Dirichlet-neumann problem for unipolar isentropic quantum drift-diffusion model

TSINGHUA SCIENCE AND TECHNOLOGY ISSN 1007-0214 21/22 pp560-569 Volume 13, Number 4, August 2008

Dirichlet-Neumann Problem for Unipolar Isentropic Quantum Drift-Diffusion Model*

CHEN Li (陈丽)**, CHEN Xiuqing (陈秀卿)†

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China; † School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract: This paper studies the existence, semiclassical limit, and long-time behavior of weak solutions to

the unipolar isentropic quantum drift-diffusion model, a fourth order parabolic system. Semi-discretization in

time and entropy estimates give the global existence and semiclassical limit of nonnegative weak solutions

to the one-dimensional model with a nonnegative large initial value and a Dirichlet-Neumann boundary con-

dition. Furthermore, the weak solutions are proven to exponentially approach constant steady state as time

increases to infinity.

Key words: quantum drift-diffusion; fourth order parabolic system; weak solution; semiclassical limit;

exponential decay

1 Introduction

In the development of the modern computer and tele-communication industries, the size of semiconductor devices has reached a quantum dominated scale in the last decades. Due to the high computational costs of microscopic models such as the Shrödinger-Poisson system, some scientists have turned to macroscopic models including the quantum drift-diffusion model (QDDM), quantum hydrodynamic model, and quantum energy transport model using momentum methods and entropy minimization methods etc. They were intro-duced to simulate the quantum effects in miniaturized semiconductor devices such as HEMT’s, MOSFET’s and RTD’s [1 2], . Some derivation of these models could be found in Jüngel[3,4] and Pinnau[5]. The quantum hy-drodynamic model and the quantum energy transport

model have been studied recently[6-8]. This paper ana-lyzes the unipolar transient isentropic quantum drift-diffusion model,

2

2

div[ ( ) ]

( )

tnn n n n V

nV n C x

γε

λ

⎧ ⎛ ⎞Δ= − ∇ +∇ + ∇ ,⎪ ⎜ ⎟

⎨ ⎝ ⎠⎪− Δ = −⎩

(1)

where the electron density, n, and the electrostatic potential, V , are unknown variables; the doping pro-file ( )C x representing the distribution of charged background ions is assumed to be independent of time, t ; 1γ > , the scaled Planck constant 0ε > , and the Debye length 0λ > .

From the mathematical point of view, the main dif-ficulty of Eq. (1) lies in the fourth order (quantum) terms. After the initial works by Jüngel and Pinnau[9,10], a series of papers have been published with homoge-neous Neumann boundary condition. Chen and Ju[11] derived better weak solutions and established the semiclassical limit (see Abdallah and Unterreiter[12]) for the stationary QDDM semiclassical limit). Chen et al.[13] established the weak solution and the long-time behavior with periodic-boundary conditions for the

Received: 2007-06-15

* Supported by the National Natural Science Foundation of China(No. 10401019)

** To whom correspondence should be addressed. E-mail: [email protected]

CHEN Li (陈丽) et al：Dirichlet-Neumann Problem for Unipolar Isentropic Quantum Drift-Diffusion Model

561

unipolar isentropic case in one dimension. They also obtained the semiclassical limit when the exponent on the pressure is less than or equal to 3/2. For the isen-tropic case, Chen and Ju[14] employed different ideas from Chen et al.[13] to prove the semiclassical limit with the exponent on the pressure of 1 3α< and its long-time behavior was established in Chen et al.[15] The quasineutral limit for bipolar QDDM was studied recently by Jüngel and Violet[16]. However, very few multidimensional QDDM results have been ob-tained[17,18].

To get more physical reasonable results, more gen-eral boundary conditions than the homogeneous Neu-mann boundary must be considered. This analysis starts from a special fourth order one-dimensional parabolic equation for (0 1)x∈ , ,

( (ln ) ) 0t xx xxn n n+ = (2) with a nonnegative large initial value and the Dirichlet-Neumann boundary condition

( 0) ( 1) 1 ( 0) ( 1) 0x xn t n t n t n t, = , = , , = , = (3) The nonnegative global existence and the exponential decay of the weak solution have already been ob-tained[19,20]. Gualdani et al.[21] generalized those results to the nonhomogeneous case. Similar results for the initial periodic-boundary problem of Eq. (2), were proved for large initial values[22] as well as for small initial values[23,24].

A positivity preserving one-dimensional global weak solution for the unipolar isothermal QDDM with large enough θ was obtained[9,10] with the mixed Dirichlet-Neumann boundary condition

D D D D, onn n F F V V Ω= , = , = ∂ ;

N0, onx x xn F V Ω= = = ∂ (4)

where 2 ln ,nF n Vn

ε θ= − + + DΩ and NΩ are

two disjoint parts of Ω∂ . In addition, they also give the approximate solutions of elliptic time discrete sys-tems with the Dirichlet-Neumann boundary condition

1 0 , onxn n V VΩ Ω= , = , = ∂ (5) However, they did not do any a priori estimates with the boundary conditions in Eq. (5) to get global solu-tions and their estimates[9,10] are not sufficient.

The main task of this paper is to study system Eq. (1)

with the boundary condition Eq. (5). We will show enough estimates to get a global weak solution, a semi-classical limit, and long-time behavior. Moreover, the time-space regularity of our weak solution is better than that of Ref. [19] on fourth order parabolic equa-tion Eq. (2) with boundary condition Eq. (3).

The main estimates in this analysis are derived from two kinds of entropy inequalities for

[ (ln 1) 1]dn n xΩ

− +∫ and ( ln ) dn n xΩ

−∫ (see the

discretized version in Eqs. (28) and (30)). In getting the estimates, the first difficulty comes from the Dirichlet-Neumann boundary. The calculations must be done carefully, especially for the terms from the boundary and the coupling with the Poisson equation. The second difficulty comes from the isentropic poten-tial, since we only have estimates for 2 ( )

|| ( ) ||T

x L Qnα

and 21

( )|| ( ) ||

Tx L Qn

α− from these entropy inequalities

which are not enough to do the semiclassical limit. Results from Chen and Ju[14] were used to get estimates for 2 ( )

|| ( ) ||T

x L Qn (1 2)α < and 2 ( )

|| ||T

x L Qn

( 2 3α ) uniformly in ε . The third difficulty oc-curs in the semiclassical limit discussion to get uni-form estimates with respect to ε since the estimate for (ln )xxn nε is no longer valid as in previous works. To have 2 ( (ln ) ) 0x x x xn nε → in some sense, look for

a key estimate for 14 nε , which together with the es-

timate for (ln )x xn nε allows us to deduce that 54 (ln )x xn nε is bounded in some norm uniformly in

ε , which shows that the quantum term vanishes in the semiclassical limit.

