∈ L∞ (0, 1),
we have ρ = eu > 0 in (0, 1), and ρ, φ ∈ H 4 (0, 1) is a solution of (1.7)–(1.11).
The constant j1 can be given explicitly (see section 2).
2. Suppose that p ∈ C 3 (0, ∞), that there exists a function A ∈ H 2 (0, 1) such
that
A > 0 in (0, 1) ,
A(0) = ρ1 ,
A(1) = ρ2 ,
Ax (0) = Ax (1) = 0
and that there is a set E ⊂ [0, 1] such that
(1.16)
j2
p (A) − 02
A
0
(
≤ 0,
> 0,
x∈E,
x ∈ [0, 1]\E .
Then if |j0 | is small enough, there exists a unique strong solution u, φ ∈
H 4 (0, 1) to (1.12)–(1.14).
Notice that we allow for non-convex pressure functions (1.6) with γ < 1 and for
non-monotone pressures satisfying (1.16). This means that the left part of (1.2)
may be not hyperbolic. The assumption (1.16) implies that the interval under
consideration may consist of subsonic, transonic and supersonic regions in the
classical sense [3]. To guarantee the well-posedness of strong solutions, we assume
a “subsonic” condition.
Finally, we notice that our estimates allow to perform the semi-classical limit
ε → 0 in (1.12)–(1.14) by employing the same techniques as in [16] (also see [7]
and Remark 3.4).
The paper is organized as follows. In section 2 our first main result is formulated
and proved. The second main result is shown in section 3.
Notation. The Lebesgue space of square integrable functions with the norm k · k
is denoted by L2 (0, 1), and H k (0, 1) or simply H k denotes the usual Sobolev space
of functions f satisfying ∂xi f ∈ L2 (0, 1), 0 ≤ i ≤ k, with the norm k · kk . In
particular, k · k0 = k · k.
440
A. JÜNGEL AND H. LI
2. Pressure functions satisfying (1.6)
In this section, we consider the steady-state solutions to the BVP (1.7)–(1.11)
when the pressure-density relation satisfies the γ-law (1.6).
Theorem 2.1. Assume that (1.6) and (1.15) hold. Let κ ∈ (0, 1). Then there exist
two constants γ0 > 0 and K(κ) > 0 such that if
r
1
−K(κ)
|j0 | ≤ e
κ T e−|γ−1|K(κ) + ε2 and |γ − 1| ≤ γ0 ,
(2.1)
2
then there exists a solution u ∈ H 4 to the BVP (1.12)–(1.13) satisfying
r
1
1
(2.2)
εkuxxk + T e−|γ−1|K(κ) + ε2 kux k ≤ K0 ,
2
2
(2.3)
|u(x)| ≤ K(κ),
where K0 is defined by (2.21) and (γ, K(κ)) is the unique solution to (2.22).
Furthermore, there are J0 , ε0 , γ1 > 0 such that if |j0 | ≤ J0 and |γ − 1| ≤ γ1 ,
the solution u is unique for any ε ∈ (0, ε0 ].
Proof. Step 1. A-priori estimates. Assume that u ∈ H 2 is a weak solution of
the boundary-value problem (BVP) (1.12)–(1.13) satisfying a-priorily that
(2.4)
−K(κ) ≤ u ≤ K(κ) .
Following [1] we introduce a function uD ∈ C 2 ([0, 1]) satisfying
(2.5)
uD (0) = u1 ,
uD (1) = u2 ,
uD,x (0) = uD,x (1) = 0 ,
with piecewise linear second order derivative
 4ζ

 µ2 (1−µ) x ,
4ζ
(2.6)
uD,xx (x) = µ2 (1−µ)
(µ − x) ,


0,
x ∈ [0, µ2 ) ,
x ∈ [ µ2 , µ] ,
x ∈ (µ, 12 ] ,
and uD,xx (x) = −uD,xx(1 − x) for x ∈ ( 12 , 1], where ζ = |u1 − u2 | and µ ∈ (0, 12 ).
Elementary computations show that
Z 1/2
Z 1
µζ
(2.7)
,
x|uD,xx (x)| dx +
(1 − x)|uD,xx (x)| dx =
1−µ
0
1/2
Z 1
8ζ 2
|uD,xx(x)|2 dx =
(2.8)
,
3µ(1 − µ)
0
Z 1
(2.9)
|uD,x (x)| dx = ζ ,
0
(2.10)
Z
1
0
|uD,x (x)|2 dx =
ζ 2 (23µ + 1)
1
≤ ζ2 .
