Approach by Galerkin-Approximation



d+1

X

j=1

|||f_d+2−j|||_{md+2−j,p,Ω}+

d+1

X

j=1

|||g_d+2−j|||_2−r

d+2−j−_p¹,p,∂Ω





(3.2.59)

holds where the constant c does not depend upon f, g, η and we use the norms depending on a parameter η ∈ C/{0}, namely let v ∈ W_p^s(Ω), g ∈ W^s−

1

p p(∂Ω) where s is an integer satisfying 1≤s≤m,

|||v|||_s,p,Ω :=kvk_Ws

p(Ω)+|η|^mskvk_Lp(Ω), (3.2.60)

|||g|||_s−1

p,p,∂Ω :=kgk

W^s−

1p p (∂Ω)

+|η|

s−1 p

m kgk_Lp(∂Ω). (3.2.61) Proof. Since the boundary problem (3.2.32) is elliptic with parameter in L from Theorem 3.2.5, the existence and the a priori estimate follow directly from Theorem2.2.1.

3.2.3 Approach by Galerkin-Approximation

We use Galerkin’s method to solve the linear system (3.2.1). The idea of this method is that one first constructs a family of approximate problems whose so-lutions satisfy certain a priori estimates. This yields a sequence of approximate solutions that have uniform bounds. Uniform bounds imply the existence of a weakly convergent subsequence. One then shows that the weak limit is the solution we seek.

Preliminaries

We introduce some basic materials on functional analysis and Sobolev spaces which will be needed in the later procedure.

Theorem 3.2.7. (Rellich selection theorem)

Let Ω⊆Rⁿ be a bounded open subset. Every bounded sequence in H₀¹(Ω)has a subsequence, which converges in the norm ofL²(Ω).

Theorem 3.2.8. (Eigenvectors of a compact, symmetric operator)

Let H be a separable Hilbert space, and suppose S : H → H is a compact and symmetric operator. Then there exists a countable orthonormal basis ofH which consists of eigenvectors ofS.

We define a bilinear form inH₀¹(Ω)∩H²(Ω) B : H₀¹(Ω)∩H²(Ω)

× H₀¹(Ω)∩H²(Ω)

→R with

B(u, v) = (u, v)_L2+ (∇u,∇v)_L2 + (4u,4v)_L2. (3.2.62) It is easy to check that B(u, v) is an inner product in H₀¹(Ω)∩H²(Ω). The norm defined by this inner product inH₀¹(Ω)∩H²(Ω) is equivalent to the norm defined by

(u, v)_H²_(Ω)= X

|α|≤2

(D^αu, D^αv) from the following fact:

Theorem 3.2.9. Suppose that f ∈ H⁻¹(Ω), Ω is C¹, u ∈ H₀¹(Ω) is a weak solution of the elliptic boundary-value problem

(4u=f, in Ω u= 0, on∂Ω

Assume finallyf ∈H^m(Ω), ∂Ω isC^m+2 (m= 0,1,2,· · ·). Then u∈H^m+2(Ω)

and we have the estimate

kuk_Hm+2(Ω)≤C4,m,Ωkfk_Hm(Ω), the constantC_4,m,Ω depends only on m, Ω.

Proof. The proof can be found in [33] pp. 323.

Additionally we record the following embedding from [33] pp. 288

Theorem 3.2.10. Let Ω be open, bounded in R^d, T > 0, m be a nonnegative interger. Assume the boundary∂Ωis smooth.

Suppose u∈L²(0, T;H^m+2(Ω)), with ∂_tu∈L²(0, T;H^m(Ω)), then u∈C([0, T];H^m+1(Ω)),

and the estimates

kuk_C([0,T];Hm+1(Ω))≤C kuk_L2(0,T;H^m+2(Ω))+k∂_tuk_L2(0,T;H^m(Ω))

holds, where C >0 depends only on T, Ω, and m.

Then we obtain the following corollary:

Theorem 3.2.11. Let u ∈ L²(0, T;H₀¹(Ω)∩H²(Ω)), ∂tu ∈ L²(0, T;L²(Ω)), then the function

t7→ k∇uk² is absolutely continuous and

d

dtk∇uk² =−2(∂_tu,4u)_L2(Ω) (3.2.63) for a.e 0≤t≤T.

Proof. FromTheorem 3.2.10 it follows direct thatu∈C([0, T];H¹(Ω)).

Fix any point t in (0, T) there exists σ > 0 such that t ∈ (ε, T −ε) for all ε≤σ. Setu^ε =η_ε∗u,η_εdenoting the usual mollifier onR¹, i.e.,η_ε=ε⁻¹η(t/ε) where

η(t) =

(ke^{−1/(1−|t|2)} if|t|<1, 0 if|t| ≥1, k >0 is selected so that

Z

R

η(t)dt= 1.We find first that

ηε(t) = 0 if |t| ≥ε, Z ε

−ε

ηε(t)dt= 1.

