Solutions to the Equations of Viscous Quantum Hydrodynamics in Multiple Dimensions

Solutions to the Equations of Viscous Quantum Hydrodynamics in Multiple Dimensions

Michael Dreher

Konstanzer Schriften in Mathematik und Informatik Nr. 215, Mai 2006

ISSN 1430–3558

c Fachbereich Mathematik und Statistik

c Fachbereich Informatik und Informationswissenschaft Universit¨at Konstanz

Fach D 188, 78457 Konstanz, Germany Email: preprints@informatik.uni–konstanz.de

WWW: http://www.informatik.uni–konstanz.de/Schriften/

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/2235/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-22352

Viscous Quantum Hydrodynamics in Multiple Dimensions

Michael Dreher^∗

Abstract

We study the viscous model of quantum hydrodynamics in a bounded domain of space di- mension 1, 2, or 3. This model is a mixed order partial differential system with nonlocal and nonlinear terms for the particle density, current density and electric potential. By a viscous regularization approach, we show existence and uniqueness of local in time solutions.

2000 Mathematics Subject Classification: 35B40, 35Q35, 76Y05

Keywords: quantum hydrodynamics, existence, uniqueness and persistence of solutions, boundary conditions of Zaremba type.

1 Introduction

Depending on the size of a semiconductor device and other physical aspects, there are several different models describing the flow of charged particles. As examples, we mention the (quantum) drift diffusion model, the (quantum) energy transport model, or the (quantum) hydrodynamic model. Derivations of such models can be found in, e.g., [10]. The quantum hydrodynamic model can be derived from the Schr¨odinger–Poisson system by WKB wave functions ([7]), or from the Wigner equations via the moment method, together with a closure of the system with the thermal equilibrium distribution ([4]). Another derivation exploits the entropy minimization principle ([8]).

Taking into account collisions of the charged particles with the background oscillators, one obtains theviscousquantum hydrodynamic model, as it can be derived from the Wigner equation using the Fokker-Planck collision operator:



















∂_tn−divJ =ν₀△n,

∂tJ −div

J⊗J n

−T∇n+n∇V +ε² 2 n∇

△√

√ n n

=ν0△J− J τ, λ²△V =n−C(x), (n, J)(0, x) = (n0, J0)(x),

(1.1)

where (t, x)∈(0,∞)×Ω, and Ω⊂R^d (d= 1,2,3) is a bounded domain with boundary Γ =∂Ω of regularityC⁴. The unknown functions are the particle densityn:R₊×Ω→R₊, the current density

∗Fachbereich Mathematik und Statistik, P.O.Box D187, Universit¨at Konstanz, 78457 Konstanz, Germany, michael.dreher@uni-konstanz.de

1

J:R₊×Ω→R^d, and the electrostatic potential V:R₊×Ω→R. The functionC: Ω→Rmodels the given profile of background charges. The (scaled) physical constants are the temperature T, the Planck constant ε, the Debye length λ, and a viscosity constant ν₀ as well as the momentum relaxation time τ, which are related to the collision operator. All these constants are supposed to be positive.

The boundary conditions for the unknowns (n, J, V) are







∂_νn(t, x) = 0, J(t, x) =J_Γ(x), (KV)(t, x) =g(x),

(t, x)∈(0,∞)×Γ, (1.2)

where ∂_ν denotes the outward normal derivative, and the last line is an abbreviation for mixed Dirichlet-Neumann conditions as follows: we assume the boundary Γ to be split into two sub- manifolds, where the boundary part ΓN is allowed to be empty.

Γ = Γ_D ∪Γ_N, Γ_D∩Γ_N =∅, Γ_D∩Γ_N =Z,

with Z being a manifold of dimension d−2 or the empty set. We are given a pair of functions g= (g_D, g_N), and the function V has to satisfy boundary conditions of Zaremba type,

( V(t, x) =gD(x) : (t, x)∈(0,∞)×ΓD,

∂_νV(t, x) =g_N(x) : (t, x)∈(0,∞)×Γ_N. We require the compatibility conditions

∂_νn₀(x) = 0, J₀(x) =J_Γ(x), x∈Γ, (1.3)

x∈Ωinf n0(x)>0. (1.4)

For transient quantum hydrodynamic models, only a few analytic results are available. We mention the existence of smooth solutions to the inviscid model (ν₀= 0) and their asymptotic behavior for large time and small initial data, as investigated in [9, 13].