2
30(1 − µ)
2
QUANTUM EULER-POISSON SYSTEMS
441
Use u − uD ∈ H02 as an admissible test function in the weak formulation of
(1.12) to obtain
Z
1 2 1
1
ε
(uxx + u2x )(u − uD )xx dx
4
2
0
Z 1
+
(T e(γ−1)u − j02 e−2u )ux (u − uD )x dx
0
+
(2.11)
=−
Z
1
0
j
τ
(eu − euD )(u − uD ) dx −
Z
1
0
Z
1
0
(euD − C)(u − uD ) dx
e−u (u − uD )x dx ,
By Cauchy’s inequality, (2.4) and (2.11), it follows for η ∈ (0, 1)
Z
Z 1
1 2
η 1 2
1
uxx dx + ε2
u2x uxx dx
ε 1−
4
2 0
8
0
Z 1 h
i
η −|γ−1|K(κ)
Te
− (1 + η)j02 e2K(κ) u2x dx
+
1−
2
0
Z 1
+
(eu − euD )(u − uD ) dx
0
Z 1
Z
1
j 2 1 2K(κ)
|uD,xx |2 dx + ε2
u2x uD,xx dx + 0
e
|uD,x |2 dx
8
4η
0
0
0
Z 1
Z 1
1
(euD − C)(u − uD ) dx
+ T e2|γ−1|K(κ)
|uD,x |2 dx +
2η
0
0
Z
j0 1 −u
e (u − uD )x dx ,
−
τ 0
1 ε2
≤
8 η
(2.12)
Z
1
Using the boundary condition (1.13), and applying (2.5) and (2.7)–(2.8), we have
as the proof of Lemma 2.1 in [16]:
Z 1
Z
1 ε2 1
ζ 2 ε2
u2x uxx dx = 0 ,
|uD,xx |2 dx ≤
,
8 η 0
3µη
0
Z
1 2 1 2
ε2 η
(2.13)
ε
kuxx k2 ,
ux uD,xx dx ≤
8
16
0
where we have used the Poincaré inequality and chosen µ = min(1/2, η/2ζ). From
(2.5) and (2.10) follows
Z
j02 1 2K(κ)
ζ2
ζ 2 2 2K(κ)
1
j0 e
≤
(2.14)
e
|uD,x |2 dx ≤
T e−|γ−1|K(κ) + ε2
4η 0
8η
8η
2
and
(2.15)
1
T e2|γ−1|K(κ)
2η
Z
1
0
|uD,x |2 dx ≤
ζ 2 2|γ−1|K(κ)
Te
.
4η
442
A. JÜNGEL AND H. LI
Since
ku − uD k ≤
1
k(u − uD )x k ,
2
we have by Cauchy’s inequality
Z 1
(euD − C)(u − uD ) dx
0
η
1 2
−|γ−1|K(κ)
≤
Te
+ ε k(u − uD )x k2
8
2
−1
1
1
T e−|γ−1|K(κ) + ε2
+
keuD − Ck2
2η
2
1
1
η
ηζ 2
T e−|γ−1|K(κ) + ε2 kux k2 +
T e−|γ−1|K(κ) + ε2
≤
8
2
16
2
−1
1
1
T e−|γ−1|K(κ) + ε2
+
(2.16)
keuD − Ck2 .
2η
2
Using (2.1) and (2.9), the last term in (2.12) can be estimated as
Z
j0 1 −u
−
e (u − uD )x dx
τ 0
Z
|j0 | K(κ) 1
≤
e
(|ux | + |uD,x |) dx
τ
0
r
1 2
1
ζκ
1
η
−|γ−1|K(κ)
2
Te
+ ε kuxk + 2 +
(2.17)
T e−|γ−1|K(κ) + ε2 .
≤
4
2
ητ
τ
2
Setting η = (1 − κ)/[2(1 + κ)] in (2.13)–(2.17) and substituting η into (2.12), we
have, estimating as in [16],
1 2
1
(1 + κ)2
(2.18)
ε kuxxk2 + T e−|γ−1|K(κ) + ε2 kux k2 ≤ a2c
K1 ,
4
2
(1 − κ)2
where K1 is given by
1
K1 = 1 + T e−|γ−1|K(κ) + ε2 + T e2|γ−1|K(κ) + T −1 e|γ−1|K(κ) ,
2
and ac a generic positive constant which only depends on ζ, T , and τ . From (2.18)
follows
r
1
1
(2.20)
εkuxxk + T e−|γ−1|K(κ) + ε2 kux k ≤ K0 ,
2
2
where
1 + κp
K0 = a c
K1 .
(2.21)
1−κ
Now, consider the equation for (γ, K(κ))
1 + κp
K2 (K(κ), γ) − |u1 | = 0 ,
K(κ) − ac
(2.22)
1−κ
(2.19)
QUANTUM EULER-POISSON SYSTEMS
443
where
K2 (K(κ), γ) = 1 + T −1 e|γ−1|K(κ) + e3|γ−1|K(κ) + T −2 e2|γ−1|K(κ) .
It has a solution
(γ, K(κ)) = (1, |u1 | + ac
1 + κp
2 + T −1 + T −2 ) .