Furthermore u^ε(t) =

Z T 0

ηε(t−y)u(y)dy= Z ε

−ε

ηε(y)u(t−y)dy, t∈(ε, T −ε) in the sense of Bochner integral with respect to H₀¹(Ω)∩H²(Ω). It is easy to calculate that

dⁿu^ε(t) dtⁿ =

Z T 0

dⁿη_ε

dtⁿ (t−y)u(y)dy for any n∈N, which implies

u^ε∈C^∞((ε, T −ε);H₀¹(Ω)∩H²(Ω))).

Choose a s ∈ (ε, T −ε), without loss of generality, let s < t. Then u^ε ∈ C^∞([s, t];H¹(Ω)). Now forx∈[s, t]

ku^ε(x)−u(x)k_H1(Ω) =

Z ε

−ε

ηε(y)u(x−y)dy− Z ε

−ε

ηε(y)u(x)dy H¹(Ω)

=

Z ε

−ε

η_ε(y)(u(x−y)−u(x))dy H¹(Ω)

≤ sup

y∈(−ε,ε)

ku(x−y)−u(x)k_H1(Ω).

Since u is uniformly continuous on [s, t] in H¹(Ω), i.e., u ∈ C([s, t];H¹(Ω)), then for anyδ > 0 there existsθ >0 independent of x∈[s, t] such that for all x1, x2 ∈[s, t] with|x₁−x2|< θ ku(x₁)−u(x2)k_H1(Ω) < δ. Thus if ε < θ, we obtain

sup

y∈(−ε,ε)

ku(x−y)−u(x)k_H1(Ω)< δ, which implies

ku^ε(x)−u(x)k_H1(Ω) < δ

for all x ∈ [s, t]. Then we conclude that u^ε converges tou in C([s, t];H¹(Ω)) asε→0.Furthermore from properties of mollifiers we obtain also

( u^ε−→u in L²_loc(0, T;H₀¹(Ω)∩H²(Ω))), (u^ε)⁰ −→u⁰ in L²_loc(0, T;L²(Ω))

asε−→0. Then d

dtk∇u^ε(t)k² = 2(∇u^ε(t),∇(u^ε(t))⁰)_L²_(Ω)=−2((u^ε(t))⁰,4u^ε(t))_L²_(Ω), consequently for anys∈(ε, T −ε)

k∇u^ε(s)k² =k∇u^ε(t)k²+ Z s

t

−2((u^ε(τ))⁰,4u^ε(τ))_L²_(Ω)

dτ. (3.2.64) Next we find

k∇u^ε(s)k²− k∇u(s)k² = Z

Ω

(∇u^ε(s)− ∇u(s))(∇u^ε(s) +∇u(s))dx

≤ k∇u^ε(s)− ∇u(s)k_L2(Ω)k∇u^ε(s) +∇u(s)k_L2(Ω)

≤ k∇u^ε(s)− ∇u(s)k²_L2(Ω)

+ 2k∇u(s)k_L2(Ω)k∇u^ε(s)− ∇u(s)k_L2(Ω).

Since u^ε(s) converges to u(s) in H¹(Ω) as ε→0, thenk∇u^ε(s)k² → k∇u(s)k² asε→0. Recalling (3.2.64), it follows

k∇u(s)k² =k∇u(t)k²+ Z s

t

−2((u(τ))⁰,4u(τ))_L2(Ω)

dτ.

Definition of Weak Solutions

Assume thatu is a classical solution to (3.2.1), then multiply (3.2.1) by (v⁰, v¹,· · · , v^d) =: (v⁰, v^d) =:v∈(C₀^∞(Ω_T))^1+d

and integrate by parts, then we obtain the following equations









 Z T

0

B1(u, v)dt= Z T

0

(F⁰, v⁰)dt, Z T

0

B₂(u, v)dt= Z T

0

(F^d, v^d)dt,

(3.2.65)

with

B₁(u, v) := (∂_tu⁰, v⁰)−ν₀(v⁰,4u⁰)−(v⁰,divu^d), B2(u, v) := (∂tu^d, v^d) +ν0(∇v^d,∇u^d) + 1

τ(u^d, v^d)−T0(∇u⁰, v^d)−²

4(4u⁰,divv^d), which give us the motivation for definition of weak solutions to (3.2.1).

Definition 3.2.2. Assume that

F⁰ ∈L²(0, T;L²(Ω)), F^d ∈L²(0, T; (H⁻¹(Ω))^d), g⁰ ∈H₀¹(Ω), g^d ∈L²(Ω)

We say a function

u∈L²(0, T; (H₀¹(Ω)∩H²(Ω))×(H₀¹(Ω))^d)∩H¹(0, T;L²(Ω)×(H⁻¹(Ω))^d) is a weak solution of the system (3.2.1) provided (3.2.65) holds for each v ∈ L²(0, T; (H₀¹(Ω))^1+d), andu(0) = (g⁰, g^d)^T.