Concerning the transient viscous model (1.1), the exponential stability of a constant steady state was proved in [6] for the one-dimensional case, and in [2] for the higher-dimensional case. The local existence and uniqueness of solutions was shown in [2] for a one-dimensional setting with insulating boundary conditions, and for the case of higher dimensions with periodic boundary conditions. It seems that the viscous transient model (1.1) in higher dimensions has been analytically investigated for the first time in [2].

A few remarks about the strategy of the approach of this paper are in order. Putting U = (n, J1, . . . , J_d)^T and observing that the quantum correction term (also called Bohm potential term) can be expressed as

ε² 2n∇

△√

√ n n

= ε²

4∇△n−ε²div (∇√

n)⊗(∇√ n)

,

we can reformulate the equations fornand J from (1.1) as

∂_tU +A(∂_x)U+ 0

G

= 0,

A(∂x) =−ν0△I_d+

















0 −∂₁ −∂₂ · · · −∂_d

ε²

4∂₁△ −T ∂₁ τ⁻¹ 0 · · · 0

ε²

4∂2△ −T ∂1 0 τ⁻¹ · · · 0 ... ... ... . .. ...

ε²

4∂_d△ −T ∂_d 0 0 · · · τ⁻¹















 ,

G=−div

J ⊗J n

+n∇V −ε²div (∇√

n)⊗(∇√ n)

. (1.5)

It turns out that the matrix Ais an elliptic differential matrix operator of mixed order in the sense of Douglis and Nirenberg, see [1] or [5]. This suggests to derivea prioriestimates in a similar fashion as for parabolic systems, after having approximated the system by an introduction of a fourth order viscous regularization.

The differences between this paper and [2] are twofold: first, we do no longer require insulating or periodic boundary conditions. And second, we greatly relax the conditions on the doping profileC, which is now merely in L²(Ω).

Acknowledgments. The author has been supported by the Deutsche Forschungsgemeinschaft (DFG), grant number 446 CHV 113/170. Moreover, I would like to thank Li Chen of Tsinghua University, Beijing, for helpful discussions.

2 Main Results

Our notations are standard: L^p denote the usual Lebesgue spaces of real-valued functions, and H^k(Ω) :=W₂^k(Ω) are the L²–based Sobolev spaces, for k∈N₀. The expression h·,·i stands for the scalar product inL²(Ω) as well as in (L²(Ω))^d.

Theorem 2.1. Suppose n0 ∈ H³(Ω), J0 ∈ H²(Ω), JΓ ∈ H^3/2(Γ), C ∈ L²(Ω), gD ∈ H^3/2(ΓD), g_N ∈H^1/2(Γ_N) the compatibility condition (1.3), and (1.4).

Then the system (1.1)with the boundary conditions (1.2)has a unique local in time solution(n, J, V) with

n∈L^∞((0, t∗), H³(Ω)), J ∈L^∞((0, t∗), H²(Ω)),

∂_tn∈L²((0, t_∗), H²(Ω)), ∂_tJ ∈L²((0, t_∗), H¹(Ω)), (n,∇n, J)∈C([0, t_∗)×Ω),

V ∈C([0, t_∗), H_loc² (Ω))∩C([0, t_∗), H¹(Ω)),

∂_tV ∈L²((0, t_∗), H¹(Ω)).

This solution persists as long as n stays positive and(n,∇n, J) are bounded in L^∞(Ω).

Remark 2.2. After slight modifications of the proof, time-dependent boundary conditions can be treated, too.

3 Proof of Theorem 2.1

We introduce a viscous regularization term γ△² into (1.1). Its purpose is to ensure the existence of approximate solutions (to be shown in the appendix); we will never use it for a prioriestimates:































∂tnγ+γ△²nγ−ν0△nγ−divJγ= 0,

∂_tJ_γ+γ△²J_γ−ν₀△J_γ+ε²

4∇△n_γ

−T∇n_γ+ 1

τJ_γ+G_γ= 0,

λ²△V_γ=n_γ−C(x), (n_γ, J_Γ)(0, x) = (n_0,γ, J_0,γ)(x),

(∂νnγ, Jγ, KVγ)(t, x) = (0, JΓ,γ(x), g(x)) on (0,∞)×Γ, (3.1)

where 0< γ <1 and Gγ =−div

J_γ⊗J_γ n_γ

+nγ∇Vγ−ε²div (∇√nγ)⊗(∇√nγ)

. (3.2)

Additionally, we require the boundary conditions

∂_ν△n_γ(t, x) = 0, △J_γ(t, x) = 0, (t, x)∈(0,∞)×Γ. (3.3) The given data (n_0,γ, J_0,γ, J_Γ,γ)∈H⁶(Ω)×H⁶(Ω)×H⁴(Γ) are suitably constructed approximations for (n₀, J₀, J_Γ); and, for γ →0, we have convergence

n_0,γ →n₀ in H³(Ω), J_0,γ →J₀ in H²(Ω), J_Γ,γ →J_Γ in H^3/2(Γ).