From now on, the discussion will be fixed on a one-dimensional space. Since all the results in this pa-per were obtained for fixed 0θ λ, > , for convenience let 1θ λ= = in the following. To search for solutions which are physically reasonable, namely solutions in which the density n is nonnegative, assume that

2n ρ= . Furthermore, let (0 1)Ω = , , 0T > be any fixed constant and (0 ]TQ T Ω= , × . Then, consider the initial Dirichlet-Neumann boundary value problem

Tsinghua Science and Technology, August 2008, 13(4): 560-569

562

22 2 2 2 2 2 2 2 2

2

0

( ) ( ) ( (ln ) ) ( ) in (0 ]2

( ) in (0 ]1 0 in (0 ]

(0 ) ( ), in

x xt x x x x x x x

xx x

xx

x

V V T

V C x TV V T

γ γ

Ω

ρ ερ ε ρ ρ ρ ρ ρ ρ ρ Ωρ

ρ Ωρ ρ Ωρ ρ Ω

⎧ ⎡ ⎤⎛ ⎞ ⎡ ⎤= − + + = − + + , , × ;⎪ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎣ ⎦⎢ ⎥⎝ ⎠⎪ ⎣ ⎦

⎪⎨− = − , , × ;⎪

= , = , = , , ×∂ ;⎪⎪ ,⋅ = ⋅⎩

(6)

where the Dirichlet data for the electric potential 2 ( )V HΩ Ω∈ . In physical applications, VΩ can be

xU where U is an applied voltage. Notations

• The Sobolev spaces are ( )m pW Ω, ( ( )mH Ω = 2 ( )mW Ω, ) and the Hölder spaces are ( )kC θ Ω, .

• A BO (or A BOO ) denotes A is continuously (or compactly) embedded in B .

• 2 ( )H Ω− denotes the dual space of 20 ( )H Ω .

The main results are stated as follows. Theorem 1.1 (Existence) Let 1 5γ< < . Suppose

that ( ) ( )C x L Ω∞∈ and 2 2

0 0(ln 1) 1ρ ρ − + , 2 20 0lnρ ρ− ∈ 1( )L Ω .

Then for any fixed 0ε > , there exists ( )Vρ, such that (1) 2 2 2 40 (0 ( )) (0 ( )) (0L T L L T H V L Tρ Ω Ω∞∈ , ; , ; , ∈ , ;∩

2( ))H Ω ,8 83 52 8 2 1(0 ( )) (0 ( )) (0L T L L T H L Tρ Ω Ω∈ , ; , ; , ;∩ ∩

2 ( ))H Ω ,10 8

5 5min{ }2 2 20ln (0 ( )) (0tL T H L Tγρ Ω ρ + ,

∈ , ; , ∂ ∈ , ;2 ( ))H Ω− ;

(2) For any 10 8

5 3max{ } 20(0 ( ))L T Hγϕ Ω− ,

∈ , ; ,

2 20

2

0

22 2

0

d =

(ln ) d d2

T

t H H

T

x x x x

t

x tΩ

ρ ϕ

ε ρ ρ ϕ

− ,∂ ,

− −

∫

∫ ∫

⟨ ⟩⟨ ⟩

2 2

0[( ) ] d d

T

x x xV x tγ

Ωρ ρ ϕ+∫ ∫ (7)

and for any 43 2(0 ( ))L T Lφ Ω∈ , ; ,

2

0 0d d [ ( )] d d

T T

x xV x t C x x tΩ Ω

φ ρ φ− = −∫ ∫ ∫ ∫ (8)

Remark 1.1 In Theorem 1.1, ( )Vρ, is a weak solution of Eq. (6) in the sense of Eqs. (7) and (8).

For the sake of explicitness, and Vρ are relabeled as and Vε ερ in the following Theorem.

Theorem 1.2 (Semiclassical limit) Let 10 2α< <

and 1 3γ< . Then for the solution obtained in Theorem 1.1, 0{( )}Vε ε ερ >, , as 0ε → , there exists a subsequence which is not relabeled such that

Case 1 1 2γ< < . 432 4 2 1, in (0 ( )) (0 ( ))n L T L L T Hερ Ω Ω, ; , ;∩ (9)

2 2, in (0 ( ))V V L T Hε Ω, ; (10) 432 0, in (0 ( ))n L T C α

ερ Ω,→ , ; (11) 6 6

3 312 , in (0 ( ))n L T Wγ γγ γερ Ω+ +,, ; (12)

43 2, in (0 ( ))V V L T Hε Ω→ , ; (13)

6 43 3min{ }2 2, in (0 ( ))t tn L T Hγ

ερ Ω+ , −∂ ∂ , ; (14) 852 2 2 2(ln ) 0, in (0 ( ))x x L T Lε εε ρ ρ Ω→ , ; (15)

and

2 200 0

4 20

2

0 0

d [( ) ] d d

(0 ( ))

d [ ( )] d ( )

T T

t x x xH H

T T

xx T

n t n nV x t

L T H

V x n C x x L Q

γ

Ω

Ω Ω

ψ ψ

ψ Ω

φ φ φ

− ,⎧ ∂ , = − + ,⎪⎪

∀ ∈ , ; ,⎨⎪− = − , ∀ ∈⎪⎩

∫ ∫ ∫

∫ ∫ ∫ ∫

⟨ ⟩⟨ ⟩

(16) here 0n .

Case 2 2 3γ . 2 6 2 2 1, in (0 ( )) (0 ( ))n L T L L T Hερ Ω Ω, ; , ;∩ (17)

3 2. in (0 ( ))V V L T Hε Ω, ; (18) 2 2 0, in (0 ( ))n L T C αερ Ω,→ , ; (19)

8 84 412 , in (0 ( ))n L T Wγ γγ γ

ερ Ω+ +,, ; (20) 2 2, in (0 ( ))V V L T Hε Ω→ , ; (21)

842 2, in (0 ( ))t tn L T Hγ

ερ Ω+ −∂ ∂ , ; (22) 852 2 2 2(ln ) 0, in (0 ( ))x x L T Lε εε ρ ρ Ω→ , ; (23)

and

2 20

84

32

0 0

20

0 0

2

d [( ) ] d d

(0 ( ));

d [ ( )] d

(0 ( ))

T T

t x x xH H

T T

xx

n t n nV x t

L T H

V x n C x x

L T L

γ

γ

Ω

Ω Ω

ψ ψ

ψ Ω

φ φ

φ Ω

−

−

,⎧ ∂ , = − + ,⎪⎪

∀ ∈ , ;⎪⎨⎪− = − ,⎪⎪ ∀ ∈ , ;⎩

∫ ∫ ∫

∫ ∫ ∫ ∫

⟨ ⟩⟨ ⟩

(24) here 0n .

