Proof of Theorem 3.3.1

Local Existence

To prove local existence in Theorem 3.3.1 we will use the iteration method and compactness arguments. The main task is to construct a sequence of ap-proximate solutions which is uniformly bounded in a certain Sobolev space in a fixed time interval. Compactness arguments then imply that there exists a limit which proves to be a local-in-time solution of (3.0.1). The first step is to linearize the system (3.0.1) around its initial state (n0, J0, V0), where V0 solves the Dirichlet problem

( λ²4V₀ =n₀(x)− C(x), V0(x)|_∂Ω =VΓ.

(3.3.1) We study the equations for the perturbation P := (P⁰, P^d) := (n−n0, J− J0), and first construct approximate solutions P_k := (P_k⁰, P_k^d) for P_k⁰ := n_k− n₀, P_k^d:=J_k−J₀ (k≥1) from a fixed-point procedure, which are expected to converge to a solutionP of the perturbed problem ask→ ∞. For this, we shall derive uniform bounds in certain Sobolev spaces on a uniform time interval and apply standard compactness arguments. A further analysis will show that (n, J) = (P⁰+n0, P^d+J0) withn >0 is the expected local (in time) solution of

the original problem (3.0.1)(V is via (3.3.1) uniquely determinate). For given (Pk−1, Vk−1)² we obtain the following linearized problems forP_k (k≥1)











∂tP_k+A(∂x)P_k =Fk−1, P_k(0, x) = 0,

P_k(t, x) = 0, on∂Ω for a.e. 0≤t≤T,

(3.3.2)

where

F_k−1:=

0 S(Pk−1)

!

−A(∂_x) n0

J0

!

(k≥2),

S(Pk−1) := div (P_k−1^d +J₀)⊗(P_k−1^d +J₀) P_k−1⁰ +n₀

!

−(P_k−1⁰ +n0)∇V_k−1 +²div

∇q

P_k−1⁰ +n₀

⊗

∇q

P_k−1⁰ +n₀

(k≥2), λ²4V_k−1 =P_k−1⁰ +n₀− C(x), Vk−1(t, x)|_∂Ω =V_Γ(x).

In the next lemma we will show that fork≥2 and any time interval [0, T], ifPk−1 satisfies (3.2.101) then Fk−1 satisfies (3.2.100).

Lemma 3.3.1. Let T >0, k≥1, ifPk−1= (P_k−1⁰ , P_k−1^d ) satisfies



















P_k−1⁰ ∈L^∞(0, T;H³(Ω))∩C([0, T], C¹(Ω))∩C([0, T], H²(Ω)), P_k−1^d ∈L^∞(0, T;H²(Ω))∩C([0, T], C(Ω))∩C([0, T], H₀¹(Ω)),

∂_tP_k−1⁰ ∈L^∞(0, T;H₀¹(Ω))∩C([0, T], L²(Ω))∩L²(0, T;H²(Ω)),

∂_tP_k−1^d ∈L^∞(0, T;L²(Ω))∩C([0, T], L²(Ω))∩L²(0, T;H₀¹(Ω)),

(3.3.3)

and

Pk−1(0, x) = (P_k−1⁰ , P_k−1^d )(0, x) = 0, (3.3.4) then Fk−1 satisfies











F_k−1⁰ ∈L^∞(0, T;H¹(Ω)), F_k−1^d ∈L^∞(0, T; (L²(Ω))^d);

F˙_k−1⁰ ∈L^∞(0, T;L²(Ω)), F˙_k−1^d ∈L^∞(0, T; (H⁻¹(Ω))^d);

F_k−1⁰ (0, x)∈H₀¹(Ω), F_k−1^d (0, x)∈L²(Ω).

(3.3.5)

2Actually it is sufficient to givePk−1as the known function since (3.3.1) holds, whereVk−1

can be considered as the unique solution of (3.3.1) for a givenPk−1⁰ .

Proof. Since F_k−1⁰ =ν04n₀+ divJ0 and thereby ˙F_k−1⁰ = 0, then from (3.0.4)

where Since Ω satisfies the cone condition under our assumptions then from the Sobolev embedding theorem the following embedding

H¹(Ω),→L⁶(Ω),→L⁴(Ω) (3.3.9) holds. From (3.3.9) we estimate K1 and K3 as follows.

K1≤C

K3 ≤C

d

X

l=1 d

X

k=1

n⁻¹₀

L^∞(Ω)k∂_xk∂xln0k_H1(Ω)k∂_xkn0k_H1(Ω)

+ n⁻¹₀

L^∞(Ω)

∂_x²_kn0

H¹(Ω)k∂_xln0k_H1(Ω)

+ n⁻²₀

L^∞(Ω)k∂_xkn0k_H1(Ω)k∂_xln0k_H1(Ω)k∂_xkn0k_H1(Ω)

! , from which we obtain

K₃≤C n⁻¹₀

L^∞(Ω)kn₀k_H3(Ω)kn₀k_H2(Ω)+ n⁻¹₀

L^∞(Ω)kn₀k_H3(Ω)kn₀k_H2(Ω)

+ n⁻²₀

L^∞(Ω)kn₀k³_H1(Ω)

!

<∞.

To estimateK₂ we go back to (3.3.1) then with (3.3.9) to find K₂ ≤

d

X

l=1

kn₀k_H1(Ω)k∂_xlV₀k_H1(Ω)

≤Ckn₀k_H1(Ω)kV₀k_H2(Ω)

≤Ckn₀k_H1(Ω)

kn₀k_L2(Ω)+kC(x)k_L2(Ω)+kV_Γk_H3/2(Ω)

<∞.

(3.3.10)

(3.3.6), (3.3.7) and (3.3.8) together with the estimations ofK1,K2,K3and our assumptions thatn₀ ∈H³(Ω), J₀∈(H²(Ω))^d yield that

F_k−1^d (0, x)∈L²(Ω).

In order to proveF_k−1^d ∈L^∞(0, T; (L²(Ω))^d), it is sufficient to show that S(Pk−1)∈L^∞(0, T; (L²(Ω))^d).

To see this we use our assumptions onPk−1 then find for a.e. t∈[0, T] Z

Ω

div (P_k−1^d +J0)⊗(P_k−1^d +J0) P_k−1⁰ +n0

!!2

dx

≤C sup

[0,T]×Ω

(P_k−1^d +J₀)⁴ inf

[0,T]×Ω

(P_k−1⁰ +n0)⁴kP_k−1⁰ +n₀k²_H1(Ω)

+C sup

[0,T]×Ω

(P_k−1^d +J0)² inf

[0,T]×Ω

(P_k−1⁰ +n₀)²kP_k−1^d +J0k²_H1(Ω)<∞,

from which it follows

div (P_k−1^d +J₀)⊗(P_k−1^d +J₀) P_k−1⁰ +n0

!