We define a functionJD,γ ∈H⁴(Ω) as solution to the elliptic boundary value problem (△J_D,γ(x) = 0 :x∈Ω,

J_D,γ(x) =J_Γ,γ(x) :x∈Γ,

and get uniform in γ estimateskJD,γk_H²_(Ω)≤C. Note that △JD,γ ∈C(Ω) due to the continuous embedding H²(Ω)⊂C(Ω), hence△J_D,γ= 0 on Γ.

According to Proposition A.1, this regularized system has a unique solution (n_γ, J_γ, V_γ) with (n_γ, J_γ)∈C([0, t_γ], H³(Ω))∩L^∞((0, t_γ), H⁴(Ω)),

(∂_tn_γ, ∂_tJ_γ)∈L^∞((0, t_γ), H²(Ω))∩L²((0, t_γ), H⁴(Ω)), (V_γ, ∂_tV_γ)∈L^∞((0, t_γ), H¹(Ω)),

and the solution persists as long as (nγ,∇nγ, Jγ) stay bounded in L^∞(Ω). The time derivatives of n_γ andJ_γ satisfy the same boundary conditions asn_γ,J_γ−J_D,γ.

Choose a numberδ₀>0, subject to the conditions δ0 ≤ 1

2 inf

x∈Ωn0(x), 2 max

kn0k_L^∞_(Ω),k∇n0k_L^∞_(Ω),kJ0k_L^∞_(Ω)

≤δ⁻¹₀ .

By the continuous embedding H³(Ω)⊂C¹(Ω), we can assume that δ₀ ≤ inf

x∈Ωn_γ(t, x), max

kn_γ(t,·)k_L^∞_(Ω),k∇n_γ(t,·)k_L^∞_(Ω),kJ_γ(t,·)k_L^∞_(Ω)

≤δ₀⁻¹, (3.4) for 0 ≤t≤t_γ; otherwise, we shrink the interval [0, t_γ]. In the following, we will prove uniform in γ estimates of (nγ, Jγ, Vγ), which will imply that the life span tγ can not tend to zero for γ going to zero. Then we will have a uniform existence interval as well as uniform estimates and can prove the convergence of a subsequence (n_γ, J_γ, V_γ)_γ by compactness arguments.

In the sequel, C will denote a generic constant which may change from line to line and is allowed to depend on δ₀ andkJ_D,γk_H²_(Ω), but notγ.

First of all: because of kg_DkH^3/2(ΓD) ≤ C, kg_NkH^1/2(ΓN) ≤ C and kn_γ(t,·)−C(·)k_L²_(Ω) ≤ C, we obtain

kVγ(t,·)k_H¹_(Ω)≤C(1 +λ⁻²), 0≤t≤tγ.

For later reference, we remark that the embedding H¹(Ω)⊂L⁴(Ω) then yields a uniform estimate of kVγ(t,·)k_L⁴_(Ω). Similarly, we have

k∂_tV_γ(t,·)k_L⁴_(Ω) ≤Ck∂_tV_γ(t,·)k_H¹_(Ω)≤Cλ⁻²k∂_tn_γk_L²_(Ω).

We multiply the parabolic equation for nγ, Jγ, respectively, with nγ or Jγ −JD,γ, respectively, integrate over Ω, and perform partial integration where appropriate:

1

2∂_tkn_γk²_L²_(Ω)+ν₀k∇n_γk²_L²_(Ω)+γk△n_γk²_L²_(Ω)− hdivJ_γ, n_γi= 0, 1

2∂_tkJ_γ−J_D,γk²_L²_(Ω)+ν₀k∇(J_γ−J_D,γ)k²_L²_(Ω)+γk△J_γk²_L²_(Ω)+ 1

τ kJ_γk²_L²_(Ω) +Thn_γ,div(J_γ−J_D,γ)i −ε²

4 h△n_γ,div(J_γ−J_D,γ)i

− 1

τ hJ_γ, J_D,γi+hG_γ, J_γ−J_D,γi= 0.