Existence and Uniqueness

Theorem 3.2.12. Assume that the given functions

F⁰ ∈L²(0, T;L²(Ω)), F^d ∈L²(0, T; (H⁻¹(Ω))^d), g⁰ ∈H₀¹(Ω), g^d ∈L²(Ω)

then (3.2.1) has a unique weak solution which satisfies the following a priori estimates

ku⁰k²_L2(0,T;H₀¹(Ω)∩H²(Ω))+ku^dk²_L2(0,T;H₀¹(Ω))+k∂_tuk²_L2(0,T;L²(Ω)×(H⁻¹(Ω))^d)

≤C

kF⁰k²_L2(0,T;L²(Ω))+kF^dk²_L2(0,T;(H⁻¹(Ω))^d)+kg⁰k²_H1

0(Ω))+kg^dk²_L2(Ω)

, where the constantC does not depend upon F.

Remark 3.2.2. From Definition (3.2.2) and Theorem3.2.10 it followsu∈ C([0, T], H¹(Ω)×(L²(Ω))^d), which allows us to interpret the initial condition.

Proof. Our idea is the Faedo-Galerkin method. More precisely, we expect a set of functions φj = φj(x) (j = 1,2, ...) in H₀¹(Ω)∩H²(Ω) such that {φ_j}^∞_j=1 is an orthonormal basis ofL²(Ω) and an orthogonal basis in H₀¹(Ω) andH₀¹(Ω)∩ H²(Ω) respectively. We take {φ_j}^∞_j=1 to be the complete set of normalized eigenfunctions for the operator−4in H₀¹(Ω). SetS the solution operator

S :L²(Ω)→L²(Ω) f 7→u=Sf, uis the weak solution of

−4u=f, u|_∂Ω = 0.

It is obvious thatS is symmetric. Furthermore from Rellich selection theorem we check that S is compact, which implies all the eigenvalues of S are real, nonzero and there are corresponding eigenfunctions suitably normalized, which make up an orthonormal basis in L²(Ω). But observe as well that for η 6= 0, we have Sf =ηf if and only if −4f = λf for λ = 1

η. Let {η_j}^∞_j=1 comprise the sequence of distinct eigenvalues ofS,Hj the eigenspace with respect toηj, we obtain thereby an orthonormal basisφ_j =φ_j(x) (j = 1,· · ·) of normalized eigenvectors inHj of L²(Ω). Actuallyφj ∈H₀¹(Ω)∩C^∞(Ω),4φ_j ∈H₀¹(Ω) and we have the following calculations

(∇φ_i,∇φ_j)_L2 =η_i(φ_i, φ_j)_L2 = 0,

−4φ_i =ηiφi =⇒ 4²φi =ηi(−4φ_i) =η_i²φi,

=⇒(4φ_i,4φ_j)_L² =η²_i(φi, φj)_L² = 0.

Therefore{φ_j}^∞_j=1is orthogonal inH₀¹(Ω)∩H²(Ω) with respect to (3.2.62). We claim further that{φ_j}^∞_j=1 is complete inH₀¹(Ω)∩H²(Ω). To see this, it suffices to verify that for u ∈ H₀¹(Ω)∩H²(Ω), B(φj, u) = 0 implies u ≡0. From the

properties of φj, B(φj, u) = (1 +ηj +η_j²)(φj, u)_L². The completeness follows immediately since {φ_j}^∞_j=1 is complete in L²(Ω). That {φ_j}^∞_j=1 is complete in H₀¹(Ω) follows from the same reason. Via normalization of {φ_j}^∞_j=1 in H₀¹(Ω)

ωj := φj

kφ_jk_H1 0(Ω)

!

we obtain an orthonormal basis {ω_j}^∞_j=1 inH₀¹(Ω).

Fix now a positive integer m. We will look for 1 +d functions u⁰_m, u^d_m :=

(u¹_m,· · · , u^d_m)^T with

u⁰_m : [0, T]→H₀¹(Ω)∩H²(Ω), (3.2.66) u^d_m : [0, T]→(H₀¹(Ω))^d, (3.2.67) of the form

u⁰_m(t) :=

m

X

j=1

d^j_u0

m(t)ωj, (3.2.68)

u^d_m(t) :=





m

X

j=1

d^j_u1 m(t)φj,

m

X

j=1

d^j_u2

m(t)φj,· · ·,

m

X

j=1

d^j_ud m(t)φj





T

, (3.2.69)

where we hope to select the coefficients d^j_u0

m(t), d^j_ul

m(t) (0 ≤ t ≤ T, l = 1, ..., d, j = 1, ..., m) so that

d^j_u0

m(0) = (g⁰, ωj)_H¹

0(Ω), d^j_ul

m(0) = (g^l, φj) (3.2.70) and















(∂_tu⁰_m, ω_j) +ν₀(∇ω_j,∇u⁰_m) + (∇ω_j, u^d_m) = (F⁰, ω_j), (∂tu^l_m, φj) +ν0(∇φ_j,∇u^l_m) +1

τ(u^l_m, φj)

−T₀(∂_xlu⁰_m, φ_j) +²

4(∂_xl4u⁰_m, φ_j) = (F^l, φ_j).