∈L^∞(0, T; (L²(Ω))^d). (3.3.11) Concerning the second term ofS(Pk−1) we have

Z It remains to give a estimation of the third term of S(Pk−1). For this we find

Z

Notice that in this equation we use the notation of (2.1.1). More precisely,

from which we infer

By a similar calculation, which is equal to

² Combining (3.3.14), (3.3.15) and (3.3.16) we conclude

F˙_k−1^d ∈L^∞(0, T; (H⁻¹(Ω))^d).

Now select two positive numbersδ0, M >0 such that δ0< inf

x∈Ωn0 (3.3.17) max k∇n₀k_L^∞_(Ω),kn₀k_L^∞_(Ω),kJ₀k_L^∞_(Ω)

< δ₀⁻¹, (3.3.18) and

kn₀k_H3(Ω) < M, kJ₀k_H2(Ω) < M. (3.3.19) Next we shall use iteration method to obtain a sequence of approximate solu-tions.

Lemma 3.3.2. Let k≥0 and P0 = 0, there exist t⁰ >0, C0 >0, C^∗ >0 and C⁰ >0 independent ofk such that in the interval [0, t⁰] P_k satisfies



















P_k⁰∈L^∞(0, t⁰;H³(Ω))∩C([0, t⁰], C¹(Ω))∩C([0, t⁰], H²(Ω)), P_k^d∈L^∞(0, t⁰;H²(Ω))∩C([0, t⁰], C(Ω))∩C([0, t⁰], H₀¹(Ω)),

∂tP_k⁰∈L^∞(0, t⁰;H₀¹(Ω))∩C([0, t⁰], L²(Ω))∩L²(0, t⁰;H²(Ω)),

∂tP_k^d∈L^∞(0, t⁰;L²(Ω))∩C([0, t⁰], L²(Ω))∩L²(0, t⁰;H₀¹(Ω)),

(3.3.20)

and the following uniform bounds



























kP_k⁰k_L∞(0,t⁰;H³(Ω))≤C⁰, kP_k^dk_L∞(0,t⁰;H²(Ω))≤C⁰, kP_k⁰k_L∞(0,t⁰;H²(Ω))≤C^∗, kP_k^dk_L∞(0,t⁰;H¹₀(Ω))≤C^∗, kP_k⁰k_L∞(0,t⁰;H₀¹(Ω))≤C₀, kP_k^dk_L∞(0,t⁰;L²(Ω))≤C₀, kP˙_k⁰k_L∞(0,t⁰;H₀¹(Ω))≤C₀, kP˙_k^dk_L∞(0,t⁰;L²(Ω))≤C₀, kP˙_k⁰k_L2(0,t⁰;H²(Ω))≤C0, kP˙_k^dk_L2(0,t⁰;H¹₀(Ω))≤C0,

(3.3.21)

and



















inf

[0,t⁰]inf

x∈Ω

(P_k⁰+n₀)> δ₀, sup

[0,t⁰]

max

k∇(P_k⁰+n₀)k_L∞(Ω),kP_k⁰+n₀k_L∞(Ω), kP_k^d+J₀k_L∞(Ω)

< δ₀⁻¹,

(3.3.22)

hold.

Remark 3.3.1. Lemma 3.3.2 interprets a uniform bounds of P_k in given re-flexive spaces, that guarantees the existence of a convergent subsequence of P_k in certain weak senses.

Proof. Throughout the proof we will use C as a generic constant which may change from line to line and depend onδ₀, Ω and all physical constants, but is independent of k.

Obviously, forP0 = 0,F0satisfies (3.2.100). Starting withP0= 0, by theorem 3.2.13 we obtain a solutionP₁ of (3.3.2) on any time interval [0, T],T >0, i.e.,

Since obviously P0 = 0 satisfies (3.3.3) and (3.3.4) then F0 satisfies(3.3.5). By Theorem 3.2.13 P1 satisfies (3.2.101).Thus we can shrink the interval [0, T] to [0, t₁] such that on this interval

Then we use mathematical induction to complete the proof. Let c^∗ >0,T >0 assume Pk−1(k≥1) satisfies



From lemma 3.3.1Fk−1 satisfies (3.3.5). Then by theorem 3.2.13 we obtainPk

in [0, T) as the solution of (3.3.2) which satisfies the regularity mentioned in (3.2.101) withu replaced byP_k. We expand (3.3.2) then obtain two equations forP_k⁰ and P_k^d for a.e. t∈[0, T]

where the following calculations

(Q_k, P_k⁰) =−ν₀(∇n₀,∇P_k⁰) + (divJ0, P_k⁰), (3.3.30)

Taking theL²(Ω) scalar product of the first equation in (3.3.27) with−² 44P_k⁰,

since∂tP_k⁰ ∈L^∞(0, T;H₀¹(Ω)) from theorem 3.2.13.

Summing up (3.3.28), (3.3.29) and (3.3.32) yields 1 where I1,I3 are defined as (2.1.1). The itemsI_k can be estimated as follows.

First by H¨older’s inequality

|I₁|= We use the assumptions (3.3.25) and (3.3.26) and Cauchy’s inequality then infer for a.e. t∈[0, T]

|I₁| ≤d²δ⁻²₀ (c^∗+M)k∇P_k^dk_L2(Ω) ≤₁k∇P_k^dk²_L2(Ω)+d⁴δ⁻⁴₀ (c^∗+M)²

4₁ ,

(3.3.38)

here

Using the assumptions 3.3.25 and (3.3.26) again we obtain

|I₃| ≤ ²

4d²δ⁻²₀ (c^∗+M)k∇P_k^dk_L2(Ω)

≤3k∇P_k^dk²_L2(Ω)+ ⁴

16d⁴δ₀⁻⁴(c^∗+M)² 43

.

(3.3.43)

In the following, we will prove uniform in k estimates of P_k, which will imply that the life span can not tend to zero for k going to infinity. Then we will have a uniform existence interval as well as uniform estimates and can prove the convergence of a subsequence P_ki by compactness argument.

Recall (3.3.30), we have T₀|(Q_k, P_k⁰)|

≤T₀ν₀k∇n₀k_L2(Ω)k∇P_k⁰k_L2(Ω)+T₀kJ₀k_L2(Ω)k∇P_k⁰k_L2(Ω)

≤T₀M(ν₀+ 1)k∇P_k⁰k_L2(Ω)

≤₄k∇P_k⁰k²_L2(Ω)+ T₀²M²(ν0+ 1)²

4₄ .