Here,k∇J_γk_L²_(Ω) stands for the Frobenius norm: k∇J_γk²=P

k,lk∂_kJ_γ,lk². The linear contribution from the quantum term is handled by means of

h△n_γ,divJ_γi=h△n_γ, ∂_tn_γi − h△n_γ, ν₀△n_γi+

△n_γ, γ△²n_γ

=−1

2∂tk∇nγk²_L²_(Ω)−ν0k△nγk²_L²_(Ω)−γk∇△nγk²_L²_(Ω), which gives us the identity

1 2∂_t

kJ_γ−J_D,γk²_L²_(Ω)+Tkn_γk²_L²_(Ω)+ ε²

4 k∇n_γk²_L²_(Ω)

(3.5) +ν₀k∇(J_γ−J_D,γ)k²_L²_(Ω)+T ν₀k∇n_γk²_L²_(Ω)+ε²

4ν₀k△n_γk²_L²_(Ω)+ 1

τ kJ_γk²_L²_(Ω) +γ

k△J_γk²_L²_(Ω)+Tk△n_γk²_L²_(Ω)+ε²

4 k∇△n_γk²_L²_(Ω)

=Thn_γ,divJ_D,γi −ε²

4 h△n_γ,divJ_D,γi+1

τ hJ_γ, J_D,γi − hG_γ, J_γ−J_D,γi.

In system (3.1), we take time derivatives, and define

n^′_γ :=∂_tn_γ, J_γ^′ :=∂_tJ_γ, G^′_γ:=∂_tG(n_γ, J_γ, V_γ).

Then we deduce, by a similar computation as before, that 1

2∂_t

J_γ^′

2

L²(Ω)+T n^′_γ

2

L²(Ω)+ε² 4

∇n^′_γ

2 L²(Ω)

(3.6) +ν₀

∇J_γ^′

2

L²(Ω)+T ν₀ ∇n^′_γ

2

L²(Ω)+ε² 4ν₀

△n^′_γ

2

L²(Ω)+ 1 τ

J_γ^′

2 L²(Ω)

+γ

△J_γ^′

2

L²(Ω)+T △n^′_γ

2

L²(Ω)+ε² 4

∇ △n^′_γ

2 L²(Ω)

=−

G^′_γ, J_γ^′ .

The scalar productshG_γ, J_γ−J_D,γi and

G^′_γ, J_γ^′

have the representations hG_γ, J_γ−J_D,γ,li=I₁+I₂+I₃

=X

l

Z

Ω

J_γ,l nγ

J_γ∇(J_γ,l−J_D,γ,l) dx+ Z

Ω

n_γ(∇V_γ)(J_γ−J_D,γ) dx +ε²X

l

Z

Ω

(∂_l√n_γ)(∇√n_γ)∇(J_γ,l−J_D,γ,l) dx, G^′_γ, J_γ^′

=I₁^′ +I₂^′ +I₃^′

=X

l

Z

Ω

∂_tJ_γ,l nγ

J_γ

(∇J_γ,l^′ ) dx+ Z

Ω

(∂_tn_γ(∇V_γ))J_γ^′dx +ε²X

l

Z

Ω

∂_t(∂_l√n_γ)(∇√n_γ)

(∇J_γ,l^′ ) dx, and the items I_k can be estimated as follows:

|I₁| ≤Ck∇(J_γ−J_D,γ)k_L²_(Ω)≤ ν₀

4 k∇(J_γ−J_D,γ)k²_L²_(Ω)+Cν₀⁻¹,

|I₂| ≤CkV_γk_H¹_(Ω)kJ_γ−J_D,γk_L²_(Ω)≤C(1 +λ⁻²)kJ_γ−J_D,γk_L²_(Ω)

≤C(1 +λ⁻⁴) +kJ_γ−J_D,γk²_L²_(Ω),

|I3| ≤Cε²k∇(Jγ−JD,γ)k_L²_(Ω)≤ ν0

4 k∇(Jγ−JD,γ)k²_L²_(Ω)+Cε⁴ν₀⁻¹. In treating the other terms I_k^′, note that

n^′_γ

L⁴(Ω) ≤ C(1 + ∇n^′_γ

L²(Ω)) due to the identity R

Ωn^′_γdx=R

∂ΩJ_Γ,γ·~νdσ and the Poincare-Sobolev inequality:

|I₁^′| ≤ ν₀ 6

∇J_γ^′

2

L²(Ω)+Cν₀⁻¹ J_γ^′

2

L²(Ω)+ n^′_γ

2 L²(Ω)

,

|I₂^′| ≤ ∇n^′_γ

L²(Ω)kVγk_L⁴_(Ω) J_γ^′

L⁴(Ω)+k∇nγk_L^∞_(Ω)k∂tVγk_L²_(Ω) J_γ^′

L²(Ω)