(3.2.71)

Thus we seek functionsu⁰_m,u^d_m of the form (3.2.68) and (3.2.69) respectively, that satisfy the ”projection” (3.2.71) of problem (3.2.1) onto the finite dimen-sional subspaces spanned by {φ_j}^m_j=1. Assuming u⁰_m, u^d_m have the structures (3.2.68), (3.2.69) respectively, we first note that

(∂tu⁰_m(t), ωj) = 1 kφ_jk²

H₀¹(Ω)

(d^j_u0

m(t))⁰, (3.2.72) (∂tu^l_m(t), φj) = (d^j_ul

m(t))⁰, (3.2.73)

since {φ_j}^∞_j=1 forms an orthonormal basis in L²(Ω). Then (3.2.71) becomes a linear system of ODE subject to the initial conditions (3.2.70), which may be

written in the form

d⁰(t) =Ad(t) +b(t), d(0) =a,

wherea is via (3.2.70) solvable. Forb(t)∈L²(0, T) this equation has a unique solution in{v∈H¹(0, T)|v(0) = 0}, more precisely,d(t) has the form

d(t) = Z t

0

e^A(t−s)b(s)ds+e^Ata.

Multiplying (3.2.71) by d^j_u0

m(t), d^j_ul

m(t) respectively, sum forj = 1,· · ·, m, l= 1,· · · , d, and then recall (3.2.68) and (3.2.69), we obtain















(∂_tu⁰_m, u⁰_m) +ν₀(∇u⁰_m,∇u⁰_m) + (∇u⁰_m, u^d_m) = (F⁰, u⁰_m), (∂_tu^d_m, u^d_m) +ν₀(∇u^d_m,∇u^d_m) +1

τ(u^d_m, u^d_m)

−T₀(∇u⁰_m, u^d_m) +²

4(∇4u⁰_m, u^d_m) = (F^d, u^d_m).

(3.2.74)

Next we will get the energy estimates of (u⁰_m, u^d_m)^T. Since−4ω_j =ηjωj, then

−4u⁰_m =

m

X

j=1

d^j_u0

m(−4ω_j) =

m

X

j=1

d^j_u0 mηjωj. Multiplying the first equation of (3.2.71) byd^j_u0

mη_j, summing up forj= 1,· · · , m yields

(∂_t∇u⁰_m,∇u⁰_m) +ν₀(4u⁰_m,4u⁰_m)−(∇4u⁰_m, u^d_m) =−(F⁰,4u⁰_m). (3.2.75) Combining (3.2.74) and (3.2.75) we obtain the following equation

1

2∂_t T₀ku⁰_mk²+²

4k∇u⁰_mk²+ku^d_mk²

+ν₀T₀k∇u⁰_mk² +²

4ν0k4u⁰_mk²+ν0k∇u^d_mk²+ 1 τku^d_mk²

=T₀(F⁰, u⁰_m) + (F^d, u^d_m)−²

4(F⁰,4u⁰_m).

(3.2.76)

Integrating (3.2.76) from 0 toT yields T0

2 ku⁰_m(T)k²+1

2ku^d_m(T)k²+²

8k∇u⁰_m(T)k² +ν₀T₀k∇u⁰_mk²_L2(0,T;L²(Ω)) +²

4ν₀k4u⁰_mk²_L2(0,T;L²(Ω))

+ν0k∇u^d_mk²_L2(0,T;L²(Ω))+ 1

τku^d_mk²_L2(0,T;L²(Ω))

= Z T

0

T₀(F⁰, u⁰_m) + (F^d, u^d_m)−²

4(F⁰,4u⁰_m)

dt +T₀

2 ku⁰_m(0)k²+1

2ku^d_m(0)k²+²

8k∇u⁰_m(0)k², which implies

T0

2 ku⁰_m(T)k²+1

2ku^d_m(T)k²+²

8k∇u⁰_m(T)k² +ν0T0k∇u⁰_mk²_L2(0,T;L²(Ω)) +²

4ν0k4u⁰_mk²_L2(0,T;L²(Ω))

+ν₀k∇u^d_mk²_L2(0,T;L²(Ω))+ 1

τku^d_mk²_L2(0,T;L²(Ω))

