Régularité en temps long - Existene des solutions

3.3 Existene des solutions

3.3.4 Régularité en temps long

Commedanslapartie3.2.1, onpeut montrergrâeà(3.3.30)et(3.3.32)quel'entropie

E ^ε

déroît vers

0

^quand

t

^tends ^vers

+ ∞

^. Considérons maintenantl'entropielinéarisée (ave les notations

n ^ε = N ^ε − N _∞ ^ε

v ^ε = V ^ε − V _∞ ^ε

^):

Des aluls similairesà lapartie 3.2.2 amènent àl'inegalité:

d

dt L ^ε (t) ≤ − C 1 L ^ε (t) + C 2 L ^ε (t) ⁴ ,

^(3.3.34)

où

C 1

^et

C 2

^sont indépendantes de

ε

^. ^Dans ^la ^partie ^3.2.2 ^on ^a ^démontré ^l'existene

t ^ε _∗ > 0

^tel ^que

L ^ε (t ^ε _∗ ) ³ ≤ C 1 /(2C 2 )

^. Âlors îl êxiste ^deux ônstantes ^positives

C

^et

λ

indépendantes de

ε

^telle ^que ^pour ^tout

t ≥ t ^ε _∗

L ^ε (t) ≤ Ce ^−λt

^. ^Le ^point ^ruial ^est ^de

trouver un tel

t ^ε _∗

^borné uniformémentpar rapportà

ε

Tout d'abord, on peut prouver failement par une argumentation par l'absurde que

pour tout

η > 0

^il ^existe

α(η) > 0

^tel^que,

Ensuite, l'inégalité(3.3.31)implique qu'il existe une onstant positive

C ₀

^telle ^que

∀ T > 0, ∃ t ^ε (T ) ∈ [0, T ]

^tel ^que

Don , en hoisissant

η = (C 1 /(2C 2 )) ^1/3

^dans ^(3.3.35) ^et

T ≥ C 0 /α(η)

^dans ^(3.3.36), ^on

déduit que le

t ^ε (T )

orrespondant peut être pris omme notre

t ^ε _∗

Finalement, sur

[T, ∞ )

^on ^a l'estimation

L ^ε (t) ≤ Ce ^−λt

^. ^Comme ^pour ^tout

t ≥ T

^on

L(t) ≤ lim inf ε→0 L ^ε (t) ≤ Ce ^−λt

⁽

L

^déni ^en ^(4.3.11)), ^on ^déduit ^que

N

^appartient ^à

C([T, ∞ ), L ² (Ω))

Referenes

[1℄ A.Arnold,P.Markowih,G.Tosani,A.Unterreiter,OngeneralizedCsiszár-Kullbak

inequalities,Monatsh. Math. 131 (2000), no. 3,235253.

[2℄ A. Arnold,P. A. Markowih, G. Tosani, A.Unterreiter, On onvex Sobolev

inequal-ities and the rate of onvergene to equilibrium for Fokker-Plank type equations,

Comm. Partial Dierential Equations26 (2001), no. 1-2, 43100.

[3℄ A. Arnold, P. Markowih, G. Tosani, On Large Time Asymptotis for

Drift-Diusion-Poisson Systems, Transport TheoryStatist. Phys. 29(2000), no. 3-5, 571

581.

[4℄ N.BenAbdallah,Weaksolutionsof theinitial-boundaryvalueproblemforthe

Vlasov-Poisson system, Math. MethodsAppl. Si. 17 (1994),no. 6,451476.

[5℄ N. Ben Abdallah,F. Méhats, N. Vauhelet, Diusivetransport of partially quantized

partile: existene, uniquenessandlongtime behavior,Pro.Edin.Math.So.(2006)

49, 513549.

[6℄ N. BenAbdallah,F. Méhats,N.Vauhelet,Anote onthe longtimebehaviour forthe

drift-diusion-Poisson system,C. R.Aad. Si. Paris, Ser. I 339 (2004), 683688.

[7℄ N. Ben Abdallah, P. Degond, On a hierarhyof marosopimodelsfor

semiondu-tors, J.Math. Phys. 37 (1996),no. 7, 33063333.

[8℄ A. Blanhet, JDolbeault,B. Perthame, Two-dimensional Keller-Segelmodel :

Opti-malritialmassandqualitative propertiesof thesolutions,preprinteremade(2006).

[9℄ P. Biler, J. Dolbeault, Long time behavior of solutions of Nernst-Plank and

Debye-Hükel drift-diusion systems, Ann. Henri Poinaré 1 (2000),no. 3,461472.

[10℄ P. Biler, J. Dolbeault, P. A. Markowih, Large time asymptotis of nonlinear

drift-diusion systems with Poisson oupling, Transport Theory Statist. Phys. 30 (2001),

no. 4-6, 521536.