+ n^′_γ

L⁴(Ω)kV_γk_L⁴_(Ω) ∇J_γ^′

L²(Ω)+kn_γk_L^∞_(Ω)k∂_tV_γk_L²_(Ω) ∇J_γ^′

L²(Ω)

≤C ∇n^′_γ

L²(Ω)(1 +λ⁻²) ∇J_γ^′

L²(Ω)+Cλ⁻² n^′_γ

L²(Ω)

J_γ^′

H¹(Ω)

≤ ν0

6 ∇J_γ^′

2

L²(Ω)+C(ν₀⁻¹+λ⁻⁴ν₀⁻¹) ∇n^′_γ

2 L²(Ω)

+C(ν₀⁻¹+λ⁻⁴ν₀⁻¹+λ⁻⁴) n^′_γ

2

L²(Ω)+ J_γ^′

2 L²(Ω),

|I₃^′| ≤Cε² ∇n^′_γ

L²(Ω)

∇J_γ^′

L²(Ω)≤ ν₀ 6

∇J_γ^′

2

L²(Ω)+Cε⁴ν₀⁻¹ ∇n^′_γ

2 L²(Ω). Plugging these estimates into (3.5) and (3.6), we then find

∂_t

kJ_γ−J_D,γk²_L²_(Ω)+Tkn_γk²_L²_(Ω)+ε²

4 k∇n_γk²_L²_(Ω)

+ν0k∇(Jγ−JD,γ)k²_L²_(Ω)+ε²ν0

4 k△nγk²_L²_(Ω)

≤C 1 +ν₀⁻¹+λ⁻⁴+ε⁴ν₀⁻¹+T +τ⁻¹

+kJ_γ−J_D,γk²_L²_(Ω),

∂_t

J_γ^′

2

L²(Ω)+T n^′_γ

2

L²(Ω)+ε² 4

∇n^′_γ

2 L²(Ω)

+ν0

∇J_γ^′

2

L²(Ω)+ 2T ν0

∇n^′_γ

2

L²(Ω)+ε²ν0

2 △n^′_γ

2 L²(Ω)

≤C ν₀⁻¹+ 1 J_γ^′

2

L²(Ω)+C ν₀⁻¹+λ⁻⁴ν₀⁻¹+λ⁻⁴ n^′_γ

2 L²(Ω)

+C ν₀⁻¹+λ⁻⁴ν₀⁻¹+ε⁴ν₀⁻¹ ∇n^′_γ

2 L²(Ω)

≤C ν₀⁻¹+ 1 + (ν₀⁻¹+λ⁻⁴ν₀⁻¹+λ⁻⁴)T⁻¹+ (ν₀⁻¹+λ⁻⁴ν₀⁻¹+ε⁴ν₀⁻¹)ε⁻²

×

×

J_γ^′

2

L²(Ω)+T n^′_γ

2

L²(Ω)+ε² 4

∇n^′_γ

2 L²(Ω)

. We can assume that

1 +ν₀⁻¹+λ⁻⁴+ε⁴ν₀⁻¹+T+τ⁻¹

tγ ≤1,

ν₀⁻¹+ 1 + (ν₀⁻¹+λ⁻⁴ν₀⁻¹+λ⁻⁴)T⁻¹+ (ν₀⁻¹+λ⁻⁴ν₀⁻¹+ε⁴ν₀⁻¹)ε⁻²

t_γ ≤1.

Making use of Gronwall’s Lemma, we then conclude that kJ_γ−J_D,γk²_L^∞_((0,t),L²_(Ω))+Tkn_γk²_L^∞_((0,t),L²_(Ω))+ε²

4 k∇n_γk²_L^∞_((0,t),L²_(Ω)) +ν0k∇(Jγ−J_D,γ)k²_L²_(Qt)+ε²ν0

4 k△nγk²_L²_(Qt)

≤C

1 +kJ0,γ−JD,γk²_L²_(Ω)+Tkn0,γk²_L²_(Ω)+ε²

4 k∇n0,γk²_L²_(Ω)