≤c₁ Z T

0

kF⁰k_H−1(Ω)ku⁰_mk_H1(Ω)+c2

Z T 0

kF^dk_H−1(Ω)ku^d_mk_H1(Ω)

+c3

Z T 0

kF⁰k_L2(Ω)k4u⁰_mk_L2(Ω)

+ max T₀

2 ,1 2,²

8

(kg⁰k²_H1

0(Ω)+kg^dk²_L2(Ω))

(3.2.77)

sinceku⁰_m(0)k²

H₀¹(Ω) ≤ kg⁰k²

H₀¹(Ω),ku^d_m(0)k²_L2(Ω)≤ kg^dk²_L2(Ω) by (3.2.70).

Applying Poincare’s inequality

ku⁰_mk²_L2(Ω) ≤c₄k∇u⁰_mk²_L2(Ω), (c₄ is a constant independent ofm) and Cauchy’s inequality we obtain

ku⁰_mk²_L2(0,T;H²(Ω))+ku^d_mk²_L2(0,T;H¹(Ω))

≤c₅

kF⁰k²_L2(0,T;L²(Ω))+kF^dk²_L2(0,T;(H⁻¹(Ω))^d)+kg⁰k²_H1

0(Ω)+kg^dk²_L2(Ω)

. (3.2.78) Fix anyh ∈L²(Ω), with khk_L2(Ω) ≤1, and write h =h¹+h², where h¹ ∈ span{ω_j}^m_j=1 and (h², ωj) = 0 (j = 1,· · ·, m). Since the functions {ω_j}^∞_j=1 are

orthogonal in L²(Ω), kh¹k_L2(Ω) ≤ khk_L2(Ω) ≤1. Utilizing (3.2.71), we deduce for a.e. 0≤t≤T that

(∂_tu⁰_m, h¹)−ν₀(h¹,4u⁰_m)−(h¹,divu^d_m) = (F⁰, h¹) (3.2.79) then we obtain

|(∂_tu⁰_m, h)|=|(∂_tu⁰_m, h¹)|=|(F⁰, h¹) +ν0(h¹,4u⁰_m) + (h¹,divu^d_m)|

≤c7(kF⁰k_L2(Ω)+k4u⁰_mk_L2(Ω)+kdivu^d_mk_L2(Ω)), which yields

k∂_tu⁰_mk_L2(Ω)≤c₇(kF⁰k_L2(Ω)+ku⁰_mk_H2(Ω)+ku^d_mk_H1(Ω)), and

Z T 0

k∂_tu⁰_mk²_L2(Ω)dt≤c₈ Z T

0

(kF⁰k²_L2(Ω)+ku⁰_mk²_H2(Ω)+ku^d_mk²_H1(Ω))dt, since (3.2.78) holds, we obtain the uniform bounds of∂tu⁰_m inL² 0, T;L²(Ω)

k∂_tu⁰_mk²_L2(0,T;L²(Ω))

≤c9 kF⁰k²_L2(0,T;L²(Ω))+kF^dk²_L2(0,T;(H⁻¹(Ω))^d)+kg⁰k²_H1

0(Ω)+kg^dk²_L2(Ω)

. (3.2.80) Next fix anyh∗∈H₀¹(Ω), with kh_∗k_H1

0(Ω) ≤1, and write h=h¹_∗+h²_∗, where h¹_∗ ∈span{ω_j}^m_j=1 and (h²_∗, ωj) = 0 (j= 1,· · ·, m). Since the functions {ω_j}^∞_j=1 are orthogonal inH₀¹(Ω), kh¹_∗k_H1

0(Ω)≤ kh∗k_H1

0(Ω) ≤1. By an analogous reason we find

k∂_tu^d_mk²_L2(0,T;(H−1(Ω))^d)

≤c10 kF⁰k²_L2(0,T;L²(Ω))+kF^dk²_L2(0,T;(H⁻¹(Ω))^d)+kg⁰k²_H1

0(Ω)+kg^dk²_L2(Ω)

. (3.2.81) Then (3.2.78), (3.2.80), and (3.2.81) are sufficient to ensure the existence of a subsequence {u⁰_m

s}^∞_s=1 ⊂ {u⁰_m}^∞_m=1 and a subsequence {u^d_m

s}^∞_s=1 ⊂ {u^d_m}^∞_m=1 and functions

u⁰ ∈L²(0, T;H₀¹(Ω)∩H²(Ω)), (3.2.82) v⁰ ∈L² 0, T;L²(Ω)

, (3.2.83)

u^d ∈L²(0, T; (H₀¹(Ω))^d), (3.2.84) v^d ∈L²(0, T; (H⁻¹(Ω))^d), (3.2.85)

such that

By (3.2.86) we obtain Z T Fix an integerN and choose functions

f ∈C¹([0, T];H₀¹(Ω))

where {d^j_f(t)}^N_j=1, {d^j_hs(t)}^N_j=1 (s = 1,2,· · ·, d) are given smooth functions.