1−κ
By the implicit function theorem, there exists a γ0 > 0 such that for |γ − 1| < γ0 ,
the equation (2.22) has a solution (γ, K(κ)).
Therefore, in view of (2.19)–(2.22), we obtain
|u(x)| ≤ |u1 | + kuxk
= |u1 | + q
K0
T e−|γ−1|K(κ)
1+κ
= |u1 | + ac
1−κ
≤ |u1 | + ac
(2.23) = K(κ) .
s
+ 12 ε2
1 + T e−|γ−1|K + 21 ε2 + T e2|γ−1|K(κ) + T −1 e|γ−1|K(κ)
T e−|γ−1|K(κ) + 12 ε2
1 + κp
K2 (K(κ), γ)
1−κ
Step 2. Existence. We apply the Leray-Schauder fixed point theorem to prove
the existence of strong solutions. Let v ∈ X := C 0,1 ([0, 1]). Consider the linear
BVP
1 2
1 2
ε uxx + λvx
− T e(γ−1)vK ux − λj02 e2vK ux
4
2
x
xx
v
e −1
j0
+λ
u + 1 − C = λ (e−vK )x ,
v
τ
u(0) = λu1 ,
u(1) = λu2 ,
ux (0) = ux (1) = 0 ,
where vK = mim{K(κ), max{−K(κ), v}} and λ ∈ [0, 1]. Define the bilinear form
Z 1
ev − 1 1 2
ε uxx ψxx + T e(γ−1)vK ux ψx + λ
uψ dx ,
a(u, ψ) =
4
v
0
for u, ψ ∈ H 2 and the functional
Z 1
j0 −vK
1 2 2
2 −2vK
vx ψx + λ(C − 1)ψ − λ e
ψx dx .
F (ψ) =
− ε λvx ψxx + λj0 e
8
τ
0
Since X ,→ W 1,∞ , a(·, ·) is continuous and coercive in H 2 , and F is linear and
continuous in H 2 , the Lax-Milgram theorem yields the existence of a solution
u ∈ H 2 . This means that the map S : X × [0, 1] → X, (v, λ) 7→ u is well
defined. Moreover, it is not difficult to see that S is continuous and compact.
444
A. JÜNGEL AND H. LI
Since S(v, 0) = 0 for all v ∈ H 2 , and by similar estimates as in Step 1 with eu
replaced by evK , we can verify that it holds for all λ ∈ [0, 1],
(2.24)
kukX ≤ c1 ,
where c1 > 0 is a constant independent of u and λ. Then the existence of u ∈ H 2
follows from the Leray-Schauder fixed point theorem. It is not difficult to prove
that indeed u ∈ H 4 (see [16] for details).
Step 3. Uniqueness. Let u, v ∈ H02 be two weak solutions of the BVP (1.12)–
(1.13), which satisfy (2.20). Using u − v ∈ H02 as an admissible test function in
the weak formulation derived for u − v, we obtain
Z
Z
1 2 1
1 2 1
2
ε
(u − v)xx dx + ε
(ux + vx )(u − v)x (u − v)xx dx
4
8
0
0
Z 1
(u − v)2x dx
+ T e(γ−1)u
0
=−
+
−
≤
Z
Z
1
T e(γ−1)u − T e(γ−1)v vx (u − v)x dx
0
1
2
Z
Z
0
1
1
0
(j02 e−2u − j02 e−2v )(u − v)xx dx
(eu − ev )(u − v) dx −
1
0
Z
1
0
T e|γ−1|K(κ) e(γ−1)u − e(γ−1)v
1
+ ε2
8
Z
1
0
j02
τ 2T
(u −
v)2xx
1
dx + 2 j04
2ε
1
Z
(e−u − e−v )(u − v)x dx
2
1
0
vx2 dx
(e−2u − e−2v )2 dx
1
(u − v)2 dx + T e−|γ−1|K(κ)
2
0
Z 1
e|γ−1|K(κ)
(e−u − e−v )2 dx ,
− e−K(κ)
+
j0
τ
Z
Z
1
0
(u − v)2x dx
0
which implies
Z
Z
1 2 1
1 2 1
2
ε
(u − v)xx dx + ε
(ux + vx )(u − v)x (u − v)xx dx
8
8
0
0
Z 1
1
+ T e−|γ−1|K
(u − v)2x dx
2
0
j02 (|γ−1|+2)K
1 4 4K(κ)
2 3|γ−1|K(κ)
2
≤
j e
+ 2 e
+ T (γ − 1) e
|vx |L∞
2ε2 0
τ T
Z 1
Z 1
×
(u − v)2 dx − e−K(κ)
(2.25)
(u − v)2 dx .
0
0
QUANTUM EULER-POISSON SYSTEMS
445
From (1.13) and (2.20), we have by Hölder’s inequality
p
1
|ux (x)| ≤ 2kuxxk · kux k ≤ θkuxx k + θ−1 kux k
2
θ
1
≤ max
,
K̃0 ,
ε θT 12 e− 12 |γ−1|K(κ)
where
K̃0 = ac
Choose
1
θ=
1 + κp
K1 .