(3.3.44)

Recall (3.3.34):

|(R_k, P_k^d)| ≤ν₀|(4J₀, P_k^d)|+1

τ|(J₀, P_k^d)|+T₀|(∇n₀, P_k^d)|

+ ²

4|(∇4n₀, P_k^d)|+|I₁|+|I₂|+|I₃|.

Using H¨older’s inequality and (3.3.19)

|(R_k, P_k^d)| ≤ν₀k4J₀k_L2(Ω)kP_k^dk_L2(Ω)+1

τkJ₀k_L2(Ω)kP_k^dk_L2(Ω)

+T₀k∇n₀k_L2(Ω)kP_k^dk_L2(Ω)+²

4k∇4n₀k_L2(Ω)kP_k^dk_L2(Ω)

+|I₁|+|I₂|+|I₃|

≤

ν0+ 1

τ +T0+ ² 4

MkP_k^dk_L2(Ω)+|I₁|+|I₂|+|I₃|.

Then from (3.3.38), (3.3.42), (3.3.43) and Cauchy’s inequality we conclude

|(R_k, P_k^d)| ≤₅kP_k^dk²_L2(Ω)+

ν₀+ 1

τ +T₀+² 4

2

M² 4₅

+1k∇P_k^dk²_L2(Ω)+d⁴δ⁻⁴₀ (c^∗+M)² 4₁

+₂kP_k^dk²_L2(Ω)+(c^∗+M)²C_λ,Ω,c² ∗,M,C(x),V_Γ

4₂

+3k∇P_k^dk²_L2(Ω)+ ⁴

16d⁴δ⁻⁴₀ (c^∗+M)² 43

.

(3.3.45)

It is easy to verify ²

4|(Q_k,4P_k⁰)| ≤ ²

4ν0|(4n₀,4P_k⁰)|+ ²

4|(divJ₀,4P_k⁰)|

≤ ²

4ν₀k4n₀k_L2(Ω)k4P_k⁰k_L2(Ω)+²

4kdivJ₀k_L2(Ω)k4P_k⁰k_L2(Ω).

(3.3.46)

By (3.3.19):

²

4|(Q_k,4P_k⁰)| ≤ ²

4M(ν0+ 1)k4P_k⁰k_L2(Ω)

≤6k4P_k⁰k²_L2(Ω)+ ⁴

16M²(ν₀+ 1)² 46

.

(3.3.47)

Combining (3.3.33), (3.3.44), (3.3.45) and (3.3.47) we obtain the following in-equality

1

2∂t T0kP_k⁰k²+²

4k∇P_k⁰k²+kP_k^dk²

+ν0T0k∇P_k⁰k² +²

4ν0k4P_k⁰k²+ν0k∇P_k^dk²+1 τkP_k^dk²

≤₄k∇P_k⁰k²+ (₂+₅)kP_k^dk²+ (₁+₃)k∇P_k^dk² +6k4P_k⁰k²+C₁,···,6,ν0,T0,τ,,λ,VΓ,C(x),Ω,δ₀,M,c^∗,C14,

(3.3.48)

C_{1,···,6,ν0,T0,τ,,λ,VΓ,C(x),Ω,δ0,M,c}^∗ >0 depends only upon the constants that oc-cur in the subscript. Notice that C_{1,···,6,ν0,T0,τ,,λ,VΓ,C(x),Ω,δ0,M,c}^∗ > 0 doesn’t depend onk.

Selecti, i= 1,· · ·,6 sufficiently small such that ₄< ν₀T₀, (₂+₅)< 1

τ, (₁+₃)< ν₀, ₆< ² 4ν₀.

Then we obtain

∂_t T₀kP_k⁰k²+²

4k∇P_k⁰k²+kP_k^dk² +C₁

T0kP_k⁰k²+²

4k∇P_k⁰k²+k4P_k⁰k²+k∇P_k^dk²+kP_k^dk²

≤ C₂, (3.3.49) where C₁ > 0 is independent of c^∗, C₂ depends on all physical known quanti-ties, the imbedding constants and c^∗, δ₀ and M. Via solving the differential inequality (3.3.49) we infer the following estimation.

T₀kP_k⁰k²_L∞(0,T;L²(Ω))+²

4k∇P_k⁰k²_L∞(0,T;L²(Ω))+kP_k^dk²_L∞(0,T;L²(Ω))

≤ C₂ Z T

0

e^C1sds.

(3.3.50)

Integrating (3.3.49) from 0 toT yields

kP_k⁰k²_L2(0,T;H²(Ω))+kP_k^dk²_L2(0,T;H¹(Ω))≤ C₃T, (3.3.51) C₃>0 depends on all physical known quantities andc^∗,δ0 and M.

Next we shall estimate ˙P_k⁰ and ˙P_k^d. Since Pk−1 satisfies the assumptions (3.3.23) and (3.3.24), then from Lemma 3.3.1 Fk−1 satisfies (3.3.5), thus the time derivatives of (P_k⁰, P_k^d) satisfy the same boundary conditions as (P_k⁰, P_k^d) from Theorem 3.2.13. Differentiating formally (3.3.27) with respect to 0 ≤ t≤T, then we obtain for a.e. 0≤t≤T







∂tP˙_k⁰−ν04P˙_k⁰−div ˙P_k^d = 0,

∂_tP˙_k^d−ν₀4P˙_k^d+ 1

τP˙_k^d−T₀∇P˙_k⁰+²

4∇4P˙_k⁰ =−S(P˙ k−1).

(3.3.52)

Notice that − Z

Ω

∂tP˙_k⁰4P˙_k⁰dx = 1

2∂tk∇P˙_k⁰k² by Theorem 3.2.11. Then we deduce, by a similar computation as before(see (3.3.33)), that

1

2∂t T0kP˙_k⁰k²+²

4k∇P˙_k⁰k²+kP˙_k^dk²

+ν0T0k∇P˙_k⁰k² +²

4ν₀k4P˙_k⁰k²+ν₀k∇P˙_k^dk²+ 1 τkP˙_k^dk²

=( ˙S(Pk−1),P˙_k^d).