We claim that functions of the form (3.2.87), (3.2.88) are dense in L² 0, T;H₀¹(Ω)

and L²(0, T; (H₀¹(Ω))^d)

respectively. To prove this let g ∈ L²([0, T], H₀¹(Ω)), ϑ > 0. The assertion follows if we can show that there is a functionp of the form (3.2.87) with

kp−gk_L2(^0,T;H0¹(Ω))< ϑ.

Let gn:=

The sequence of functions



converges pointwise to kg(t)k²_H1 0(Ω)

sincegm is an interior point of{f ∈L²(0, T;H₀¹(Ω))|kf−gk_L2(0,T;H₀¹(Ω)) < ϑ}.

Using the dense embedding C₀^∞((0, T)) ,→ L²((0, T)) then for each aj (j = 1,· · · , m) we can select a sequence (a_j,n)_n⊂C₀^∞((0, T)) such that a_j,n→a_j in L²((0, T)) forn→ ∞.

fn:=

m

X

j=1

aj,n(t)wj −→gm

inL²(0, T;H₀¹(Ω)) since

kf_n−g_mk_L2(0,T;H₀¹(Ω)) =

m

X

j=1

(a_j,n−a_j)w_j

L²(0,T;H₀¹(Ω))

≤

m

X

j=1

k(a_j,n−a_j)w_jk_L2(0,T;H1 0(Ω))=

m

X

j=1

ka_j,n−a_jk_L2((0,T)). Thus there existsM ∈Nsuch that for alln≥M fn belongs to

{f ∈L²(0, T;H₀¹(Ω))|kf −gmk_L2(0,T;H₀¹(Ω))< r}.

Consequently from (3.2.91)kf_n−gk_L2(0,T;H₀¹(Ω))< ϑ.

The system (3.2.89) holds for all f ∈L² 0, T;H₀¹(Ω)

and h^d∈L²(0, T; (H₀¹(Ω))^d), since functions of the form (3.2.87), (3.2.88) are dense in

L² 0, T;H₀¹(Ω)

and L²(0, T; (H₀¹(Ω))^d) respectively.

In order to proveu⁰(0) =u^d(0) =g, we first note from (3.2.86) that



















u⁰_ms *^∗ u⁰ weakly star in L²(0, T;H⁻¹(Ω)),

∂tu⁰_ms *^∗ ∂tu⁰ weakly star in L² 0, T;H⁻¹(Ω) , u^d_ms *^∗ u^d weakly star in L²(0, T; (H⁻¹(Ω))^d),

∂tu^d_ms *^∗ ∂tu^d weakly star in L²(0, T; (H⁻¹(Ω))^d),

(3.2.92)

then from TheoremB.0.7 we obtain

(u⁰_ms, u^d_ms)(0)*^∗(u⁰, u^d)(0) weakly star in H⁻¹(Ω)×(H⁻¹(Ω))^d. By the uniqueness of the limit, we get

u(0) =g.

Let (u⁰₁, u^d₁) and (u⁰₂, u^d₂) be two weak solutions of (3.2.1), define u⁰₄ =u⁰₁−u⁰₂, u^d₄=u^d₁ −u^d₂

then we get the system

∂_tu⁰₄−ν₀4u⁰₄−divu^d₄ = 0, (3.2.93)

∂_tu^d₄−ν₀4u^d₄+²

4∇4u⁰₄−T₀∇u⁰₄+ 1

τu^d₄ = 0 (3.2.94) with

(u⁰₄, u^d₄)(0) =0, (u⁰₄, u^d₄)|_∂Ω=0.

To get the uniqueness we first note− Z

Ω

∂tu⁰₄4u⁰₄dx= 1

2∂tk∇u⁰₄k²by theorem 3.2.11.

Taking the L²(Ω) scalar product of (3.2.93) with −²

44u⁰₄ then for a.e. 0 ≤ t≤T

1 2

²

4∂_tk∇u⁰₄k²+²

4ν₀k4u⁰₄k²−²

4(∇4u⁰₄, u^d₄) = 0. (3.2.95) Taking the L²(Ω) scalar product of (3.2.93) with T0u⁰₄; (3.2.94) with u^d₄ we obtain for a.e. 0≤t≤T

1

2T₀∂_tku⁰₄k²+ν₀T₀k∇u⁰₄k²−T₀(divu^d₄, u⁰₄) = 0, (3.2.96) and

1

2∂_tku^d₄k²+1

τku^d₄k+ν₀k∇u^d₄k²+T₀(divu^d₄, u⁰₄) +²

4(∇4u⁰₄, u^d₄) = 0.