Remark 1.2 In Theorem 1.2, ( )n V, is a weak solution to the initial Dirichlet-Neumann boundary


563

problem of the classical drift-diffusion model (DDM)

[( ) ]

( )t x x x

xx

n n nVV n C x

γ⎧ = + ,⎪⎨− = −⎪⎩

(25)

in the sense of Eq. (16). Then the weak solution of the QDDM in Eq. (6) converges to that of DDM in Eq. (25) as 0ε → .

Theorem 1.3 (Long-time behavior) Let ( ) 1C x ≡ and 2 2

0 0(ln 1) 1ρ ρ − + , 2 2 10 0ln ( )Lρ ρ Ω− ∈ . Then the

weak solution ρ obtained in Theorem 1.1 satisfies 0

2 0( )( ) 1 e (0 ]M t

Lt C t T

Ωρ −,⋅ − , ∀ ∈ , (26)

where 12 2 2 1

0 0 0 ( )( ln 1)

LM

Ωε ρ ρ −= − + and 0C =

121

2 20 0 ( )(ln 1) 1

L Ωρ ρ − + .

Remark 1.3 Let 0c > be a constant. For the

boundary condition 0 , in (0 ] ,xc V V TΩρ ρ Ω= , = , = , × ∂

we have similar results as in Theorems 1.1 and 1.2 as well as in Theorem 1.3 with 2( )C x c≡ .

2 Approximate Problem

This section introduces the approximation problem of Eq. (6). Let 0τ > such that T Nτ= (without loss of

generality, otherwise, let 1TNτ⎡ ⎤= +⎢ ⎥⎣ ⎦

). Hence N =

( )N τ ∈N depends only on τ . Divide the time inter-val (0 ]T, by 1(0 ] (( 1) ]N

kT k kτ τ=, = − ,∪ . For any 1 2k N= , ,..., , given 1kρ − , solve the following problem

2 2 22 2 2 2 2 2 2 21

2

( )( ) ( ) ( (ln ) ) ( ) ( ) in

2

( ) ( ) in 1 ( ) 0 on

k x xk kk k x k k x k k x x x k x k k x

xk x x

k x x k

k k x k

V V

V C xV V

γ γ

Ω

ρρ ρ εε ρ ρ ρ ρ ρ ρ ρ Ωτ ρ

ρ Ωρ ρ Ω

−⎧ ⎡ ⎤⎛ ⎞ ⎡ ⎤−

= − + + = − + + , ;⎪ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎣ ⎦⎪ ⎢ ⎥⎝ ⎠⎣ ⎦⎨− = − , ;⎪⎪ = , = , = , ∂⎩

(27)

Theorem 2.1 Let 11 0 ( ) ( ) ,2

C x Lγ α Ω∞< <∞, < < , ∈

and 01 1( ) min 0k kC α

Ωρ Ω ρ,− −∈ , > .Then there exists a

constant 0kc > such that Eq. (27) has a solution 2 2( ) ( ) ( )k kV C Hαρ Ω Ω,, ∈ × and 0k kcρ > .

Proof Similar to the proof of Jüngel and Pinnau[9]. Remark 2.1 In fact, 4 ( )k Hρ Ω∈ by standard el-

liptic estimates.

3 Uniform Estimates of the Approximate Solution

Suppose 00 ( )C αρ Ω,∈ and ( ) ( )C x L Ω∞∈ satisfy

the assumption of Theorem 2.1 for 1k = , i.e.,

0min 0Ω ρ > . Then use Theorem 2.1 iteratively to obtain a sequence of approximate solutions ( )k kVρ , ∈

4 2( ) ( )H HΩ Ω× , 1 2k N= , , ,… . This section focuses on the uniform estimates for the approximate solution. From now on, C (or )Cε is asssumed to be con-stant dependent only on

( )( )

LT C

Ω∞, ⋅ ,

12 20 0 ( )

(ln 1) 1L Ω

ρ ρ − + , 12 20 0 ( )

lnL Ω

ρ ρ− (and ε ).

Lemma 3.1 1 12 2 2 2

0 0[ (ln 1) 1]d 2 ( ) dk k k x xx xρ ρ ε τ ρ− + + | | +∫ ∫

221 12 1 2

0 0

2 ( ) d 4 ( ) d3

k xk k x

k

x xγρε τ γτ ρ ρρ

−+ | | +∫ ∫

1 14 2 21 10 0

d [ (ln 1) 1]dk k kx xτ ρ ρ ρ− − − + +∫ ∫ 1 2

0dkC x Cτ ρ τ+∫ (28)

Proof Multiplying the first equation of Eq. (27) by 2ln kρ , then

2 21 210

1 2 2 2 2 2

0

ln d

( )( ) ( ) ln d .

k kk

k x xk k x k k x k

k x x

x

V xγ

ρ ρ ρτ

ρε ρ ρ ρ ρ

ρ

−−=

⎛ ⎞− + +⎜ ⎟

⎝ ⎠

∫

⎡ ⎤∫ ⎢ ⎥⎣ ⎦

Since ( ) (ln 1) 1f s s s= − + is convex in (0 ),∞ , 2 21 12 2 21

0 0

1 2 21 10

1ln d [ (ln 1) 1]d

1 [ (ln 1) 1]d .