1−κ
3
1
T 2 e− 2 |γ−1|K(κ)
K̃0
and
0 < ε ≤ ε0 =:
3
T 2 e− 2 |γ−1|K(κ)
K̃02
to obtain
θ
1
≥
1
1
−
ε
2
2
θT e |γ−1|K(κ)
and
|ux (x)| ≤
(2.26)
θ
1 1 1
K̃0 ≤ T 2 e− 2 |γ−1|K(κ) .
ε
ε
From this estimate and Cauchy’s inequality follows that the left-hand side of (2.25)
is bounded from below by
Z
Z 1
1 2 1
1
(u − v)2xx dx + ε2
(ux + vx )(u − v)x (u − v)xx dx
ε
8
8
0
0
Z
1 1 −|γ−1|K(κ)
Te
(u − v)2x dx
+
2 0
Z
Z
1 2 1
1 −|γ−1|K(κ) 1
2
≥
ε
(u − v)xx dx + T e
(u − v)2x dx .
16
4
0
0
On the other hand, by the implicit function theorem, there is a γ1 ≤ γ0 such that
for |γ − 1| < γ1 it holds
|γ − 1|2 e(2|γ−1|+1)K(κ) ≤
(2.27)
1 −2 2
T ε ,
8
which implies
2 2
1
T (γ − 1)2 e2|γ−1|K(κ) ≤ e−K(κ) .
ε2
4
Thus, there exists J0 such that if
(2.28)
(2.29) j02 ≤ J02 =:
(
min e
−2K(κ)
κ Te
−|γ−1|K(κ)
)
√
1 2
2 − 5 K(κ) 1 2 −(|γ−1|+3)K(κ)
, Tτ e
+ ε ,
εe 2
,
2
2
4
446
A. JÜNGEL AND H. LI
then the first integral on the right-hand side of (2.25) is bounded, in view of (2.26)
and (2.28), by
1
j02 (|γ−1|+2)K(κ)
4 4K(κ)
2 3|γ−1|K(κ)
2
−K(κ)
j
e
+
e
+
T
(γ
−
1)
e
|v
|
−
e
∞
x
L
2ε2 0
τ 2T
Z 1
×
(u − v)2 dx
0
1
j02 (|γ−1|+2)K(κ)
2 2
4 4K(κ)
2 2|γ−1|K(κ)
−K(κ)
j
e
e
T
+
+
(γ
−
1)
e
−
e
≤
0
2ε2
τ 2T
ε2
Z 1
(u − v)2 dx
×
0
(2.30)
Z
1 −K(κ) 1
≤− e
(u − v)2 dx .
4
0
Therefore, the weak solution is unique if both (2.27) and (2.29) holds. The proof
of Theorem 2.1 is complete.
As in [16], we can conclude from Theorem 2.1 the following result.
Theorem 2.2. Assume that (1.6) and (1.15) hold. Then there exist two constants
γ0 > 0 and K(κ) > 0 (κ ∈ (0, 1)) such that if
r
1
−K(κ)
|j0 | ≤ e
κ T e−|γ−1|K(κ) + ε2 and |γ − 1| ≤ γ0 ,
(2.31)
2
then there is a solution (ρ, φ) ∈ H 4 × H 2 to the BVP (1.7)–(1.11) such that
(2.32)
ρ ≥ ρ̄ =: e−K(κ) > 0 ,
where (γ, K(κ)) solves the equation (2.22). Moreover, if |j0 |, ε, and |γ − 1| are
small enough, the solution is unique.
3. Pressure functions satisfying (1.16)
In this section, we consider the BVP (1.12)-(1.13) (and (1.7)–(1.11)) with pressure functions satisfying the condition (1.16).
Set uD := ln A. Then euD = A. We have the following theorem.
Theorem 3.1. Assume that (1.15) and (1.16) holds and that p ∈ C 3 (0, ∞). For
κ ∈ (0, 1) assume that it holds
1 2
min A(x)2
(3.1)
κε + p0 (A(x)) > j02 .
4
x∈[0,1]
QUANTUM EULER-POISSON SYSTEMS
447
Then the BVP (1.12)–(1.13) has a unique solution u ∈ H 4 provided that kA0 k1 +
kA − Ck is sufficiently small. Moreover, it holds
A∗ ku − uD k2 + A0 kuxx k2 + ε4 k(uxxx, uxxxx)k2
Z
j2
+
p0 (A2 ) − 04 (u − uD )2x dx ≤ Kc δ0 ,
A
I\E
(3.2)
where
A∗ = min A(x), δ0 = kA0 k1 + kA − Ck ,
x∈[0,1]
1 2
A0 = min
κε + p0 (A) − j02 A−2 > 0 ,
x∈[0,1] 4
(3.3)
(3.4)
and Kc > 0 is a constant depending on A, τ and j0 .