(3.2.97)

Summing (3.2.95)-(3.2.97) yields 1

2∂t

T0ku⁰₄k²+²

4k∇u⁰₄k²+ku^d₄k²

+ν0T0k∇u⁰₄k²+²

4ν0k4u⁰₄k² +ν0k∇u^d₄k²+ 1

τku^d₄k²= 0,

(3.2.98) thenu⁰₄=u^d₄ = 0.

Regularity of the Weak Solution

Theorem 3.2.13. Concerning the linear problem











∂tu(t, x) +A(∂x)u(t, x) =F(t, x), (t, x)∈[0, T)×Ω u(0, x) = 0,

u|_∂Ω= 0 for a.e. t∈[0, T].

(3.2.99)

Let T >0 and











F⁰ ∈L^∞(0, T;H¹(Ω)), F^d∈L^∞(0, T; (L²(Ω))^d);

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

F⁰(0, x)∈H₀¹(Ω), F^d(0, x)∈L²(Ω),

(3.2.100)

then the weak solutionu= (u⁰, u^d)of (3.2.99)has the following global regularity



















u⁰ ∈L^∞(0, T;H³(Ω))∩C([0, T], C¹(Ω))∩C([0, T], H²(Ω)), u^d∈L^∞(0, T;H²(Ω))∩C([0, T], C(Ω))∩C([0, T], H₀¹(Ω)),

∂_tu⁰ ∈L^∞(0, T;H₀¹(Ω))∩C([0, T], L²(Ω))∩L²(0, T;H²(Ω)),

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

(3.2.101)

and the a priori estimates

ku⁰k²_L∞(0,T;H³(Ω))+ku^dk²_L∞(0,T;H²(Ω))

≤C kF⁰k²_L∞(0,T;H¹(Ω))+kF^dk²_L∞(0,T;L²(Ω))

+kF⁰(0,·)k²_H1(Ω)+kF^d(0,·)k²_L2(Ω)

!

+C

kF˙⁰k²_L∞(0,T;L²(Ω))+kF˙^dk²_L∞(0,T;H⁻¹(Ω))

Z T 0

e^Csds,

(3.2.102)

and

ku˙⁰k²_L∞(0,T;H¹(Ω))+ku˙^dk²_L∞(0,T;L²(Ω))

≤C

kF⁰(0,·)k²_H1(Ω)+kF^d(0,·)k²_L2(Ω)

+C

kF˙⁰k²_L∞(0,T;L²(Ω))+kF˙^dk²_L∞(0,T;H⁻¹(Ω))

Z T 0

e^Csds

(3.2.103)

hold, where the constantC depends only onΩand the coefficients ofA(∂_x), but doesn’t depend upon F.

Proof. The usual way of proving regularity for solutions of problem (3.2.99) is to first establish temporal regularity and then use elliptic estimates for the operator A(∂_x) to show spatial regularity. More precisely, we can formally differentiate equation (3.2.99) with respect to time. This yields











∂_tu˙ +A(∂_x) ˙u= ˙F ,

˙

u(0, x) =F(0, x),

u|˙ _∂Ω = 0 for a.e. t∈[0, T].

(3.2.104)

We now consider ˙u as a new variableQ= (Q⁰, Q^d), then from theorem 3.2.12 Q∈L²(0, T;V)∩H¹(0, T; (H⁻¹(Ω))^1+d)∩C([0, T], L²(Ω)) solves











∂tQ+A(∂x)Q= ˙F , Q(0, x) =F(0, x),

Q|_∂Ω= 0 for a.e. t∈[0, T].

(3.2.105)

Below we shall prove that actually Q= ˙u. For this set Z(t) :=

Z t 0

Qdτ (3.2.106)

in the sense of the Bochner integral. Then we have Z T

0

kZ(t)k²_Vdt= Z T

0

Z t 0

Qdτ

2

V

dt≤ Z T

0

Z T 0

kQk²_Vdτ dt <∞

=⇒Z(t)∈L²(0, T;V).

We conclude that ˙Z =Q in the sense of distribution and A(∂x)Z(t) =

Z t 0

A(∂x)Qdτ

=F(t)−Q(t).

With W denotingZ−u, then W ∈L²(0, T;V). We have ˙Z(t) = ˙W(t) + ˙u(t), combining ˙Z(t) =Q(t) we thus obtain

(W˙ +A(∂x)W = 0, W(0) = 0.

(3.2.107) Since (3.2.107) has only one solution and 0 is a solution, we conclude W = 0, which implies that the times derivatives of u satisfies the same boundary

conditions as u, namely ˙u|_∂Ω = u|_∂Ω = 0 for a.e. 0 ≤ t ≤ T. Using a same calculation as (3.2.76) on (3.2.105):

1

2∂t T0kQ⁰k²+²

4k∇Q⁰k²+kQ^dk²

+ν0T0k∇Q⁰k² +²

4ν₀k4Q⁰k²+ν₀k∇Q^dk²+1 τkQ^dk²

=T0( ˙F⁰, Q⁰) + ( ˙F^d, Q^d)−²

4( ˙F⁰,4Q⁰).