(3.3.53)

The scalar product −( ˙S(Pk−1),P˙_k^d) has the representation

−( ˙S(Pk−1),P˙_k^d) =I₁⁰ +I₂⁰ +I₃⁰, (3.3.54)

where Then by Cauchy’s inequality

|Z₁| ≤⁰₁

Furthermore

Combining (3.3.61), (3.3.62) and (3.3.63) we obtain

|I₁⁰| ≤(⁰₁+⁰₂+⁰₃) where the constant C⁰

1,⁰₂,⁰₃,Ω,δ0,c^∗>0 depends only on⁰₁, ⁰₂, ⁰₃,Ω, δ₀, c^∗.

Recall (3.3.40), (3.3.41) and use the imbedding

H¹(Ω),→L⁴(Ω) (3.3.65)

where let C14 denote the imbedding constant of (3.3.65), then under the as-sumptions (3.3.25) we find

|I₂⁰| ≤C₁₄² kP˙_k^dk_L2(Ω)kV_k−1k_H2(Ω)kP˙_k−1⁰ k_H1(Ω)

+C₁₄² kP˙_k^dk_L2(Ω)kP_k−1⁰ +n0k_H1(Ω)kV˙k−1k_H2(Ω)

≤c^∗C₁₄² C_λ,Ω,c^∗_,M,C(x),VΓkP˙_k^dk_L2(Ω)

+c^∗(c^∗+M)C₁₄² C_Ω,λkP˙_k^dk_L2(Ω), which implies

|I₂⁰| ≤⁰₄kP˙_k^dk²_L2(Ω)+(c^∗)²C₁₄⁴ C_λ,Ω,c² ∗,M,C(x),V_Γ

4⁰₄

+⁰₅kP˙_k^dk²_L2(Ω)+(c^∗(c^∗+M))²C₁₄⁴ C_Ω,λ² 4⁰₅

≤(⁰₄+⁰₅)kP˙_k^dk²_L2(Ω)+C⁰

4,⁰₅,λ,Ω,c^∗,M,C(x),VΓ,C14.

(3.3.66)

Now we considerI₃⁰. By a similar computation asI₁⁰ we obtain

I₃⁰ =² 4

X

i,j=1,···,d

∂i(P_k−1⁰ +n0)∂j(P_k−1⁰ +n0) P_k−1⁰ +n₀

!0

, ∂xj

P˙_k^d

i

!

=²

4(Z₁+Z₂+Z₃), where we define

Z₁: = X

i,j=1,···,d

(∂xiP_k−1⁰ )⁰∂xj(P_k−1⁰ +n0) P_k−1⁰ +n0

, ∂_xj P˙_k^d

i

!

, (3.3.67)

Z₂: = X

i,j=1,···,d

∂_xi(P_k−1⁰ +n₀)(∂_xjP_k−1⁰ )⁰ P_k−1⁰ +n₀ , ∂xj

P˙_k^d

i

!

, (3.3.68)

Z₃: =− X

i,j=1,···,d

(P_k−1⁰ )⁰∂_xi(P_k−1⁰ +n₀)∂_xj(P_k−1⁰ +n₀) (P_k−1⁰ +n0)² , ∂_xj

P˙_k^d

i

! . (3.3.69) Under the assumptions (3.3.25) and (3.3.26) the items Z_l, l = 1,2,3 can be estimated as follows.

|Z₁| ≤ X Then by Cauchy’s inequality

|Z₁| ≤⁰₆ Using Cauchy’s inequality yields

|Z₃| ≤⁰₈

Combining (3.3.70), (3.3.71) and (3.3.72) we obtain

|I₃⁰| ≤ ²

where the constant C⁰

6,⁰₇,⁰₈,,Ω,δ0,c^∗ > 0 depends only on ⁰₆, ⁰₇, ⁰₈, ,Ω, δ0, c^∗. Recall (3.3.53), (3.3.54), (3.3.64), (3.3.66) and (3.3.73) then

1 imbedding constants and all physical known quantities.

Select⁰_l,(l= 1,· · · ,8) sufficiently small such that quanti-ties, the imbedding constants and c^∗, δ0 and M. Via solving the differential inequality (3.3.75) we infer the following estimation.

T₀kP˙_k⁰k²_L∞(0,T;L²(Ω))+²

and

hold. kS(P_k−1)(0)k can be estimated via a same procedure as (3.3.7)-(3.3.10) from which we obtain that kS(P_k−1)(0)k is bounded by a constant depending only upon δ₀, M, C₁₄, C₁₆ where C_1s, s = 4,6 denotes the imbedding constant of H¹(Ω),→L^s(Ω) respectively. Thus

kP˙_k^d(0,·)k² ≤C_{ν0,T0,τ,,δ0,M,C14,C16}. (3.3.78) Together with (3.3.76) and (3.3.77) we obtain

T0kP˙_k⁰k²_L∞(0,T;L²(Ω))+²

Integrate (3.3.75) from 0 to T:

C₁⁰ T0kP˙_k⁰k²_L2(0,T;L²(Ω))+²

RecallTheorem3.2.9

Fix now a positive numberγ >0 and define

C₀ := max

We recall (3.3.23)-(3.3.26) and conclude that ifPk−1 defined in [0, t^∗) satisfies





then first from (3.3.85) and (3.3.86) together withLemma3.3.1 andTheorem 3.2.13 the solution Pk of (3.3.2) in [0, t^∗) satisfies

second from (3.3.50), (3.3.80), (3.3.81), (3.3.83) and (3.3.84)Pk satisfies



which implies

kP_k⁰k_L∞(0,t^∗;H²(Ω))≤C^∗, kP_k^dk_L∞(0,t^∗;H₀¹(Ω))≤C^∗. (3.3.97) To get more estimates first from (3.3.2) we conclude that for a.e. t∈[0, t^∗]Pk

solves

(A(∂x)Pk=Fk−1−P˙k, P_k(x) = 0, on ∂Ω,

(3.3.98) where Fk−1 −P˙k ∈ H₀¹(Ω)×(L²(Ω))^d because of the assumptions (3.3.85), (3.3.86) together with Lemma 3.3.1 and (3.3.89). Using Theorem 3.2.6 we have the following a priori estimate forP_k:

kP_k⁰k_L∞(0,t^∗;H³(Ω))+kP_k^dk_L∞(0,t^∗;(H²(Ω))^d)

≤C_A(∂x) kF_k−1⁰ k_L∞(0,t^∗;H₀¹(Ω))+kF_k−1^d k_L∞(0,t^∗;L²(Ω))

+kP˙_k⁰k_L∞(0,t^∗;H₀¹(Ω))+kP˙_k^dk_L∞(0,t^∗;(L²(Ω))^d)

,

(3.3.99)

the constantC_A(∂x)>0 depends only on the coefficients of operatorA(∂_x).