Remark 3.2. (1) We call the main assumption (3.1) a “subsonic” condition for
the quantum Euler-Poisson system. When ε = 0, the assumption (3.1) is exactly
the subsonic condition for the classical hydrodynamic model [4].
(2) One can verify that the assumption (3.1) can be replaced by
1 2
(3.5)
κε + meas(E) min(p0 (A) − j02 A−2 ) > 0 , κ ∈ (0, 1) ,
x∈E
4
in order to obtain the existence and uniqueness of strong solutions. Here, we recall
that the region E ⊂ [0, 1] is defined such that it holds p0 (A)A2 − j02 < 0 in E.
Proof. We prove Theorem 3.1 by the same steps as Theorem 2.1.
Step 1. The a-priori estimates. Let u ∈ H 4 be a solution of the BVP (1.12)–
(1.13) satisfying
(3.6)
uD − δ 1 ≤ u ≤ u D + δ 1 ,
where δ1 > 0 is chosen such that
4
5
(3.7)
A∗ ≤ e−δ1 euD ≤ eu ≤ eδ1 euD ≤ A∗ ,
5
4
2
(3.8)
max (|p00 (A)|A + 2j02 A−2 )δ1 ≤ (1 + θ)A0 ,
x∈[0,1]
9
(3.9)
ln
max
25
16
∗
25 A∗ ≤y≤ln 16 A
(|p000 (ey )|e2y + |p00 (ey )|ey + 4j02 e−2y )δ12 ≤
4
(1 + θ)A0 .
9
where A∗ = maxx∈[0,1] A(x), A∗ = minx∈[0,1] A(x), and θ = 1−κ
1+κ (then κ =
Assume that δ0 = kA − Ck + k(Ax , Axx )k is so small that it holds
(3.10)
∞
1−θ
1+θ ).
|uD,x |∞ + |uD,xx|∞ ≤ θ ,
where | · |∞ denotes the L norm.
Taking (u − uD ) ∈ H02 as an admissible test function in the weak formulation
of (1.12), we have, by Cauchy’s inequality and (3.7), that
Z 1
1
1
1
1 − θ − |uD,xx |∞ ε2
u2xx dx
4
2
2
0
448
A. JÜNGEL AND H. LI
+
(3.11)
Z
1
≤
A∗
1
u
(p (e ) −
0
Z
0
1
j02 e−2u )ux (u
1
(A − C) dx + ε2
8θ
2
0
Z
1
− uD )x dx + A∗
2
1
j0
|uD,xx | dx −
τ
2
0
Z
1
0
Z
(u − uD )2 dx
1
0
e−u (u − uD )x dx,
where we have used the facts that
Z
(3.12)
1
0
u2x uxx dx = 0 ,
Z
1
0
u2x dx ≤
Z
1
0
u2xx dx .
The last term in (3.11) can be estimated as
j0
−
τ
Z
1
e
−u
0
(3.13)
1
Z
j0 −uD −(u−uD ) j0 1 −u
e
(u − uD )x dx = e
e uD,x dx
+
τ
τ 0
x=0
Z
j0 1 −uD
(e
− e−u )uD,x dx ≤ Kc δ0 ,
=−
τ 0
where here and in the following Kc > 0 is a generic constant depending on A, τ
and j0 .
By Taylor’s expansion and Cauchy’s inequality, the second term on the left-hand
side of (3.11) can be estimated as
Z
1
0
(p0 (eu ) − j02 e−2u )ux (u − uD )x dx
=
Z
1
A−2 (p0 (A)A2 − j02 )(u2x − ux uD,x ) dx
0
+
Z
1
0
p00 (A)A + 2j02 A−2 (u − uD )ux (u − uD )x dx
Z
1 1 000 y 2y
+
p (e )e + p00 (ey )ey − 4j02 e−2y ux (u − uD )x (u − uD )2 dx
2 0
Z 1
≥
A−2 (p0 (A)A2 − j02 )(u2x − ux uD,x ) dx
0
−
−
9
max |(p00 (A)|A + 2j02 A−2 δ1
8 x∈[0,1]
Z
1
0
u2x dx
9
|p000 (ey )|e2y + |p00 (ey )ey | + 4j02 e−2y δ12
max 25
16
16 ln 25 A∗ ≤y≤ln 16 A∗
− K c δ0 ,
Z
1
0
u2x dx
QUANTUM EULER-POISSON SYSTEMS
449
where y = u + θ1 (u − uD ) for some θ1 ∈ (0, 1). This implies, using (3.7)–(3.9) and
(3.12),
Z 1
(p0 (eu ) − j02 e−2u )ux (u − uD )x dx
0
≥
(3.14)
Z
1
0
×
Z
1
A−2 (p0 (A)A2 − j02 )(u2x − ux uD,x ) dx − (1 + θ)A0
2
1
0
u2xx dx − Kc δ0 .