k kk k k

k k

x x

x

ρ ρ ρ ρ ρτ τ

ρ ρτ

−

− −

−− + −

− +

∫ ∫

∫

Integrating by parts with the boundary condition 2ln 0k Ωρ ∂| = ,

1 2 2 2 2 2

0

( )( ) ( ) ln dk xx

k k x k k x kk x x

V xγρε ρ ρ ρ ρ

ρ⎡ ⎤⎛ ⎞− + + =⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫

112 2 1 20 0

( ) ( ) d 4 ( ) dk xxk x k k x

k x

x xγρε ρ γ ρ ρρ

−⎛ ⎞ − | | −⎜ ⎟⎝ ⎠

∫ ∫

1 12 1 21 20 0

( ) ( ) d 4 ( ) dk x k x k k xV x I x Iγρ γ ρ ρ−= − | | +∫ ∫


564

where since ( ) 0k x Ωρ ∂| = and 1k Ωρ ∂| = ,

12 21 0

21 12 2 2

0 0

1 12 2 2 3

0 0

221 12 2 2

0 0

( )( ) d

( ) ( )2 ( ) d 2 d

2 12 ( ) d [( ) ] d3

2 ( )2 ( ) d d3

k x xk xx

k

k x x k xk x x

k

k x x k x xk

k xk x x

k

I x

x x

x x

x x

ρε ρ

ρ

ρ ρε ρ ε

ρ

ε ρ ε ρρ

ρε ρ ερ

= − =

− | | − =

− | | − =

− | | −

∫

∫ ∫

∫ ∫

∫ ∫

and 12 1 2

2 0 0

11 20 0

1 1 2

0 0

1 2 2

0

( ) ( ) d

( ) ( ) d

( ) d ( ) d

( 1)( ( ) )d .

k x k k k x x

k x k k x x

k x x k k x x

k k

I V V x

V V x

V x V x

C x x

ρ ρ

ρ

ρ

ρ ρ

= − | + =

− | + =

− + =

− −

∫

∫

∫ ∫

∫

Thus, 1 2 2

0

1 2 21 10

221 12 2 2

0 0

1 11 2 2 2

0 0

1 [ (ln 1) 1]d

1 [ (ln 1) 1]d

( )22 ( ) d d3

4 ( ) d ( 1)( ( ) )d

k k

k k

k xk x x

k

k k x k k

x

x

x x

x C x xγ

ρ ρτ

ρ ρτ

ρε ρ ερ

γ ρ ρ ρ ρ

− −

−

− + −

− +

− | | − −

| | + − −

∫

∫

∫ ∫

∫ ∫

(29) which gives the discretized version of the first entropy inequality Eq. (28).

Lemma 3.2 1 12 2 2 2

0 0

1 12 2 2 21 10 0

( ln )d 2 (ln ) d

4 ( ) d ( ln )d

k k k xx

k k x k k

x x

x xγ

ρ ρ ε τ ρ

γτ ρ ρ ρ ρ−− −

− + | | +

| | − +

∫ ∫

∫ ∫

1

0ln dkC x Cτ ρ τ| | +∫ (30)

Proof Multiplying the first equation of Eq. (27) by

2

11kρ

− ,

2 21 120

11 dk k

k

xρ ρτ ρ

−− ⎛ ⎞− =⎜ ⎟⎝ ⎠

∫

1 2 2 2 220

( ) 1( ) ( ) 1 d .k xxk k x k k x

k kx x

V xγρε ρ ρ ρρ ρ

⎡ ⎤⎛ ⎞ ⎛ ⎞− + + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦

∫

Since ( ) lnf s s s= − is convex in (0 ),∞ ,

2 211

20

1 12 2 2 21 10 0

11 d

1 ( ln )d ( ln )d .

k k

k

k k k k

x

x x

ρ ρτ ρ

ρ ρ ρ ρτ

−

− −

− ⎛ ⎞−⎜ ⎟⎝ ⎠

⎡ ⎤− − −⎢ ⎥⎣ ⎦

∫

∫ ∫

Integrating by parts with 2

11 0k Ω

ρ∂

− = ,

1 2 2 2 220

21 12 2 2 240 0

21 1 2 21 220 0

( ) 1( ) ( ) 1 d

( ) ( )d 4 ( ) d

( )( ) d 4 ( ) d

k xxk k x k k x

k kx x

k xx k xk k k x

k kx

k xk x k k x

k

V x

x x

V x I x I

γ

γ

γ

ρε ρ ρ ρ

ρ ρ

ρ ρε ρ γ ρ ρ

ρ ρ

ργ ρ ρ

ρ

−

−

⎡ ⎤⎛ ⎞ ⎛ ⎞− + + − =⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞ − | | −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

= − | | +

∫

∫ ∫

∫ ∫Since

2 ( )( ) ( ) ( )k x x

k k k xxx k x k xxk x

ρρ ρ ρ ρ ρ

ρ⎛ ⎞

= − =⎜ ⎟⎝ ⎠

2 21 [ (ln ) ]2 k k xx xρ ρ ,

further calculations of 1I and 2I with the boundary conditions 2( ) 0k x Ωρ ∂| = and 2ln 0k Ωρ ∂| = give

212 2 21 40

2 2 2 212 240

12 2 2 2 2

0

12 2

0

( )1 [ (ln ) ] d2

( ) 2( )1 (ln ) d21 (ln ) [(ln ) (ln ) ]d2

2 (ln ) d

k xk k xx x

k

k k xx k xk xx

k

k xx k xx k x

k xx

I x

x

x

x

ρε ρ ρ

ρ

ρ ρ ρε ρ

ρ

ε ρ ρ ρ

ε ρ

= =

−− =

− − =

− | |

∫

∫

∫

∫

and 1 12 2

2 0 0

1 2 2

0

( ) (ln ) d ( ) ln d

( ( ) )ln d .

k x k x k xx k

k k

I V x V x

C x x

ρ ρ

ρ ρ

= − = =

−

∫ ∫

∫

Then 1 12 2 2 2

1 10 0

1 12 2 2 2

0 0

1 12 2 2 2

0 0

1 ( ln )d ( ln )d

2 (ln ) d 4 ( ) d

( ( ) ) ln d 2 (ln ) d

k k k k

k xx k k x

k k k xx

x x

x x

C x x x

γ

ρ ρ ρ ρτ

ε ρ γ ρ ρ

ρ ρ ε ρ

− −

−

⎡ ⎤− − −⎢ ⎥⎣ ⎦

− | | − | | +

− − | | −

∫ ∫

∫ ∫

∫ ∫

1 12 2

0 04 ( ) d ln dk k x kx C x Cγγ ρ ρ ρ−| | + | | +∫ ∫ (31)

Thus, the discretized version of the second entropy inequality Eq. (30) holds.

Definition 3.1 ( ) ( ), for (( 1) ]kt x x x t k kτρ ρ Ω τ τ, ∈ , ∈ − , , ( ) ( ), for (( 1) ],kV t x V x x t k kτ Ω τ τ, ∈ , ∈ − ,

.