Since F_k−1⁰ =ν04n₀+ divJ0 then

kF_k−1⁰ k_L∞(0,t^∗;H₀¹(Ω))≤ν0kn₀k_H3(Ω)+kJ₀k_H2(Ω) ≤M(ν0+ 1). (3.3.100) Moreover

F_k−1^d =S(Pk−1) +ν₀4J₀−1

τJ₀+T₀∇n₀−²

4∇4n₀, which can be estimated as

kF_k−1^d k_L^∞_(0,t^∗_;L2(Ω))

≤ kS(P_k−1)k_L∞(0,t^∗;L²(Ω))+ν₀kJ₀k_H2(Ω)+ 1

τkJ₀k_L2(Ω)

+T₀kn₀k_H1(Ω)+²

4kn₀k_H3(Ω)

≤M

ν0+1

τ +T0+² 4

+kS(P_k−1)k_L^∞_(0,t^∗_;L2(Ω)).

(3.3.101)

For a.e. 0≤t≤t^∗, kS(P_k−1)k_L2(Ω)≤

div (P_k−1^d +J0)⊗(P_k−1^d +J0) P_k−1⁰ +n0

! L²(Ω)

+

(P_k−1⁰ +n0)∇V_k−1 L²(Ω)

+

²div

∇q

P_k−1⁰ +n₀

⊗

∇q

P_k−1⁰ +n₀ L²(Ω)

=:M₁+M₂+M₃.

• Estimates of M₁. Combining (3.3.19), (3.3.87), (3.3.88) and (3.3.102) we decuce

M₁ ≤2dδ⁻²₀ (C^∗+M) +dδ⁻⁴₀ (C₀+M). (3.3.103) From (3.3.87) and (3.3.41) we infer

M₂ ≤C₁₄² (C₀+M)C_{λ,Ω,C0,M,C(x),VΓ}. (3.3.104)

• Estimates of M₃.

we use a similar reasoning asM₁ to find 4

Thus combining (3.3.103)-(3.3.105) we conclude that for a.e. t∈[0, t^∗]

kS(P_k−1)k_L2(Ω)≤C_λ,,Ω,C0,C^∗_{,δ0,M,C(x),VΓ,C14}. (3.3.106) Together with (3.3.91), (3.3.99), (3.3.100) and (3.3.101) we deduce the following estimation.

kP_k⁰k_L^∞_(0,t^∗_;H3(Ω))+kP_k^dk_L∞(0,t^∗;(H²(Ω))^d)

≤C_{ν0,λ,,τ,Ω,C0,C}^∗_{,δ0,M,C(x),VΓ,C14} =:C⁰,

(3.3.107) C⁰ >0 depends upon all known physical quantities, the imbedding constant of H¹(Ω),→L⁴(Ω),δ0, M and C0, C^∗, but is independent of kand t^∗.

Now we make a summary. Given a function Pk−1; three positive constants C₀, C^∗, C⁰ which are defined in (3.3.83), (3.3.96), (3.3.107) respectively and depend only upon all physical constants, the corresponding imbedding constants C₁₄, C₁₆, the initial functions n₀, J₀ and the boundary conditions (n_Γ, J_Γ, V_Γ);

a time interval [0, t^∗) which satisfies (3.3.84). IfPk−1 satisfies (3.3.85), (3.3.86), (3.3.87), (3.3.88), (3.3.102) and



then we obtainPk via solving (3.3.2), andPk satisfies



















P_k⁰ ∈L^∞(0, t^∗;H³(Ω))∩C([0, t^∗], C¹(Ω))∩C([0, t^∗], H²(Ω)), P_k^d ∈L^∞(0, t^∗;H²(Ω))∩C([0, t^∗], C(Ω))∩C([0, t^∗], H₀¹(Ω)),

∂_tP_k⁰ ∈L^∞(0, t^∗;H₀¹(Ω))∩C([0, t^∗], L²(Ω))∩L²(0, t^∗;H²(Ω)),

∂_tP_k^d ∈L^∞(0, t^∗;L²(Ω))∩C([0, t^∗], L²(Ω))∩L²(0, t^∗;H₀¹(Ω)),

(3.3.109)

and



























kP_k⁰k_L∞(0,t^∗;H³(Ω))≤C⁰, kP_k^dk_L∞(0,t^∗;H²(Ω)) ≤C⁰, kP_k⁰k_L^∞_(0,t^∗_;H2(Ω))≤C^∗, kP_k^dk_L∞(0,t^∗;H₀¹(Ω)) ≤C^∗, kP_k⁰k_L∞(0,t^∗;H₀¹(Ω))≤C₀, kP_k^dk_L∞(0,t^∗;L²(Ω)) ≤C₀, kP˙_k⁰k_L∞(0,t^∗;H₀¹(Ω))≤C₀, kP˙_k^dk_L∞(0,t^∗;L²(Ω)) ≤C₀, kP˙_k⁰k_L2(0,t^∗;H²(Ω))≤C₀, kP˙_k^dk_L2(0,t^∗;H₀¹(Ω)) ≤C₀.

(3.3.110)

It is pointed out that up to now we have not derived (3.3.22) yet. In order to let Pk satisfy (3.3.22) we can shrink the time interval [0, t^∗) into [0, tk) such that



















inf

[0,tk]inf

x∈Ω

(P_k⁰+n₀)> δ₀, sup

[0,tk]

max

k∇(P_k⁰+n₀)k_L∞(Ω),kP_k⁰+n₀k_L∞(Ω), kP_k^d+J₀k_L∞(Ω)

< δ₀⁻¹.

(3.3.111)

This process can be realized because of (3.3.17), (3.3.18) and (3.3.109). Under the conditions (3.3.109), (3.3.110), (3.3.111) and via shrinking the time interval into [0, t_k+1) the solutionP_k+1satisfies the same estimates asP_kbut in [0, t_k+1).

More precisely, we have derived a sequence{P_k}^∞_k=1 in [0, tk) which satisfies the uniform bounds (3.3.110) and (3.3.111).

Next we are able to show that to guarantee (3.3.111) t_k will not converge to zero for k→ ∞. For 0≤t1 ≤t2 ≤tk, recallnk =P_k⁰+n0, Jk =P_k^d+J0, we deduce the H¨older estimates

kn_k(t1,·)−n_k(t2,·)k_H2(Ω)≤ Z t2

t1

kn⁰_k(s,·)k_H2(Ω)ds

≤ |t₁−t2|^1/2kP˙_k⁰k_L2(0,t_k;H²(Ω)),

which allows us to estimate P_k in C^1/2([0, t_k], C(Ω)). Fix a number β with 0< β < 1

2 then by Sobolev’s embedding theorem

kP_k(t1)−Pk(t2)k_C(Ω)≤CβkP_k(t1)−Pk(t2)k_H2−β(Ω), (3.3.112)

here letCβ denote the imbedding constant depending only on Ω. Interpolation yields

kP_k(t1)−P_k(t2)k_H2−β(Ω)

≤C_interkP_k(t₁)−P_k(t₂)k^β/2_L2(Ω)kP_k(t₁)−P_k(t₂)k^(2−β)/2_H2(Ω) .