,

J_γ^′

2

L^∞((0,t),L²(Ω))+T n^′_γ

2

L^∞((0,t),L²(Ω))+ε² 4

∇n^′_γ

2

L^∞((0,t),L²(Ω))

+ν₀ ∇J_γ^′

2

L²(Qt)+ 2T ν₀ ∇n^′_γ

2

L²(Qt)+ε²ν0

2 △n^′_γ

2 L²(Qt)

≤C

J_γ^′(0,·)

2

L²(Ω)+T

n^′_γ(0,·)

2

L²(Ω)+ε² 4

∇n^′_γ(0,·)

2 L²(Ω)

,

where 0≤t≤t_γ andQ_t:= (0, t)×Ω. The time derivatives att= 0 can be estimated as follows:

J_γ^′(0,·)

L²(Ω)≤γkJ0,γk_H⁴_(Ω)+ν0kJ0,γk_H²_(Ω)+ ε²

4 kn0,γk_H³_(Ω)+C(δ0)T +C

kJ_0,γk_H¹_(Ω)+kn_0,γk_H¹_(Ω)

+Cλ⁻²+Cε²kn_0,γk_H²_(Ω),

n^′_γ(0,·)

H¹(Ω) ≤γkn0,γk_H⁵_(Ω)+ν0kn0,γk_H³_(Ω)+kJ0,γk_H²_(Ω).

We may assume that the numbers γ are chosen in such a way that

γ→0lim

γkJ_0,γk_H⁴_(Ω)+γkn_0,γk_H⁵_(Ω)

= 0.

With this choice of γ, the following uniform in γ estimates have been obtained:

kn_γk_{C([0,tγ]×Ω)}+k∇n_γk_{C([0,tγ]×Ω)}+kJ_γk_{C([0,tγ]×Ω)}≤C₀, kJ_γk_L²_((0,tγ),H¹_(Ω))+kn_γk_L²_((0,tγ),H²_(Ω))≤C₀, J_γ^′

L^∞((0,tγ),L²(Ω))+ n^′_γ

L^∞((0,tγ),H¹(Ω))≤C₀,

J_γ^′

L²((0,tγ),H¹(Ω))+ n^′_γ

L²((0,tγ),H²(Ω))≤C₀,

where the constant C₀ is allowed to depend on all physical constants,δ₀, and the normskn₀kH³(Ω), kJ₀kH²(Ω),kJ_D,γk_H²_(Ω), but notγ.

To get more estimates, we consider certain elliptic boundary value problems:

Definition 3.1. Forf ∈L²(Ω) andν₀, γ >0, let u=Q_γf ∈H²(Ω) be the solution to the problem ((ν0−γ△)u(x) =f(x) :x∈Ω,

∂_νu(x) = 0, :x∈Γ,

Lemma 3.2. The operator Qγ is a bounded endomorphism on L²(Ω)and on H¹(Ω), kQ_γfk_L²_(Ω) ≤ 1

ν0kfkL²(Ω), k∇Q_γfk_L²_(Ω) ≤ 1

ν0k∇fkL²(Ω).

The operator Q_γ△, first defined on H²(Ω), extends to a continuous endomorphism on L²(Ω):

kQ_γ△fk_L²_(Ω) ≤ 1

γkfk_L²_(Ω), f ∈L²(Ω).

Moreover, Q_γ△is non-positive:

hQ_γ△f , fi ≤0, f ∈L²(Ω).

Proof. Write u = Q_γf. Taking the L²(Ω) scalar product of the equation (ν₀ −γ△)u = f with u (with △u) and performing appropriate integrations by parts gives the first estimate (second estimate).

Let 0≤λ₁≤λ₂ ≤. . . denote the eigenvalues of− △with homogeneous Neumann boundary condi- tions on Γ, counted according to multiplicity, and call the associated eigenfunctions ϕj, normalized by kϕ_jk_L²_(Ω) = 1. Then (ϕ₁, ϕ₂, . . .) is an orthonormal basis of L²(Ω), and the space X of finite linear combinations is a sub-space of H²(Ω) and dense in L²(Ω). Writef ∈X asf =PN

j=1αjϕj.

Then we have Qγ△f =

N

X

j=1

−λ_j

ν₀+γλ_jαjϕj, kQ_γ△fk²_L²_(Ω)=

N

X

j=1

λ²_j

(ν0+γλj)²|α_j|² ≤ 1 γ²

N

X

j=1

|α_j|² = 1

γ² kfk²_L²_(Ω), hQ_γ△f , fi=

N

X

j=1

−λ_j

ν₀+γλ_j|α_j|² ≤0.