565

where 4 2( ) ( ) ( )k kV H Hρ Ω Ω, ∈ × is the solution in Theorem 2.1, 1 2k N= , , ,… .

Remark 3.1 In Definition 3.1, 40 ( ) ( )t Hτρ Ω< ,⋅ ∈ for any (0 ]t T∈ , .

Using Lemmas 3.1 and 3.2, now obtain the uniform estimates in τ for τρ and Vτ .

Theorem 3.1 Let 1 γ< < ∞. Then

2 2 2

2

2

82 2 1 4 13

14

8 4 2

2(0 ( )) ( ) ( )

2

( )( )

(0 ( )) (0 ( )) (0 ( ))

(0 ( )) (0 ( ))

( ) ( )

( ) (ln )

T T

T

T

x xL T L L Q L Q

xxx L Q

L Q

L T H L T W L T H

L T L L T HV

γ γτ τ τ τΩ

ττ τ

τ

τΩ Ω Ω

τ τΩ Ω

ρ ρ ρ ρ

ρε ε ρ ρ

ρ

ε ρ

ε ρ ε

∞

,∞

∞

−, ;

, ; , ; , ;

, ; , ;

+ + +

+ +

+

+ +

∩ ∩

1 2 20(0 ( )) (0 ( ))

ln lnL T L L T H

Cτ τΩ Ωρ ε ρ∞ , ; , ;

+ (32)

Furthermore, if 1 5γ< < ,

10 1015 5

2

(0 ( ))L T WC

γ γ

γτ ε

Ωρ

,+ +, ;

(33)

In particular, if 1 2γ< < ,

42 1 4 13

2

(0 ( )) (0 ( )) (0 ( ))L T H L T L L T Hτ τΩ Ω Ωρ ρ

∞, ; , ; , ;+ +∩

2 2(0 ( ))L T HV Cτ Ω, ;

(34)

and 6 61

3 3

2

(0 ( ))L T WC

γ γ

γτ

Ωρ

,+ +, ;

(35)

else if 2 3γ ,

2 13 6 2

2 2(0 ( ))(0 ( )) (0 ( )) L T HL T L L T Lτ τ ΩΩ Ω

ρ ρ∞ , ;, ; , ;

+ +∩

3 2(0 ( ))L T HV Cτ Ω, ;

(36)

and 8 81

4 4

2

(0 ( ))L T WC

γ γ

γτ

Ωρ

,+ +, ;

(37)

Proof By Lemma 3.1, the Gronwall inequality and the inequality (ln 1) 3 ( 0)x x x x− + ∀ > , then

2(0 ( ))L T LCτ Ω

ρ ∞ , ;, so that

2 2 2

2

2

(0 ( )) ( )( )

( )( )

T

T

xxL T H L Q

L Q

Cγ ττ τΩ

τ

ρε ρ ρ ε

ρ, ;+ +

and hence,

2

2

2

( )( )

( )(ln ) ( )

T

T

xxx xxL Q

L Q

ττ τ τ

τ

ρε ρ ρ ε ρ

ρ= −

2

2

2

( )( )

( )( ) .T

T

xxx L Q

L Q

Cττ

τ

ρε ρ ε

ρ+

By the Gagliardo-Nirenberg inequality,

314 4

2 2( ) ( ) ( )L H LCτ τ τΩ Ω Ω

ρ ρ ρ∞ ,

3 14 4

2 2( ) ( ) ( )( )x L H L

Cτ τ τΩ Ω Ωρ ρ ρ∞ ,

1 12 2

2 2 2( ) ( ) ( )( )x L H L

Cτ τ τΩ Ω Ωρ ρ ρ

and then

14

8

31 14 4 4

2 2 2

(0 ( ))

(0 ( )) (0 ( ))

L T L

L T H L T LC C

τ Ω

τ τΩ Ω

ε ρ

ε ρ ρ

∞

∞

, ;

, ; , ;,

83

3 14 4

2 2 2

(0 ( ))

(0 ( )) (0 ( ))

( )xL T L

L T H L T LC C

τΩ

τ τΩ Ω

ε ρ

ε ρ ρ

∞

∞

, ;

, ; , ;,

4 2

1 12 2

2 2 2

(0 ( ))

(0 ( )) (0 ( ))

( )

.

x L T L

L T H L T LC C

τ Ω

τ τΩ Ω

ε ρ

ε ρ ρ ∞

, ;

, ; , ;

Therefore, 42

(0 ( ))L T LCτ Ω

ε ρ ∞, ;. Consequently,

4 2(0 ( ))L T HVτ Ω

ε, ;

C by standard elliptic estimates.

With the Gronwall inequality and the inequality ln ln ( 0)x x x x| | − ∀ > , then from Lemma 3.2

1(0 ( ))ln

L T LCτ Ω

ρ ∞ , ;. Consequently,

2 2 20

2( ) (0 ( ))

( ) ln .T

x L Q L T HCγ

τ τ τ Ωρ ρ ε ρ−

, ;+

This finishes the proof of Eq. (32). If 1 5γ< < , using the Gagliardo-Nirenberg

inequality, 1 45 5

10 2 2( ) ( ) ( )L H LCτ τ τΩ Ω Ω

ρ ρ ρ ,

and hence, from Eq. (32), 1 45 5

10 2 2 2( ) (0 ( )) (0 ( )).

TL Q L T H L T LC Cτ τ τ εΩ Ω

ρ ρ ρ ∞, ; , ;

Therefore, 10 5 10

5

2 2 2( )

( ) ( ) TT T

L QL Q L Q

C C Cγ γ

γ γ γτ τ τ ερ ρ ρ

+=

and 10 10 2

5

2( )

( ) ( )( ) 2 ( )

TT T

x x L QL Q L Qγ γ

γ γ γτ τ τρ ρ ρ

+=

10 2( ) ( )2 ( )

T TxL Q L Q

Cγ γτ τ ερ ρ ,

so that Eq. (33) is true. If 1 2γ< < , then 2 1(0 ( ))L T H

Cτ Ωρ

, ; from Eq. (32)

and Corollary 3.9 in Chen and Ju[14]. In fact,

22 2

( ) 0 { 1}

2

0 { 1}

2( 2) 2

0 { 1}

2( 1) 2

0 { 1}

( ) ( ) d d

( ) d d

( ) d d

( ) d d

T

T

x xL Q

T

x

T

x

T

x

x t

x t

x t

x t

τ

τ

τ

τ

τ τΩ ρ

τΩ ρ

γτ τΩ ρ

γτ τΩ ρ

ρ ρ

ρ

ρ ρ

ρ ρ

∩ <

∩

−

∩ <

−

∩

= | | +

| |

| | +

| |

∫ ∫

∫ ∫

∫ ∫

∫ ∫


566

2 22 2 1 2

( ) ( )( ) ( ) .