Furthermore, it follows from (1.16)
Z 1
(p0 (A) − j02 A−2 )(u2x − ux uD,x ) dx
0
Z 1
≥ (1 + θ) min(p0 (A)A2 − j02 )A−2
u2x dx + (1 − θ)
x∈E
0
Z
×
(p0 (A)A2 − j02 )A−2 u2x dx
I\E
−
1
max |(p0 (A) − j02 A−2 )|
θ x∈[0,1]
0
2
Z
1
0
|uD,x |2 dx
j02 )A−2
Z
1
≥ (1 + θ) min (p (A)A −
u2xx dx + (1 − θ)
x∈[0,1]
0
Z
1+κ
Kc δ 0 ,
×
(p0 (A)A2 − j02 )A−2 u2x dx −
1
−κ
I\E
(3.15)
where we have used
Z
E
u2x dx ≤
Z
1
0
The estimates (3.15) and (3.14) yield
Z 1
(p0 (eu ) − j02 e−2u )ux (u − uD )x dx
u2xx dx .
0
(3.16)
1
1
≥(1 + θ) min (p (A)A −
− (1 + θ)A0
2
x∈[0,1]
0
Z
2
j
1+κ
+ (1 − θ)
p0 (A) − 02 u2x dx −
Kc δ 0 .
A
1−κ
I\E
0
2
j02 )A−2
Z
u2xx dx
Z
1
0
u2xx dx
Substituting (3.13) and (3.16) into (3.11) and using (3.10), we have
(3.17)
A0 kuxxk2 + A∗ ku − uD k2
Z
1+κ
+
(p0 (A)A2 − j02 )A−2 (u − uD )2x dx ≤
Kc δ 0 ,
1−κ
I\E
where we recall that A∗ and A0 are given by (3.3) and (3.4), respectively.
450
A. JÜNGEL AND H. LI
Now, we turn to higher order estimates. Let u ∈ H 4 be a solution to the BVP
(1.12). Multiply this equation with ε2 uxxxx and integrate over (0, 1) to obtain
Z
1 4 1 2
ε
[uxxxx − (ux uxxx + u2xx )uxxxx] dx
4
0
Z 1
Z 1
2
0 u
2 2u
2
=ε
((p (e ) − j0 e )ux )x uxxxx dx − ε
(eu − euD )uxxxx dx
0
(3.18)
− ε2
Z
0
1
0
j0
(A − C)uxxxx dx − ε2
τ
Z
1
e−u ux uxxxx dx .
0
Due to (1.13), there are y1 , y2 , y3 , y4 ∈ (0, 1) such that
ux (y1 ) = uxx (y2 ) = uxx (y3 ) = uxxx (y4 ) = 0 ,
and
(3.19)
u2xx (x) +
Z
1
0
u2xx dx ≤
Z
1
0
u2xxx dx ,
Z
1
0
u2xxx dx ≤
Z
1
0
u2xxxx dx .
Thus, it follows from (3.17), (3.19) and Hölder’s inequality
Z 1
(ux uxxx + u2xx )uxxxx dx
0
(3.20)
≤ |ux |∞ kuxxxk · kuxxxxk + |uxx |∞ kuxx k · kuxxxxk
Z 1
1+κ
≤
Kc δ 0
u2xxxx dx .
1−κ
0
Then, we obtain from (3.18), in view of (3.7), (3.17), (3.20), (3.19), and Cauchy’s
inequality, that
Z 1
Z 1
4
2
[(p0 (eu ) − j02 e−2u )2 u2xx + (p00 (eu )eu + 2j02 e−2u )2 u4x ] dx
ε
uxxxx dx ≤ Kc
0
(3.21)
0
+ Kc
Z
1
0
[(u − uD )2 + (A − C)2 + u2x ] dx ≤
1+κ
Kc δ 0 ,
1−κ
provided that δ0 is small enough. By (3.19) and (3.21), we have
Z 1
1+κ
(3.22)
ε4
Kc δ 0 .
[u2xxx + u2xxxx ] dx ≤
1
−κ
0
The combination of (3.17) and (3.22) finally leads to
A∗ ku − uD k2 + A0 kuxx k2 + ε4 k(uxxx , uxxxx)k2
Z
1+κ
(3.23)
+
(p0 (A2 )A2 − j02 )A−2 (u − uD )2x dx ≤
Kc δ 0 .