(3.2.108)

Define the energy

E(t) := 1 2

T0kQ⁰k²+²

4k∇Q⁰k²+kQ^dk²

,

then by H¨older’s inequality and Cauchy’s inequality and the assumption that F˙⁰ ∈L^∞(0, T;L²(Ω)), F˙^d ∈L^∞(0, T; (H⁻¹(Ω))^d)

we obtain the following type of differential inequality:

d

dtE(t) +C₁E(t)≤C₂, ∀t≥0, whereC2>0 depends only on

kF˙⁰k_L∞(0,T;L²(Ω))+kF˙^dk_L∞(0,T;(H⁻¹(Ω))^d). Solving this differential inequality leads to

E(t)≤E(0)e^−C1t+C2

C₁, ∀t≥0.

Since

E(0) = 1 2

T0kQ⁰(0)k²+²

4k∇Q⁰(0)k²+kQ^d(0)k²

= 1 2

T0kF⁰(0,·)k²+ ²

4k∇F⁰(0,·)k²+kF^d(0,·)k²

, we conclude

k∂_tu⁰k_L∞(0,T;H₀¹(Ω))+k∂_tu^dk_L^∞_(0,T;L2(Ω))≤C, C >0 depends only on F(0, x) and ˙F.

From theorem 3.2.5 and 2.2.1 we have the estimate

ku⁰k_L^∞_(0,T;H3(Ω))+|η|^3/2ku⁰k_L^∞_(0,T;L2(Ω))+ku^dk_L^∞_(0,T;H2(Ω))

+|η|ku^dk_L∞(0,T;L²(Ω))

≤c kF⁰k_L∞(0,T;H¹(Ω))+|η|^1/2kF⁰k_L∞(0,T;L²(Ω))+kF^dk_L∞(0,T;L²(Ω))

+k∂_tu⁰k_L∞(0,T;H₀¹(Ω))+|η|^1/2k∂_tu⁰k_L∞(0,T;L²(Ω))+k∂_tu^dk_L∞(0,T;L²(Ω))

≤C,

C >0 depends only onF,F(0, x) and ˙F. For 0≤t⁰ ≤t⁰⁰≤T, the H¨older estimates

ku⁰(t⁰,·)−u⁰(t⁰⁰,·)k_H2(Ω)≤ Z t⁰⁰

t⁰

k∂_tu⁰k_H2(Ω)dt

≤ |t⁰−t⁰⁰|^1/2k∂_tu⁰k_L2(0,T;H²(Ω))

holds. Fix a number β with 0< β < 1

2 then by Sobolev’s embedding theorem ku^d(t⁰)−u^d(t⁰⁰)k_C(Ω)≤Cku^d(t⁰)−u^d(t⁰⁰)k_H2−β(Ω), (3.2.109) here letC denote the embedding constant. Interpolation yields

ku^d(t⁰)−u^d(t⁰⁰)k_H2−β(Ω)

≤Cku^d(t⁰)−u^d(t⁰⁰)k^β/2_L2(Ω)ku^d(t⁰)−u^d(t⁰⁰)k^(2−β)/2_H2(Ω) .

(3.2.110) Finally

ku^d(t⁰)−u^d(t⁰⁰)k_L2(Ω)≤ Z t⁰⁰

t⁰

k∂_tu^dk_L2(Ω)dt

≤ |t⁰−t⁰⁰|k∂_tu^dk_L∞(0,T;L²(Ω))

(3.2.111) together with (3.2.109) and (3.2.110) yields

ku^d(t⁰)−u^d(t⁰⁰)k_C(Ω)

≤C|t⁰−t⁰⁰|^β/2k∂_tu^dk^β/2

L^∞(0,T;L²(Ω))ku^dk^(2−β)/2

L^∞(0,T;H²(Ω)).

(3.2.112) In like manner

k∇u⁰(t⁰)− ∇u⁰(t⁰⁰)k_C(Ω)

≤C|t⁰−t⁰⁰|^β/2k∂_tu⁰k^β/2

L^∞(0,T;H₀¹(Ω))ku⁰k^(2−β)/2_L∞(0,T;H³(Ω)).

(3.2.113) Then

(u⁰ ∈C([0, T], C¹(Ω)), u^d ∈C([0, T], C(Ω)).

Thus we have completed the proof of the theorem.

Im Dokument Solutions to the Multi-Dimensional Viscous Quantum Hydrodynamic Equations for Semiconductors (Seite 54-73)