T Tx xL Q L Q

Cγ γτ τ τ τρ ρ ρ ρ− −+

Using the Gagliardo-Nirenberg inequality, 1 12 2

1 2( ) ( ) ( )L H LCτ τ τΩ Ω Ω

ρ ρ ρ∞

and then 1 12 2

4 2 1 2(0 ( )) (0 ( )) (0 ( )).

L T L L T H L T LC Cτ τ τΩ Ω Ω

ρ ρ ρ∞ ∞, ; , ; , ;

Therefore, 4 2

13

2 2(0 ( ))(0 ( ))

.L T LL T H

Cτ τ ΩΩρ ρ ∞, ;, ;

+

Consequently, 2 2(0 ( ))L T HV Cτ Ω, ;

by standard ellip-

tic estimates, so Eq. (34) is true. Using the Gagliardo- Nirenberg inequality again,

1 23 3

6 1 2( ) ( ) ( )L H LCτ τ τΩ Ω Ω

ρ ρ ρ

such that 1 23 3

6 2 1 2( ) (0 ( )) (0 ( )).

TL Q L T H L T LC Cτ τ τΩ Ω

ρ ρ ρ ∞, ; , ;

Thus, 6 3 6

3

2 2 2( )

( ) ( ) TT T

L QL Q L Q

C C Cγ γ


+=

and 6 6 2

3

2( )

( ) ( )( ) 2 ( )

TT T

x x L QL Q L Qγ γ


+=

6 2( ) ( )2 ( ) .

T TxL Q L Q

Cγ γτ τρ ρ

This implies Eq. (35). If 2 3γ , then 2 1

2(0 ( ))L T H

Cτ Ωρ

, ; in view of

Eq. (32) and Corollary 3.9 in Chen and Ju[14]. In fact, 2

2 2

2 2( )

2 2

0 { 1}

2 2

0 { 1}

2( 2) 2

0 { 1}

2( 1) 2

0 { 1}

2 2 1 2( ) ( )

( )

4 ( ) d d

4 ( ) d d

4 ( ) d d

4 ( ) d d

4 ( ) 4 ( ) .

T

T T

x L Q

T

x

T

x

T

x

T

x

x xL Q L Q

x t

x t

x t

x t

C

τ

τ

τ

τ

τ

τ τΩ ρ

τ τΩ ρ

γτ τΩ ρ

γτ τΩ ρ

γ γτ τ τ τ

ρ

ρ ρ

ρ ρ

ρ ρ

ρ ρ

ρ ρ ρ ρ

∩ <

∩

−

∩ <

−

∩

− −

=

| | +

| |

| | +

| |

+

∫ ∫

∫ ∫

∫ ∫

∫ ∫

Also from the Gagliardo-Nirenberg inequality, 2 13 3

1 12 2 2

( ) ( ) ( )L H LCτ τ τΩ Ω Ω

ρ ρ ρ∞ , 1 23 3

2 1 12 2 2

( ) ( ) ( )L H LCτ τ τΩ Ω Ω

ρ ρ ρ , 1 12 2

4 1 12 2 2

( ) ( ) ( )L H LCτ τ τΩ Ω Ω

ρ ρ ρ

which yields 2 13 3

3 2 1 12 2 2

(0 ( )) (0 ( )) (0 ( ))L T L L T H L T LC Cτ τ τΩ Ω Ω

ρ ρ ρ∞ ∞, ; , ; , ;,

1 23 3

6 2 2 1 12 2 2

(0 ( )) (0 ( )) (0 ( ))L T L L T H L T LC Cτ τ τΩ Ω Ω

ρ ρ ρ ∞, ; , ; , ;,

1 12 2

4 2 1 12 2 2

( ) (0 ( )) (0 ( )).

TL Q L T H L T LC Cτ τ τΩ Ω

ρ ρ ρ ∞, ; , ;

Therefore, 8 4 4

4

2 2 2( )

( )( ) TTT

L QL QL Q

C C Cγγ


+=

and 8 8 2

4

2( )

( ) ( )( ) 2 ( ) =

TT T

x x L QL Q L Qγ γ


+

224

2( )( )

2 ( ) ,TT

x L QL QC

γγ

τ τρ ρ

which proves Eq. (37). Moreover, 3 2(0 ( ))L T HV Cτ Ω, ;

by standard elliptic estimates.

4 Existence of Weak Solution

In this section, let τρ and Vτ be the functions in Definition 3.1 which satisfy Theorem 3.1. Using a compactness argument and the Aubin-Lions lemma[25], the following convergence of τρ and Vτ is proved to complete the proof of Theorem 1.1.

Definition 4.1 Define the difference quotient of 2τρ as

2 21

2

20

( ) ( ) , for (( 1) ]( )( ), for 0.

k k

t

x x x t k kt xx x t

ττ

ρ ρΩ τ τρ τ

ρ Ω

−⎧ −∈ , ∈ − , ,⎪∂ , ⎨

⎪ ∈ , =⎩

Theorem 4.1 Let 10 2α< < and 1 5γ< < . Then

for any fixed 0ε > , as 0τ → , there exists a subse-quence of 2

0{( )}tV ττ τ τ τρ ρ >, ,∂ which is not relabeled,

such that 2 2, in (0 ( ))L T Hτρ ρ Ω, ; (38)

2, in (0 ( ))L T Lτρ ρ Ω∞∗ , ; (39) 4 2, in (0 ( ))V V L T Hτ Ω, ; (40)

2 8, in (0 ( )) 23

pV V L T H pτ Ω ⎛ ⎞→ , ; ∀ <⎜ ⎟⎝ ⎠

(41)

832 2 8 2 1, in (0 ( )) (0 ( ))L T L L T Hτρ ρ Ω Ω, ; , ;∩ ∩

85 2(0 ( ))L T H Ω, ; (42)