(3.3.113)

Finally

kP_k(t₁)−P_k(t₂)k_L2(Ω)≤ Z t2

t1

k∂_tP_kk_L2(Ω)dt

≤ |t₁−t2|k∂_tPkk_L^∞_(0,tk;L2(Ω))

(3.3.114)

together with (3.3.112) and (3.3.113) yields kP_k(t₁)−P_k(t₂)k_C(Ω)

≤2^(2−β)/2C_βCinterk∂_tP_kk^β/2_L∞(0,tk;L²(Ω))kP_kk^(2−β)/2_L∞(0,tk;H²(Ω))|t₁−t2|^β/2. (3.3.115) Thus under (3.3.110)

kn_k(t₁,·)−n_k(t₂,·)k_C(Ω)

≤2^(2−β)/2CβCinterkP˙_k⁰k^β/2_L∞(0,tk;L²(Ω))kn_kk^(2−β)/2_L∞(0,tk;H²(Ω))|t₁−t2|^β/2

≤2^(2−β)/2C_βC_interC₀^β/2(C^∗+M)^(2−β)/2|t₁−t₂|^β/2.

(3.3.116)

kJ_k(t1,·)−Jk(t2,·)k_C(Ω)

≤2^(2−β)/2C_βC_interkP˙_k^dk^β/2_L∞(0,tk;L²(Ω))kJ_kk^(2−β)/2_L∞(0,tk;H²(Ω))|t₁−t₂|^β/2

≤2^(2−β)/2C_βC_interC₀^β/2(C⁰+M)^(2−β)/2|t₁−t₂|^β/2.

(3.3.117)

By a similar reasoning

k∇n_k(t1,·)− ∇n_k(t2,·)k_C(Ω)

≤2^(2−β)/2C_βC_interk∇P˙_k⁰k^β/2_L∞(0,tk;L²(Ω))kn_kk^(2−β)/2_L∞(0,tk;H³(Ω))|t₁−t₂|^β/2

≤2^(2−β)/2C_βCinterC₀^β/2(C⁰+M)^(2−β)/2|t₁−t2|^β/2.

(3.3.118)

Since the right-hand sides of (3.3.116), (3.3.117) and (3.3.118) are uniformly bounded with respect to k, there is a time interval [0, t⁰] with 0< t⁰ ≤t^∗ such that for all k = 1,2, ..., (3.3.20), (3.3.21) and (3.3.22) hold. We have finished the proof of Lemma 3.3.2.

To obtain a local-in-time solution the convergence property of the total sequence {P_k}^∞_k=0 needs to be considered. For this we have

Lemma 3.3.3. There exits t∗∈(0, t⁰)such that {P_k}^∞_k=0 is a Cauchy sequence inL^∞(0, t∗;H¹(Ω)×(L²(Ω))^d). Moreover there exists aP^∗∈L^∞(0, t∗;H¹(Ω)×

(L²(Ω))^d) such that Pk converges to P^∗ in L^∞(0, t∗;H¹(Ω)×(L²(Ω))^d).

Proof. Recall (3.3.2) the following system











∂t(P_k+1−P_k) +A(∂x)(P_k+1−P_k) =F_k−Fk−1, (P_k+1−P_k)(0, x) = 0,

(P_k+1−P_k)(t, x) = 0, on ∂Ω for a.e. 0≤t≤t⁰, holds in the time interval [0, t⁰]. By a similar calculations as (3.3.28)-(3.3.33):

1

2∂_t T₀kP_k+1⁰ −P_k⁰k²+²

4k∇(P_k+1⁰ −P_k⁰)k²+kP_k+1^d −P_k^dk² +ν₀T₀k∇(P_k+1⁰ −P_k⁰)k²+²

4ν₀k4(P_k+1⁰ −P_k⁰)k² +ν0k∇(P_k+1^d −P_k^d)k²+1

τkP_k+1^d −P_k^dk²

=T₀(F_k⁰−F_k−1⁰ , P_k+1⁰ −P_k⁰) + (F_k^d−F_k−1^d , P_k+1^d −P_k^d)

−²

4(F_k⁰−F_k−1⁰ ,4(P_k+1⁰ −P_k⁰)).

By H¨older’s inequality 1

2∂t T0kP_k+1⁰ −P_k⁰k²+²

4k∇(P_k+1⁰ −P_k⁰)k²+kP_k+1^d −P_k^dk² +ν₀T₀k∇(P_k+1⁰ −P_k⁰)k²+²

4ν₀k4(P_k+1⁰ −P_k⁰)k² +ν0k∇(P_k+1^d −P_k^d)k²+ 1

τkP_k+1^d −P_k^dk²

≤T₀kF_k⁰−F_k−1⁰ k_L2(Ω)kP_k+1⁰ −P_k⁰k_L2(Ω)+kF_k^d−F_k−1^d k_H⁻¹_(Ω)×

× kP_k+1^d −P_k^dk_H1

0(Ω)+²

4kF_k⁰−F_k−1⁰ k_L2(Ω)k4(P_k+1⁰ −P_k⁰)k_L2(Ω), then by Cauchy’s inequality

∂_t T₀kP_k+1⁰ −P_k⁰k²+²

4k∇(P_k+1⁰ −P_k⁰)k²+kP_k+1^d −P_k^dk²

≤C

kF_k⁰−F_k−1⁰ k²_L2(Ω)+kF_k^d−F_k−1^d k²_H−1(Ω)

for someC >0 depending only on the coefficients of operatorA(∂x). Then we

To estimate L1 we selectg∈(H₀¹(Ω))^d withkgk_(H1

Now considerL2. Set4₃ :=Vk−Vk−1, then

n_k∇V_k−nk−1∇V_k−1=n_k∇V_k−n_k∇V_k−1+n_k∇V_k−1−nk−1∇V_k−1

=nk∇4₃+∇V_k−14₂, which yields

L₂≤ kn_k∇4₃k_L2(Ω)+k∇V_k−14₂k_L2(Ω)