[11℄ J. A. Carrillo, G. Tosani, Asymptoti

L ¹

^-deay ^of ^solutions ^of ^the ^porous ^medium

equationto self-similarity, Indiana Univ. Math. J. 49 (2000),no. 1,113142.

[12℄ I. Csiszár, Information-type measures of dierene of probability distributions, Stud.

[13℄ M. Del Pino, J. Dolbeault, Best onstants for Gagliardo-Nirenberg inequalities and

appliations to nonlinear diusions,J. Math. Pures Appl. (9) 81 (2002), no. 9, 847

875.

[14℄ M. Del Pino, J. Dolbeault, Generalized Sobolev inequalities and asymtoti behaviour

in fast diusion and porous medium problems, preprint eremade, 1999.

[15℄ J.Dolbeault,Stationarystatesinplasmaphysis: Maxwelliansolutionsofthe

Vlasov-Poisson system, Math. Models Methods Appl. Si. 1 (1991), no. 2, 183208.

[16℄ H. Gajewski,On Existene,Uniqueness and Asymptoti Behavior of Solutions of the

Basi Equations for Carrier Transport in Semiondutors, Z. Angew. Math. Meh.

65 (1985)2, 101-108.

[17℄ H. Gajewski, K. Gröger, Semiondutor equations for variable mobilities based on

Boltzmann statistis or Fermi-Dira statistis, Math. Nahr., 140 (1989), 7-36.

[18℄ I. Gasser, C. D. Levermore, P. A. Markowih, C. Shmeiser, The initial time layer

problem and the quasineutral limit in the semiondutor drift-diusion model,

Euro-pean J. Appl. Math. 12 (2001), no. 4,497512.

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LetureNotesinMathematis1563,E.Fabesetal.(Eds.)DirihletForms,Springer

Berlin-Heidelberg, 1993.

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Equations of Paraboli Type, Translations of Mathematial Monographs, A.M.S.,

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Springer-Verlag, Vienna, 1990.

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Numerial ComputationSeries 3,BoolePress, 1983.

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Théorie

L ²

^pour ^le ^système ^DDSP

(dérive-diusion-Shrödinger-Poisson)

Joint workwithN.Ben Abdallah 1

andF. Méhats 2

4.1 Introdution and main result

In this hapter, we present the analysis of the DDSP system introduedin the beginning

of thisPhDinthease wherethediusionmatrix isassumedtobethe identity. Wereall

that this system models the transportof apartilesystem whihis partiallyquantized in

one diretion (denoted by

z

⁾ ^and ^whih, ⁱⁿ ^the ^transport ^diretion ^denoted ^by

x

^, ^has ^a

diusivemotion. Thesystemisatequilibriumintheonned diretionwithaloalFermi

level

ǫ F

^whih ^depends ^on^the ^transport^variable

x

^. ^The ^variable

x

îs âssumed ^to^lie ⁱⁿâ

boundedregulardomain

ω ∈ R ²

while

z

^belongs^to^the ^interval

(0, 1)

^. ^The^spatial^domain

isthen

Ω = ω × (0, 1)

^. ^At^a^time

t

^and^a^position

(x, z)

^,^we^reall ^that^the ^partile^density

N (t, x, z)

^is ^given ^by ^:

N (t, x, z) = N s (t, x) X +∞

k=1

e ⁻ ^ǫ ^k ^(t,x)

Z (t, x) | χ k (t, x, z) | ² ,

^(4.1.1)

where the repartition funtion

Z

^is ^dened ^by

Z (t, x) =

X +∞

k=1

e ⁻ ^ǫ ^k ^(t,x) ,

^(4.1.2)

MIP,UMR5640,UniversitéPaulSabatier,Toulouse.

IRMAR, UniversitéRennes1,Rennes.

70 Chapter 4. Théorie

L ²

^pour ^le^système ^DDSP

and

(χ k , ǫ k )

îs ^the ômplete ^set ôf eigenfuntions and eigenvalues of the Shrödinger op-erator in the

z

^variable

 



 

− 1

2 ∂ _z ² χ _k + V χ _k = ǫ k χ _k (k ≥ 1), χ k (t, x, · ) ∈ H ₀ ¹ (0, 1),

Z 1 0

χ k χ ℓ dz = δ kℓ .

(4.1.3)

The eletrostati potential

V

îs â^solution ôf ^the ^Poisson êquation

− ∆ _x,z V = N.

^(4.1.4)

The surfae density

N s

^satises^the drift-diusionequation

∂ t N s − div x ( ∇ ^x N s + N s ∇ ^x V s ) = 0,

^(4.1.5)

where the eetive potential

V s

^is ^given ^by

V s = − log X

k

e ⁻ ^ǫ ^k .

^(4.1.6)

The unknowns of the overall system are the surfae density

N s (t, x)

^, ^the eigenenergies

ǫ k (t, x)

^,^the eigenfuntions

χ k (t, x, z)

^and ^theeletrostatipotential

V (t, x, z)

^. ^The ^F^ermi

level

ǫ F

^is ^determined^by^:

ǫ F (t, x) = log N s (t, x)

Z (t, x) .