2 2 0 8, in (0 ( )) 23

pL T C pατρ ρ Ω, ⎛ ⎞→ , ; ∀ <⎜ ⎟

⎝ ⎠ (43)

852 2 1 5, in (0 ( )) if 1

4L T C α

τρ ρ Ω γ, ⎛ ⎞→ , ; <⎜ ⎟⎝ ⎠

(44)

2 2 1 5 8, in (0 ( )) if 5 14 5

qL T C qατρ ρ Ω γ, ⎛ ⎞→ , ; < < , ∀ < <⎜ ⎟

⎝ ⎠

(45) 10 10

5 512 2 , in (0 ( ))L T Wγ γγ γτρ ρ Ω+ +,

, ; (46)


567

2 20ln ln , in (0 ( ))L T Hτρ ρ Ω, ; (47)

10 85 5min{ }2 2 2, in (0 ( ))t t L T Hγτ

τρ ρ Ω+ , −∂ ∂ , ; (48) 852 2 2(ln ) (ln ) , in (0 ( ))xx xx L T Lτ τρ ρ ρ ρ Ω, ; (49)

where 0ρ . Furthermore, we have the following estimates uniformly in ε .

Case 1 1 2γ< < . 2 2 1 4

4 2 213

(0 ( )) (0 ( )) (0 ( ))

2(0 ( ))(0 ( ))

L T L L T H L T L

L T HL T HV

Ω Ω Ω

ΩΩ

ρ

ρ

∞ ∞, ; , ; , ;

, ;, ;

+

+ +∩ ∩

54

6 6 81 23 3 5

2 2 2

(0 ( ))(0 ( ))(ln )xx

L T LL T WC

γ γ

γ

ΩΩρ ε ρ ρ

,+ + , ;, ;

+

(50) Case 2 2 3γ .

1 2 1 6 2 3

3 2 8 814 4

2

(0 ( )) (0 ( )) (0 ( )) (0 ( ))

2 (0 ( ))

(0 ( ))

L T L L T H L T L L T L

L T HL T W

Vγ γ

Ω Ω Ω Ω

γΩ

Ω

ρ

ρ

∞ ∞

,+ +

, ; , ; , ; , ;

, ;, ;

+

+ +∩ ∩ ∩

54

825

2 2

(0 ( ))(ln )xx

L T LC

Ωε ρ ρ

, ; (51)

where C is a constant independent of ε . Proof Remark 3.1 and Eq. (32) are used to deduce

that there exists 0ρ satisfying Eqs. (38) and (39).

Let 10 8min5 5

rγ

⎧ ⎫= ,⎨ ⎬+⎩ ⎭ and 0 1τ< < be fixed.

Then for any 0 h τ< < ,

2 22 2 2 2

(0 ( )) ( )0dr

T hr rh hL T h H H

tτ τ τ τΩ Ωπ ρ ρ π ρ ρ− −

−

, − ;− = − =∫

2

12 2

1 ( )1

,N

rk k H

k

hΩ

ρ ρ −

−

+=

−∑

where ( )( ) ( )h f t f t hπ = + . Moreover, for any 1 k 1N − ,

22

2 221

1 1 ( )( )

(ln )k kk k xx L

HΩ

Ω

ρ ρρ ρ

τ −

++ +

−+

12 2

1 1 1 ( )( ) ( )k x k k x L

VγΩ

ρ ρ+ + ++ (52)

Then Eqs. (32) and (33) are used to show that 1

2 2

1

2

12 2 2 2

1(0 ( )) ( )1

2(0 ( ))

( (ln )

r

r

rr

Nr

h k kL T h H Hk

xx L T L

h

h

τ τ Ω Ω

τ τ Ω

π ρ ρ ρ ρ

ρ ρ

− −

−

+, − ;=

, ;

⎛ ⎞− −⎜ ⎟⎝ ⎠

+

∑

1

12 2

(0 ( ))( ) ( ) ) r

rx x L T LV C hγ

τ τ τ εΩρ ρ

, ;+ (53)

Moreover, Eq. (32) implies 8 8

1 23 5

2 2

(0 ( )) (0 ( ))L T H L T HCτ τ ε

Ω Ωρ ρ

, ; , ;+ (54)

Combining this with Eq. (53) and the compact imbed-dings 1 0( ) ( )H C αΩ Ω,OO and 2 1( ) ( )H C αΩ Ω,OO gives Eqs. (43)-(45) by the Aubin-Lions lemma.

Furthermore, Eq. (41) comes from Eq. (43) by standard elliptic estimates, while Eq. (40) can be obtained di-rectly from Eq. (32). Equations (32) and (43) imply Eq. (42), Eq. (47), and, hence, Eq. (49). By Eq. (52) and the proof of Eq. (53), 2

2(0 ( ))rt L T H

ττ Ωρ −, ;

∂ Cε . The

mean value theorem of differentials and Eq. (43) show that 2 2 in ( )t t TQτ

τρ ρ ′∂ ∂ ,D where ( )TQ′D repre-sents the set of all distributions on TQ . Hence Eq. (48) is true. Equation (46) is obvious from Eqs. (33) and (43). The weakly lower semi-continuity of the norm, the Gagliardo-Nirenberg inequality and Theorem 3.1 lead to Eqs. (50) and (51).

Proof of Theorem 1.1 Theorem 2.1 and Theorem 4.1 lead to Eqs. (7) and (8). The initial value is satis-fied in the sense of 2( )H Ω− since

10 85 51 min{ } 2 0 2(0 ( )) ([0 ] ( ))W T H C T Hγ θΩ Ω+, , − , −, ; , ;O

for 5 310 80 min{ }γθ − , . The boundary data is obvious

since 2 20ln (0 ( ))L T Hρ Ω∈ , ; .

5 Semiclassical Limit

Consider the semiclassical limit of the weak solution ( Vρ, ) obtained in Theorem 1.1. The solution depends on ε , to be precise, and is relabeled as ( Vε ερ , ) throughout this section. Therefore, for any ϕ∈

10 85 3max{ } 2

0(0 ( ))L T Hγ Ω− ,, ; ,

2 20

22 2 2

0 0d (ln ) d d

2T T

t xx xxH Ht x tε ε εΩ

ερ ϕ ρ ρ ϕ− ,∂ , = − −∫ ∫ ∫⟨ ⟩⟨ ⟩

2 2

0[( ) ( ) ] d d

T

x x xV x tγε ε εΩ

ρ ρ ϕ+∫ ∫ (55)

2

0 0( ) d d [ ( )] d d

T T

xxV x t C x x tε εΩ Ωφ ρ φ− = − ,∫ ∫ ∫ ∫

43 2(0 ( ))L T Lφ Ω∀ ∈ , ; (56)

and the following estimates are uniform in ε . Case 1 1 2γ< < .