≤ kn_kk_L∞(Ω)k4₃k_H1(Ω)+k∇V_k−1k_L4(Ω)k4₂k_L4(Ω)

≤δ⁻¹₀ k4₃k_H1(Ω)+C₁₄² kV_k−1k_H2(Ω)k4₂k_H1(Ω). Since4₃ solves

(λ²44₃=4₂, 4₃|_∂Ω= 0 recall (3.3.41) we obtain

L2 ≤Ck4₂k_H1(Ω), (3.3.124) C > 0 is independent of k, t⁰, but depends upon C0. The representation and estimates of L₃ follows from a similar process as L₁. Precisely, let ∂_xln_k (∂xsnk,∂xlnk−1,∂xsnk−1) replace (Jk)l ((Jk)s,(Jk−1)l,(Jk−1)s) in (3.3.122) then from (3.3.123)

L₃ ≤2δ₀⁻⁴

k∇P_k⁰− ∇P_k−1⁰ k_L2(Ω)+kP_k⁰−P_k−1⁰ k_L2(Ω)

≤4δ₀⁻⁴kP_k⁰−P_k−1⁰ k_H1(Ω).

(3.3.125) Combining (3.3.123), (3.3.124) and (3.3.125) we conclude

kS(P_k)−S(Pk−1)k²_L∞(0,t⁰;H⁻¹(Ω))

≤C

k4₁k²_L2(Ω)+k4₂k²_H1(Ω)

≤C

kP_k⁰−P_k−1⁰ k²_L∞(0,t⁰;H¹(Ω))+kP_k^d−P_k−1^d k²_L∞(0,t⁰;L²(Ω))

,

(3.3.126)

whereC >0 is independent ofk andt⁰. Recall (3.3.119), (3.3.120) and (3.3.121):

kP_k+1⁰ −P_k⁰k²_L∞(0,t⁰;H¹(Ω))+kP_k+1^d −P_k^dk²_L∞(0,t⁰;L²(Ω))

≤Ct⁰

kP_k⁰−P_k−1⁰ k²_L∞(0,t⁰;H¹(Ω))+kP_k^d−P_k−1^d k²_L∞(0,t⁰;L²(Ω))

. Define a Banach spaceB:=L^∞(0, t⁰;H¹(Ω))×L^∞(0, t⁰; (L²(Ω))^d) with norm

k(x₁, x₂)k_B =q

kx₁k²_{L∞(0,t0;H1(Ω))}+kx₂k²

L^∞(0,t⁰;(L²(Ω))^d,

then

kP_k+1−Pkk_B ≤C(t⁰)^1/2kP_k−Pk−1k_B.

We shrink [0, t⁰] into [0, t∗] such thatC(t∗)^1/2 <1 sinceC is independent of the time interval. Consequently let k > l,

kP_k−P_lk_B ≤ kP₁−P0k_B

k−1

X

j=l

(C(t∗)^1/2)^j.

Hence {P_k = (P_k⁰, P_k^d)}^∞_k=0 is a Cauchy sequence in B, and therefore there exists a point P^∗ = ((P^∗)⁰,(P^∗)^d) ∈ L^∞(0, t∗;H¹(Ω))×L^∞(0, t∗; (L²(Ω))^d) withP_k→P^∗ in L^∞(0, t∗;H¹(Ω))×L^∞(0, t∗; (L²(Ω))^d).

Recall the uniform bounds of {P_k⁰}^∞_k=0 it follows

Lemma 3.3.4. The limit P^∗ in Lemma 3.3.3 satisfies P^∗∈A^∗₁∩A^∗₂∩A^∗₃

with

∂tP^∗∈L²(0, t∗;H²(Ω))×L²(0, t∗; (H¹(Ω))^d) where

A^∗₁ : =C([0, t∗];H²(Ω))×C([0, t∗]; (H¹(Ω))^d), A^∗₂ : =L^∞(0, t∗;H³(Ω))×L^∞(0, t∗; (H²(Ω))^d), A^∗₃ : =C([0, t∗];C¹(Ω))×C([0, t∗]; (C(Ω))^d).

Proof. Since the embeddings H³(Ω) ,→ H²(Ω), H²(Ω) ,→ H¹(Ω) are com-pact, (P_k⁰, P_k^d) (k = 0,1,· · ·) are bounded in L^∞(0, t∗;H³(Ω)× (H²(Ω))^d) and ( ˙P_k⁰,P˙_k^d) (k = 0,1,· · ·) are bounded in L²(0, t∗;H¹(Ω)×(L²(Ω))^d) from Lemma 3.3.2, then Aubin’s Lemma (Corollary 4 in [72]) yields {P_k⁰}^∞_k=0 and {P_k^d}^∞_k=0 are relatively compact in C([0, t∗], H²(Ω)) and C([0, t∗], H¹(Ω)) re-spectively, i.e., there are a subsequence

{(P_k⁰_s, P_k^d_s)}^∞_s=0⊆ {(P_k⁰, P_k^d)}^∞_k=0 and a function

P_S= (P_S⁰, P_S^d)∈C([0, t∗];H²(Ω))×C([0, t∗]; (H¹(Ω))^d) (3.3.127) such that

(P_k⁰_s, P_k^d_s)→(P_S⁰, P_S^d) in C([0, t∗];H²(Ω))×C([0, t∗]; (H¹(Ω))^d). (3.3.128)

Furthermore from the uniform bounds ofPk(see Lemma 3.3.2) there exists a subsequence{(P_k⁰

sm, P_k^d sequence of simple functions{f_j} such that

j→∞lim

where {A^j₁,· · · , A^j_k} is a finite collection of mutually disjoint subsets of (0, t∗) Recall (3.3.131) it follows

j→∞lim Combining (3.3.133) and (3.3.134) we obtain

ω

From a same reasoning ω

From (3.3.130) then it follows ω from a similar reasoning.

As derived above

P_k⁰_sm −→P_S⁰, inC([0, t∗];H²(Ω)) as m→ ∞, P_k^d_sm −→P_S^d, inC([0, t∗]; (H¹(Ω))^d) as m→ ∞,

we infer

P_k⁰_sm *^∗P_S⁰, inL^∞(0, t∗;H²(Ω)) asm→ ∞, P_k^d_sm *^∗P_S^d, inL^∞(0, t∗; (H¹(Ω))^d) as m→ ∞.

Since (H¹(Ω))^∗⊂(H²(Ω))^∗⊂(H³(Ω))^∗, then from (3.3.129) P_k⁰_sm *^∗ P_M⁰ , inL^∞(0, t∗;H²(Ω)) asm→ ∞, P_k^d_sm *^∗ P_M^d, inL^∞(0, t∗; (H¹(Ω))^d) as m→ ∞.