A density argument concludes the proof of Lemma 3.2.

Then we have, for a.e. t∈(0, t_γ), the representation (△n_γ)(t, x) =Q_γ(∂_tn_γ−divJ_γ)(t, x).

The parabolic equation for J_γ can be expressed as (ν₀−γ△)△J_γ− ε²

4∇(△n_γ) =∂_tJ_γ+ 1

τJ_γ−T∇n_γ+G_γ, from which we deduce that

(ν₀−γ△)△J_γ+ ε²

4∇(Q_γ(divJ_γ)) =∂_tJ_γ+ 1

τJ_γ−T∇n_γ+G_γ+ε²

4∇Q_γ(∂_tn_γ) =:R_γ. We take the L²(Ω) scalar product of R_γ with△J_γ ∈H²(Ω)∩H₀¹(Ω):

hR_γ,△J_γi=ν₀k△J_γk²_L²_(Ω)−γ

△²J_γ,△J_γ + ε²

4 h∇(Q_γ(divJ_γ)),△J_γi

=ν₀k△J_γk²_L²_(Ω)+γk∇△J_γk²_L²_(Ω)−ε²

4 hQ_γ(divJ_γ),div△J_γi

=ν₀k△J_γk²_L²_(Ω)+γk∇△J_γk²_L²_(Ω)−ε²

4 hdivJ_γ, Q_γ(△divJ_γ)i

≥ν₀k△J_γk²_L²_(Ω)+γk∇△J_γk²_L²_(Ω), due to Lemma 3.2. Consequently, we arrive at

k△J_γk_L²_(Ω)≤ν₀⁻¹kR_γk_L²_(Ω).

A careful analysis ofR_γ showsR_γ∈L^∞((0, t_γ), L²(Ω)) with uniform inγ estimate, whence kJ_γk_L^∞_((0,tγ),H²_(Ω))≤C₀.

Going back to △n_γ = Q_γ(∂_tn_γ−divJ_γ) with (∂_tn_γ −divJ_γ) ∈ L^∞((0, t_γ), H¹(Ω)) we then find

△nγ ∈L^∞((0, tγ), H¹(Ω)), and therefore kn_γk_L^∞_((0,tγ),H³_(Ω)) ≤C₀,

uniformly in γ.

Eventually, we are in a position to show thattγ can not tend to zero forγ→0: for 0≤t^′ ≤t^′′≤tγ, we obtain the H¨older estimates

n_γ(t^′,·)−n_γ(t^′′,·)

H²(Ω)≤ Z _t^′′

t^′

n^′_γ(t,·)

H²(Ω) dt≤ |t^′−t^′′|^1/2 n^′_γ

L²((0,tγ),H²(Ω)),

which enables us to estimate nγ in C^1/2([0, tγ], C(Ω)). Next, fix a numberα with 0< α < ¹₂. By Sobolev’s embedding theorem and interpolation,

J_γ(t^′,·)−J_γ(t^′′,·)

C(Ω)≤C

J_γ(t^′,·)−J_γ(t^′′,·)

H^2−α(Ω)

≤C

J_γ(t^′,·)−J_γ(t^′′,·)

α/2 L²(Ω)

J_γ(t^′,·)−J_γ(t^′′,·)

(2−α)/2 H²(Ω)

≤C|t^′−t^′′|^α/2 J_γ^′

α/2

L^∞((0,tγ),L²(Ω))kJ_γk^(2−α)/2_L^∞_((0,tγ),H²_(Ω)),

∇n_γ(t^′,·)− ∇n_γ(t^′′,·)

C(Ω)≤C|t^′−t^′′|^α/2 n^′_γ

α/2

L^∞((0,tγ),H¹(Ω))kn_γk^(2−α)/2_L^∞_((0,tγ),H³_(Ω)).

The right-hand sides are bounded uniformly with respect to γ. Then these H¨older estimates give us an estimate from below for the earliest time t_∗ at which the solution (n_γ, J_γ) is able to violate the conditions (3.4). As a consequence, t_γ≥t_∗>0, for 0< γ <1.