2 2 1 4

4 2 213

(0 ( )) (0 ( )) (0 ( ))

2(0 ( ))(0 ( ))

L T L L T H L T L

L T HL T HV

ε Ω Ω Ω

ε ε ΩΩ

ρ

ρ

∞ ∞, ; , ; , ;

, ;, ;

+

+ +

∩ ∩

54

6 6 81 23 3 5

2 2 2

(0 ( ))(0 ( ))(ln )xx

L T LL T WC

γ γ

γε ε ε

ΩΩρ ε ρ ρ

,+ + , ;, ;

+

(57) Case 2 2 3γ .

1 2 1 6 2 3

3 2 8 81 4 4

2

(0 ( )) (0 ( )) (0 ( )) (0 ( ))

2(0 ( ))

(0 ( ))

L T L L T H L T L L T L

L T HL T W

Vγ γ

ε Ω Ω Ω Ω

γε εΩ

Ω

ρ

ρ

∞ ∞

,+ +

, ; , ; , ; , ;

, ;, ;

+

+ +

∩ ∩ ∩


568

54

825

2 2

(0 ( ))(ln )xx

L T LCε ε

Ωε ρ ρ

, ; (58)

where C is a constant independent of ε . Proof of Theorem 1.2 Case 1 1 2γ< < . By Eq. (57),

44 2 13

2

(0 ( )) (0 ( ))L T L L T HCε

Ω Ωρ

, ; , ;∩ (59)

so that there exists 0n satisfying Eq. (9). Since Eqs. (55) and (57) imply

6 4min{ } 23 3

2

(0 ( ))t

L T HC

γε

Ωρ

, −+ , ;∂ ,

the Aubin-Lions lemma gives Eq. (11) and, hence, Eqs. (12) and (14) as well as Eq. (13) by standard elliptic estimates. In addition, Eq. (10) and Eq. (15) can be obtained directly from Eq. (57). Letting 0ε → in Eqs. (55) and (56) finishes the proof of this part.

Case 2 2 3γ . The proof is similar to the above.

6 Long-Time Behavior

This section analyzes the long-time behavior of Eq. (6) by employing the idea of Jüngel and Toscani[20].

Lemma 6.1 Let ( ) 1C x ≡ . Then the approximate solution, kρ , in Theorem 2.1 satisfies

(1) 2 2 2 21 1[ (ln 1) 1]d [ (ln 1) 1]dk k k kx x

Ω Ωρ ρ ρ ρ− −− + − − + +∫ ∫

22 2 2(ln ) d 0

2 k k xx xΩ

ε τ ρ ρ| |∫ (60)

2 2 2 21 1( ln )d ( ln )dk k k kx x

Ω Ωρ ρ ρ ρ− −− − − +∫ ∫

22 2(ln ) d 0

2 k xx xΩ

ε τ ρ| |∫ (61)

(2) 2 2[ (ln 1) 1]dk k xΩρ ρ − +∫

12 20 0 ( ) 2 2 2

ln 1(ln ) d

4L

k k xx xΩ

Ω

ρ ρρ ρ

− +| |∫ (62)

Proof Since 22

2 2 2

22 42

2

222

( )1 (ln ) d ( ) d4

( ) ( )2( ) d d d3

( )1( ) d d ,3

k xk k xx k xx

k

k x k xk xx

k k

k xk xx

k

x x

x x x

x x

Ω Ω

Ω Ω Ω

Ω Ω

ρρ ρ ρρ

ρ ρρρ ρ

ρρρ

| | = − =

| | + − =

| | +

∫ ∫

∫ ∫ ∫

∫ ∫

we used integration by parts for the last term with ( ) 0k xρ = on Ω∂ . Then, Eq. (29) comes from

2 2 2 21 1

22 2 2 1 2

2 2

1 1[ (ln 1) 1]d [ (ln 1) 1]d

(ln ) d 4 ( ) d2( 1)(1 )d

k k k k

k k xx k k x

k k

x x

x x

x

Ω Ω

γ

Ω Ω

Ω

ρ ρ ρ ρτ τ

ε ρ ρ γ ρ ρ

ρ ρ

− −

−

− + − − +

− | | − | | +

− −

∫ ∫

∫ ∫∫

and then 2 2 2 2

1 11 1[ (ln 1) 1]d [ (ln 1) 1]dk k k kx x

Ω Ωρ ρ ρ ρ

τ τ − −− + − − + +∫ ∫2

2 2 2 2 2(ln ) d ( 1) d 0,2 k k xx kx x

Ω Ω

ε ρ ρ ρ| | − −∫ ∫

which implies Eq. (60). Moreover, Eq. (31) implies 2 2 2 2

1 11 1( ln )d ( ln )dk k k kx x

Ω Ωρ ρ ρ ρ

τ τ − −− − − +∫ ∫

22 2 2 2(ln ) d (1 ) ln d 0,

2 k xx k kx xΩ Ω

ε ρ ρ ρ| | −∫ ∫

such that Eq. (61) is true. See Lemmas 3.1-3.3 in Jüngel and Toscani[20] for the proof of Eq. (62).

Lemma 6.2 Let ( ) 1C x ≡ and 2 20 0(ln 1) 1ρ ρ − + , 2

0ρ − 2 10ln ( ) .Lρ Ω∈ Then the weak solution, ρ, obtained

in Theorem 1.1 satisfies 0

122 2 2 2

0 0 ( )[ (ln 1) 1]d (ln 1) 1 e M t

Lx

ΩΩρ ρ ρ ρ −− + − + ,∫

(0 ],t T∀ ∈ , where 1

2 2 2 10 0 0 ( )

( ln 1)L

MΩ

ε ρ ρ −= − + .

Proof Repeating the arguments of Jüngel and Toscani[20], finishes the proof by Lemma 6.1.

Proof of Theorem 1.3 From the inequality 2( 1) (ln 1) 1 0a a a a− − + , ∀ > , deduce that 2( 1)ρ −

2 2(ln 1) 1ρ ρ − + and, hence, Eq. (26) with the aid of Lemma 6.2.

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Dirichlet-neumann problem for unipolar isentropic quantum drift-diffusion model