1
−κ
I\E
Step 2. Existence. It is not difficult to prove that there exists a solution u ∈ H 4
to the BVP (1.12)–(1.13). The argument is similar to that used in section 2
QUANTUM EULER-POISSON SYSTEMS
451
based on the Leray-Schauder fixed point theorem. The function space is X :=
C 0,1 ([0, 1]). The corresponding linear BVP is
1
1 2
ε (uxx + λvx2 )xx − (p0 (evK )ux − λj02 e−2vK vx )x
4
2
ev − 1
j0
+ λ(
u + 1 − C) = λ (e−vK )x ,
v
τ
u(0) = λu1 , u(1) = λu2 , ux (0) = ux (1) = 0 ,
where λ ∈ [0, 1], v ∈ X and vK = mim{δ1 ln A∗ , max{−δ1 ln A∗ , v}} with δ1
chosen such that (3.7)–(3.9) hold. The bilinear form and functional are defined
respectively by
Z 1
ev − 1
1 2
0 vK
a(u, ψ) =
ε uxxvxx + p (e )ux ψx + λ
uψ dx
4
v
0
and
F (ψ) =
Z
1
0
1
j0
− ε2 λvx2 ψxx − λj02 e−2vK vx ψx + λ(C − 1)ψ − λ e−vK ψx dx .
8
τ
where vK . We omit the details.
Step 3. Uniqueness. Let u, v ∈ H 4 be two solutions to the BVP (1.12)–(1.13)
satisfying (3.23). Using u − v ∈ H02 as an admissible test function in the weak
formulation derived for u − v, we have
Z
Z 1
1 2 1
1
(u − v)2xx dx + ε2
(ux + vx )(u − v)x (u − v)xx dx
ε
4
8
0
0
Z 1
+
(p0 (eu ) − j02 e−2u )ux − (p0 (ev ) − j02 e−2v )vx (u − v)x dx
0
(3.24)
=−
Z
1
0
(eu − ev )(u − v) dx +
j0
τ
Z
1
0
(e−u − e−v )x (u − v) dx .
The last term in (3.24) can be estimated as
Z
j0 1 −u
(e − e−v )x (u − v) dx
τ 0
Z
1
j0 1 −u
(e − e−v )(u − v)ux + e−v [(u − v)2 ]x dx
=−
τ 0
2
r
Z 1
1 + κ Kc
(3.25)
(u − v)2 dx ,
δ0
≤
1 − κ A0
0
where we have used
(3.26)
|(ux , vx )|∞ ≤
r
1 + κ Kc
δ0 ,
1 − κ A0
452
A. JÜNGEL AND H. LI
derived from (3.23), and Kc > 0 is a generic constant depending on A, τ , and j0 .
From (3.7), (3.24), and (3.25) follows
Z
Z 1
1 2 1
1
ε
(u − v)2xx dx + ε2
(ux + vx )(u − v)x (u − v)xx dx
4
8
0
0
Z 1
+
(p0 (eu ) − j02 e−2u )ux − (p0 (ev ) − j02 e−2v )vx (u − v)x dx
0
!Z
r
1
1 + κ Kc
3
(u − v)2 dx
≤−
A∗ −
δ0
4
1 − κ A0
0
Z 1
1
≤ − A∗
(3.27)
(u − v)2 dx ,
2
0
provided that δ0 is small enough.
By (3.7), (3.26), Hölder’s inequality and
Z 1
Z 1
2
(u − v)x dx ≤
(3.28)
(u − v)2xx dx ,
0
0
we have
1 2
ε
4
≥
(3.29)
Z
1
0
(u −
v)2xx
1
dx + ε2
8
Z
1
(ux + vx )(u − v)x (u − v)xx dx
!
r
Z 1
1 + κ Kc
δ0
1−
(u − v)2xx dx .
1 − κ A0
0
1 2
ε
4
0
By (1.16), (3.7) and (3.28), the third term on the left-hand side of (3.27) can
be estimated as follows, using an approach similar to (3.14):
Z 1
(p0 (eu ) − j02 e−2u )ux − (p0 (ev ) − j02 e−2v )vx (u − v)x dx
0
=
Z
1
0
+
(p0 (eu ) − j02 e−2u )(u − v)2x dx
Z
1
0
(p0 (eu ) − p0 (ev ) − j02 e−2u + j02 e−2v )vx (u − v)x dx
0
+
I\E
1
0
j02 )A−2
1
Z
≥ min(p (A)A −
(u − v)2x dx
x∈E
0
Z
+
(p0 (A)A2 − j02 )A−2 (u − v)2x dx
Z
2
p0 (eu ) − j02 e−2u − p0 (A) + j02 A−2 (u − v)2x dx
− |vx |∞
Z
1
0
|p0 (eu ) − p0 (ev ) − j02 e−2u + j02 e−2v k(u − v)x | dx
≥ min(p0 (A)A2 − j02 )A−2
x∈E
Z
1
0
(u − v)2xx dx
QUANTUM EULER-POISSON SYSTEMS
+
−
Z
I\E
r
453
(p0 (A)A2 − j02 )A−2 (u − v)2x dx
1 + κ Kc
δ0
1 − κ A0
Z
1
0
[(u − v)2 + (u − v)2xx ] dx
Z 1
(u − v)2xx
≥ min(p0 (A)A2 − j02 )A−2
x∈E
0
Z
+
(p0 (A)A2 − j02 )A−2 (u − v)2x dx
I\E
(3.30)
1
− A∗
3
Z
1
0
(u − v)2 dx −
r
1 + κ Kc
δ0
1 − κ A0
Z
1
0
(u − v)2xx dx ,
provided that δ0 is small enough.