^(4.1.7)

This will be useful for the study of global equilibria. The system (4.1.3)(4.1.2) is

om-pleted with the initialondition

N s (0, x) = N _s ⁰ (x)

^(4.1.8)

and with the following boundary onditions:

( N s (t, x) = N b (x), V (t, x, z) = V b (x, z),

^for

x ∈ ∂ω, z ∈ (0, 1),

∂ z V (t, x, 0) = ∂ z V (t, x, 1) = 0,

^for

x ∈ ω.

(4.1.9)

InappliationliketheDouble-Gatetransistor[4℄,thefrontier

∂ω × [0, 1]

^inludes^the^soure

and the drainontats aswellasinsulatingorartiialboundary. On the otherhand

ω × { 0 }

^and

ω ×{ 1 }

^represent^the^gateôntats⁽ⁱⁿâddition^to^possibleînsulatingboundaries).

Mixed type boundary onditions are then to be presribed. The boundary onditions

(4.1.9) do not take into aount this omplexity and are hosen for the mathematial

onveniene: ellipti regularity properties of the Poisson equation (4.1.4) are needed in

Dans le document Modélisation mathématique du transport diffusif de charges partiellement quantiques. (Page 66-72)

Régularité en temps long

3.3 Existene des solutions

3.3.4 Régularité en temps long

E ε

0

t

+ ∞

n ε = N ε − N ∞ ε

v ε = V ε − V ∞ ε

d

dt L ε (t) ≤ − C 1 L ε (t) + C 2 L ε (t) 4 ,

C 1

C 2

ε

t ε ∗ > 0

L ε (t ε ∗ ) 3 ≤ C 1 /(2C 2 )

C

λ

ε

t ≥ t ε ∗

L ε (t) ≤ Ce −λt

t ε ∗

ε

η > 0

α(η) > 0

C 0

∀ T > 0, ∃ t ε (T ) ∈ [0, T ]

η = (C 1 /(2C 2 )) 1/3

T ≥ C 0 /α(η)

t ε (T )

t ε ∗

[T, ∞ )

L ε (t) ≤ Ce −λt

t ≥ T

L(t) ≤ lim inf ε→0 L ε (t) ≤ Ce −λt

L

N

C([T, ∞ ), L 2 (Ω))

L 1

L 2

z

x

ǫ F

x

x

ω ∈ R 2

z

(0, 1)

Ω = ω × (0, 1)

t

(x, z)

N (t, x, z)

N (t, x, z) = N s (t, x) X +∞

k=1

e − ǫ k (t,x)

Z (t, x) | χ k (t, x, z) | 2 ,

Z

Z (t, x) =

X +∞

k=1

e − ǫ k (t,x) ,

L 2

(χ k , ǫ k )

z

 



 

− 1

2 ∂ z 2 χ k + V χ k = ǫ k χ k (k ≥ 1), χ k (t, x, · ) ∈ H 0 1 (0, 1),

Z 1 0

χ k χ ℓ dz = δ kℓ .

V

− ∆ x,z V = N.

N s

∂ t N s − div x ( ∇ x N s + N s ∇ x V s ) = 0,

V s

V s = − log X

k

e − ǫ k .

N s (t, x)

E ^ε

n ^ε = N ^ε − N _∞ ^ε

v ^ε = V ^ε − V _∞ ^ε

dt L ^ε (t) ≤ − C 1 L ^ε (t) + C 2 L ^ε (t) ⁴ ,

t ^ε _∗ > 0

L ^ε (t ^ε _∗ ) ³ ≤ C 1 /(2C 2 )

t ≥ t ^ε _∗

L ^ε (t) ≤ Ce ^−λt

t ^ε _∗

C ₀

∀ T > 0, ∃ t ^ε (T ) ∈ [0, T ]

η = (C 1 /(2C 2 )) ^1/3

t ^ε (T )

t ^ε _∗

L ^ε (t) ≤ Ce ^−λt

L(t) ≤ lim inf ε→0 L ^ε (t) ≤ Ce ^−λt

C([T, ∞ ), L ² (Ω))

L ¹

L ²

ω ∈ R ²

e ⁻ ^ǫ ^k ^(t,x)

Z (t, x) | χ k (t, x, z) | ² ,

e ⁻ ^ǫ ^k ^(t,x) ,

L ²

2 ∂ _z ² χ _k + V χ _k = ǫ k χ _k (k ≥ 1), χ k (t, x, · ) ∈ H ₀ ¹ (0, 1),

− ∆ _x,z V = N.

∂ t N s − div x ( ∇ ^x N s + N s ∇ ^x V s ) = 0,

e ⁻ ^ǫ ^k .

N s (0, x) = N _s ⁰ (x)