Having secured a uniform existence interval, we can now show the convergence of the sequence (n_γ, J_γ, V_γ)_γ forγ →0. We have the uniform bounds

kn_γk_L^∞_(0,t∗),H³(Ω)≤C, k∂_tn_γk_L^∞_(0,t∗),H²(Ω)≤C,

and the compact embedding H³(Ω) ⊂ H²(Ω). Then Aubin’s Lemma [12, Corollary 4] shows that a subsequence of (nγ)γ (which we will not relabel) converges to a limit n in the space C([0, t_∗], H²(Ω)). Similarly, we can prove the convergence of a subsequence (J_γ)_γ to a limit J in the spaceC([0, t_∗], H¹(Ω)). By interpolation, we then have the convergences

n_γ →n in C([0, t_∗], H^3−δ(Ω)), δ >0, J_γ →J in C([0, t_∗], H^2−δ(Ω)), δ >0.

In particular, there is uniform convergence

(n_γ,∇n_γ, J_γ)→(n,∇n, J) in C(Q_∗), Q_∗ := (0, t_∗)×Ω,

which guarantees that the limit functions n and J satisfy the initial conditions from (1.1) and the boundary conditions from (1.2).

Obviously, we have the following weak convergences, too:

∂_tn_γ ⇀ ∂_tn in L²((0, t_∗), H²(Ω)), ∂_tJ_γ⇀ ∂_tJ in L²((0, t_∗), H¹(Ω)), nγ⇀^∗n in L^∞((0, t∗), H³(Ω)), Jγ⇀^∗ J in L^∞((0, t∗), H²(Ω)).

Moreover, a sub-sequence (V_γ)_γconverges to a limitV in the spaceC([0, t_∗], H_loc² (Ω)); and this limit V solves the Poisson equationλ²△V =n−C(x).

In a next step, we show that (n, J, V) solves (1.1). We choose a test function ϕ∈C₀^∞(Q∗), and it follows that

Z Z

Q^∗ −ϕ_tn_γ+γ(△²ϕ)n_γ−ν₀(△ϕ)n_γ+ (∇ϕ)J_γ

dxdt= 0, Z Z

Q^∗

−ϕtJγ+γ(△²ϕ)Jγ−ν0(△ϕ)Jγ−ε²

4(∇△ϕ)nγ+T(∇ϕ)nγ+ϕGγ

dxdt= 0.

Note thatG_γapproachesGin the norm of the spaceC([0, t_∗], L²(Ω)). Sendingγto zero and making use of the uniform convergence of (n_γ, J_γ)_γ to (n, J) we deduce that (n, J) are solutions to (1.1).

It only remains to show the uniqueness of the solutions. Let (n¹, J¹, V¹) and (n², J², V²) be two solutions of (1.1), (1.2) with regularity as in Theorem 2.1. Define

n_∆=n¹−n², J_∆=J¹−J², V_∆=V¹−V², G_∆=G¹−G², where G¹ and G² are given in (1.5). Then we get the system























∂_tn_∆−ν₀△n_∆−divJ_∆= 0,

∂tJ∆−ν0△J∆+ε²

4∇△n∆−T∇n∆+ 1

τJ∆+G∆= 0, λ²△V_∆=n_∆, (n_∆, J_∆)(0, x) = 0,

(∂_νn_∆, J_∆(t, x), KV_∆)(t, x) = 0, (t, x)∈(0, t_∗)×Γ.

Similarly as in (3.5), we can prove 1

2∂_t

kJ_∆k²L²(Ω)+Tkn_∆k²L²(Ω)+ε²

4 k∇n_∆k²L²(Ω)

+ν₀k∇J_∆k²L²(Ω)+T ν₀k∇n_∆k²L²(Ω)+ε²

4ν₀k△n_∆k²L²(Ω)+ 1

τ kJ_∆k²L²(Ω)

=− hG_∆, J_∆i,

and we estimate, after some calculations,

| hG∆, J∆i | ≤C J^j

L^∞(Ω), n^j

L^∞(Ω), ∇n^j

L^∞(Ω), ∇V^j

L^∞(Ω)

×

×

k∇J_∆kL²(Ω)kJ_∆kL²(Ω)+kJ_∆kL²(Ω)(kn_∆kL²(Ω)+k∇V_∆kL²(Ω))

+k∇J_∆k_L²_(Ω)kn_∆k_H¹_(Ω) . By Young’s inequality, we then have

T

2∂_tkn_∆k²L²+ε²

2∂_tk∇n_∆k²L²+1

2∂_tkJ_∆k²L² ≤C

kJ_∆k²L² +kn_∆k²H¹

.

Now it suffices to exploit Gronwall’s lemma for the conclusion n_∆ ≡ 0, J_∆ ≡ 0, which completes the proof of Theorem 2.1.