This implies

P_S⁰=P_M⁰ , P_S^d=P_M^d. (3.3.139) Fix 0< γ < 1

2 and anyt∈(0, t∗) we deduce from the interpolation inequality (C.0.1) that

kP_k⁰

sm −P_M⁰ k_H3−γ(Ω)≤CinkP_k⁰

sm −P_M⁰ k^(3−γ)/3_H3(Ω) kP_k⁰

sm −P_M⁰ k^γ/3_L2(Ω), then by Lemma 3.3.2

kP_k⁰

sm −P_M⁰ k_H3−γ(Ω)≤Cin2^(3−γ)/3C^0(3−γ)/3kP_k⁰

sm −P_M⁰ k^γ/3_L2(Ω). (3.3.140) Similarly

kP_k^d_sm −P_M^dk_H^2−γ_(Ω)≤Cin2^(2−γ)/2C^0(2−γ)/2kP_k^d_sm −P_M^dk^γ/2_L2(Ω). (3.3.141) From (3.3.140), (3.3.141) and Sobolev embedding theorem we infer

(P_k⁰_sm,∇P_k⁰

sm, P_k^d_sm)→(P_M⁰ ,∇P_M⁰ , P_M^d) in C([0, t∗]×Ω). (3.3.142) FromLemma 3.3.3 the subsequence {P_ksm}^∞_m=1 converges toP^∗ in

L^∞(0, t∗;H¹(Ω))×L^∞(0, t∗; (L²(Ω))^d).

This implies

P_ksm *^∗ P^∗ inL^∞(0, t∗;H¹(Ω))×L^∞(0, t∗; (L²(Ω))^d).

From (3.3.129) and the embeddingsL²(Ω)⊂(H²(Ω))^∗, (H¹(Ω))^∗ ⊂(H³(Ω))^∗ we conclude

Pk_sm *^∗ (P_M⁰ , P_M^d) in L^∞(0, t∗;H¹(Ω))×L^∞(0, t∗; (L²(Ω))^d).

By the uniqueness of the limit

P^∗ = (P_M⁰ , P_M^d).

Furthermore using (3.3.127), (3.3.139), (3.3.129) and (3.3.142) we complete the proof.

Remark 3.3.2. This lemma ensures that the limit function (n, J) := P^∗ + (n₀, J₀) satisfies the initial conditions (3.0.1) and the boundary conditions from (3.0.3).

It is also obvious that the sequence {V_k}^∞_m=0 converges to a limit V in the space C([0, t∗];H²(Ω)) which solves the Possion equation λ²4V = n− C(x)

Then we find

Going back to (3.3.144) we find

C(ϕ) >0 is a constant depending upon ϕ. Substitute (3.3.145) into (3.3.146), UsingLemma 3.3.3 we conclude

By a similar calculation it is easy to verify that ²

Thus we conclude thatP^∗ is the solution we seek.

Uniqueness

Let (n¹, J¹, V¹) and (n², J², V²) be two solutions of (3.0.1)-(3.0.4). Put n4=n¹−n², J4 =J¹−J², V4 =V¹−V², then we obtain the system



Similarly as (3.3.28)-(3.3.33) we obtain 1

2∂t

T0kn₄k²+²

4k∇n₄k²+kJ₄k²

+ν0T0k∇n₄k² +²

4ν₀k4n₄k²+ν₀k∇J₄k²+1 τkJ₄k²

=(F₁−F₂, J4)≤ kF₁−F₂k_H−1(Ω)kJ₄k_H1 0(Ω).

(3.3.147)

We go back to the proof ofLemma3.3.3. By a similar calculations as (3.3.121)-(3.3.126) we deduce

kF₁−F₂k²_H−1(Ω) ≤C(kn₄k²_H1(Ω)+kJ₄k²_L2(Ω)). (3.3.148) (3.3.147) and (3.3.148) together with Cauchy’s inequality yield

1 2∂t

T0kn₄k²+²

4k∇n₄k²+kJ₄k²

≤C(kn₄k²_H1(Ω)+kJ₄k²_L2(Ω)).

(3.3.149)

Applying Gronwall’s lemma on (3.3.149) then n4≡J4 ≡0.

Global Existence and

Exponential Decay on a Torus

We assume that the doping profile of background charges is constant, denoted by C₀. Then it is easy to verify that (C₀,0,0) is a steady state of (3.0.1) on a torus T^dor on a bounded domain Ω under insulating boundary conditions. In order to extend the local solution of (3.0.1)-(3.0.3) globally in time we need to establish uniform bounds. Here we only consider the situation when the initial data is close to the steady state (C₀,0,0).

4.1 Reformulation

We consider the viscous QHD model (3.0.1) on ad-dimensional torus T^d. The local existence of solutions to the viscous model of quantum hydrodynamics on a torus is proved in [17]. We will study the situation when the initial data are assumed in a small neighborhood of the steady state (C₀,0,0). The idea of extending the local classical solution globally in time is first to establish uniform estimates, and then to use the usual continuity argument.

Suppose

x∈Ωinf n0(x)>0, (4.1.1) Z

Ω

(n₀− C₀)dx= 0. (4.1.2)

The second condition (4.1.2) is necessary, otherwise the Poisson equation for V would not be solvable from Green’s formulas. Note that (4.1.1) and (4.1.2) imply C₀ >0.

In the following, we use the abbreviation

u=n− C₀, u0 =n0− C₀. (4.1.3) Now we reformulate the original problem (3.0.1) into an equivalent one with respect to the classical solution (u, J, V) which describes the perturbation to

105

the steady state (C₀,0,0).

Differentiate (4.1.4) with respect to t, take divergence of (4.1.5), add the resul-tants, then substitute (4.1.8) (4.1.9) into it, we obtain a nonlinear fourth-order wave equation for u

utt+ 1

Introduce the perturbations of the steady-state (C₀,0,0) u=n− C₀, J =J−0, V =V −0.

Then, using (4.1.4)-(4.1.6) and (4.1.10), the evolution equations for (u, J, V) read as follows

where

H:=G+T₀∇u−(u+C₀)∇V −² 4∇4u.

The initial data are given:

u(0, x) =u0(x), ut(0, x) =u1(x), J(0, x) =J0(x) (4.1.12) with

u1(x) := divJ0+ν04n₀.

Im Dokument Solutions to the Multi-Dimensional Viscous Quantum Hydrodynamic Equations for Semiconductors (Seite 73-117)