Substituting (3.29) and (3.30) into (3.27) leads to
Z 1
1 2
0
2 −2
κε + p (A) − j0 A
(u − v)2xx dx
min
x∈[0,1] 4
0
!Z
r
1
1 2
1 + κ Kc
+
ε (1 − κ) −
δ0
(u − v)2xx dx
4
1 − κ A0
0
Z 1
1
(3.31)
(u − v)2 dx ≤ 0 ,
+ A∗
6
0
which implies that u = v in (0, 1) if δ0 is so small that
r
1 2
1 + κ Kc
ε (1 − κ) −
δ0 > 0
4
1 − κ A0
and if the condition
1 2
0
2 −2
(3.32)
>0
min
κε + p (A) − j0 A
x∈[0,1] 4
holds. The proof of Theorem 3.1 is completed.
The existence and uniqueness of stationary solutions of (1.7)–(1.11) follows
immediately from Theorem 3.1:
Theorem 3.3. Assume that (1.15) and (1.16) hold and that p ∈ C 3 (0, ∞). For
κ ∈ (0, 1), assume that it holds
1 2
0
2
(3.33)
κε + p (A(x)) > j02 .
min A(x)
4
x∈[0,1]
Then, the BVP (1.7)–(1.11) has a unique solution (ρ0 , φ0 ) ∈ H 4 × H 2 such that
(3.34)
A∗ kρ0 − Ak2 + A0 kρ0xx k2 + ε4 k(ρ0xxx , ρ0xxxx)k2 + kφ0x k21 ≤ K̃c δ0 ,
provided that kA0 k1 + kA − Ck is sufficiently small. The constant A0 is defined in
(3.4) and K̃c > 0 is a constant depending on j0 , τ and A.
454
A. JÜNGEL AND H. LI
Remark 3.4. The estimates of Theorem 2.1 and Theorem 3.1 show that one can
pass to the limit ε → 0 in the quantum Euler-Poisson system (similarly as in [16])
to obtain a solution to the classical Euler-Poisson system:
(3.35)
(3.36)
∂t j +
∂t ρ + j x = 0 ,
√
( ρ)xx
j2
ε2
j
=− ,
+ p(ρ) − ρφx − ρ
√
ρ
2
ρ
τ
x
x
(3.37)
λ2 φxx = ρ − C(x)
in
[0, 1] × (0, T ) ,
The solution of this system (together with appropriate boundary conditions) is
classical because the condition (2.1) or (3.1) reduces to the classical subsonic condition for (3.35)–(3.37). For the mathematical analysis on the Euler-Poisson system,
we refer to [2, 17, 24] and references therein.
The following theorem is important for the stability analysis of stationary solutions obtained by Theorem 3.3 (see [22] for details).
Theorem 3.5. Let (ρ0 , j0 , φ0 ) the unique strong solution given by Theorem 3.3.
√
Let ω0 = ρ0 . Then (ω0 , j0 , φ0 ) is the unique solution of the following BVP
(3.38)
jx = 0,
2
(3.39)
(
(3.40)
j
j
1 2 2 wxx 2
2
− ,
+
p(w
))
=
w
φ
+
ε w
x
x
2
w
2
w x τ
φxx = w2 − C(x) ,
with boundary conditions
√
(3.41)
w(0) = ρ1 ,
(3.42)
w(1) =
φ(0) = 0,
√
ρ2 ,
wx (0) = wx (1) = 0,
φ(1) = Φ0 .
Moreover, it holds
(3.43)
kω0 −
√ 2
Ak + kω0x k23 + kφ0x k21 ≤ C̃0 δ0 ,
where C̃2 > 0 is a constant depending on A, j0 , τ and ε.
Acknowledgements. The first author acknowledges partial support from the
Deutsche Forschungsgemeinschaft, grants JU 359/3 (Gerhard-Hess Award) and
JU 359/5 (Priority Program “Multi-Scale Problems”), and by the EU Project
“Hyperbolic and Kinetic Equations’, grant HPRN-CT-2002-00282. The second acknowledges the partial support from the Wittgenstein Award 2000 of P. Markowich
funded by the Austrian FWF and from the Austrian-Chinese Scientific-Technical
Collaboration Agreement.
QUANTUM EULER-POISSON SYSTEMS
455
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† Fachbereich
Mathematik und Informatik, Universität Mainz
Staudingerweg 9, 55099 Mainz, Germany
E-mail: juengel@mathematik.uni-mainz.de
∗ Department of Mathematics, Capital Normal University
Beijing 100037, P. R. China
E-mail: lihl@math.sci.osaka-u.ac.jp