The A-priori Estimates

In view of the uniform a priori estimates of (u, J, V), we are able to extend the local classical solution globally in time and prove its exponential convergence to the stationary solution (C₀,0,0).

Assume that for any givenT >0, there is a classical solution (u, J, V) of the IVP (4.1.11)-(4.1.12) satisfying the regularity conditions

(u, J, V)∈S_u×S_J×S_V, (4.2.1) where

S_u:=H¹(0, T;H⁵(T^d))\

L²(0, T;, H⁶(T^d)), S_J :=C([0, T], H⁴(T^d))\

L²(0, T;, H⁵(T^d)), SV :=C([0, T], H⁴(T^d)).

Sobolev embedding theorem provides

H¹(0, T;H⁵(T^d)),→C([0, T], H⁵(T^d)).

Assume that

δT := max

0≤t≤T

kuk_H5(T^d)+kJk_H4(T^d)

. (4.2.2)

By the Sobolev embedding theorem

H⁵(T^d)×H⁴(T^d)×H⁴(T^d)−→C³(T^d)×C²(T^d)×C²(T^d),

it is easy to verify that if δT is sufficiently small, there are constants φ−, φ+

such that

0< φ−≤u+C₀≤φ₊.

In the following we assume that δT is sufficiently small such that the above estimate holds. And we will use C as a generic constant which may change from line to line and is not allowed to depend on timet∈[0, T].

Lemma 4.2.1. Given the multi-indexα satisfies0≤α≤4, then the following inequality holds

d

dtkJk²_|α|+CkJk²_|α|+1 ≤Ck(u, u_t,∇u,4u)k²_|α|. (4.2.3) Proof. Setting ˆu:=D^αu, ˆJ :=D^αJ, one obtains the following equation

Jˆ_t−ν₀4Jˆ+1

τJˆ=D^αG+T∇ˆu−D^α((u+C₀)∇V)− ²

4∇4ˆu. (4.2.4) Take the inner product between (4.2.4) and ˆJ, and integrate by parts overT^d:

1 2

d

dtkJˆk²+ 1

τkJkˆ ²+ν0k∇Jkˆ ² (4.2.5)

=T(∇ˆu,J) +ˆ ²

4(4ˆu,div ˆJ) +L₁+L₂+L₃, (4.2.6) here define

L1 : =−X

k,l

Z

T^d

(∂kJˆl)D^α

JlJk

u+C₀

dx,

L₂ : =−² 4

X

k,l

Z

T^d

(∂_kJˆ_l)D^α

(∂_ku)(∂_lu) u+C₀

dx,

L₃ : =− Z

T^d

J(Dˆ ^α((u+C₀)∇V))dx.

Next we estimate the integrals L1, L2, L3. The constants in the following computations may change from one line to another, and can depend on the order of differentiation |α|, the space dimension dandδ_T, but are independent of the time T. Recall the embedding H^|α|(T^d) ⊂ L^∞(T^d), and We will make free use of the following Moser-type calculus inequalities [55, 61, 76]

kD^α(f g)k ≤C(kfk_L^∞kD^αgk+kD^αfkkgk_L^∞), kD^α(f g)−f D^αgk ≤C(kgk_L^∞kD^αfk+kD^α−1gkkfk_L^∞).

Then we conclude that

|L₁| ≤X

k,l

k∂_kJˆlk_L2k(u+C₀)⁻¹JlJkk_H|α|

≤Ck∇Jˆk_L2

X

k,l

k(u+C₀)⁻¹Jlk_L^∞kJ_kk_H|α|+k(u+C₀)⁻¹Jlk_H|α|kJ_kk_L^∞ .

Since the following estimates hold Concerning the second integral, we have

|L₂| ≤CX Similarly to the computations ofL₁, we deduce

|L₂| ≤Cσ2k∇Jk²_H|α| +C(δ²_T +δ⁴_T)

σ₂ k∇uk²_H|α|. (4.2.8) Concerning the third integral, we have

|L₃| ≤ kJˆk_L2k(u+C₀)∇Vk_H|α| The itemsσi are arbitrarily selected positive numbers.

Recall (4.2.5), substitute (4.2.7), (4.2.8) and (4.2.9) into (4.2.5), then sum up for all derivatives whose order are less than |α|, we obtain (4.2.3), that completes the proof of lemma 4.2.

Next we estimate kuk_H|α|+2 in terms ofkJk_H|α|

Lemma 4.2.2. We set Y(x, t) := Proof. From (4.1.10) ˆu satisfies the equation

ˆ

Taking the inner product between (4.2.11) and ˆu+ (τ + 1)ˆut, then integrating the resulting equation by parts overT^d yields

d

then we obtain

Note that there exists a constantκ, such that SelectδT,σ∗ sufficiently small such that

min then there is a positive constantβ₁, such that

d

which implies (4.2.10).

4.3 Global Existence and Exponential Decay

Now we are in a position to show the following

Theorem 4.3.1. Assume that (4.1.1) and (4.1.2) hold, Let(C₀,0,0) be a sta-tionary solution of the initial-value problem (3.0.1) on a torus T^d. Assum that the initial datum (u₀, J₀) ∈H⁶(T^d)×H⁵(T^d), then there is a number m₁ >0 such that if

ku₀k²_H6(T^d)+kJ₀k²_H5(T^d)≤m1,

the (classical) solution (u, J, V) of (4.1.11)-(4.1.12)exists globally in time and satisfies

kuk²_H5(T^d)+kJk²_H4(T^d)+kVk²_H4(T^d)≤C

ku₀k²_H6(T^d)+kJ₀k²_H5(T^d)

e^−c0t for all t≥0. Here,C >0 and c0 >0 are constants independent of time t.

Proof. lemma 4.2 yields

CδTkJk²_|α|+1≤CδTk(u, u_t,∇u,4u)k²_|α|−CδT

d

dtkJk²_|α|. (4.3.1) Substituting (4.3.1) into (4.2.10) yields

d

dt(Y +Cδ_TkJk²_|α|) +CY ≤Cδ_Tk(u, u_t,∇u,4u)k²_|α|. Choose δ_T sufficiently small such that

d

dt(Y +Cδ_TkJk²_|α|) +CY ≤0, (4.3.2) then add (4.3.1) and (4.3.2), we obtain

d

dt(Y +Cδ_TkJk²_|α|) +CY +Cδ_TkJk²_|α|+1 ≤Cδ_Tk(u, u_t,∇u,4u)k²_|α|,

=⇒d

dt(Y +CδTkJk²_|α|) +CY +CδTkJk²_|α|≤CδTk(u, u_t,∇u,4u)k²_|α|. Choosing δ_T sufficiently small, then there is a constant C such that

d

dt(Y +CδTkJk²_|α|) +CY +CδTkJk²_|α|≤0. (4.3.3) Via solving the differential inequality (3.2.74) we deduce

Y +CδTkJk²_|α|≤(Y0+CδTkJ₀k²_|α|)e^−β2t, (4.3.4)

hereY0:=Y(0, x), the initial value of Y. Recall (4.1.12) we infer

ku_t(0,·)k²_|α|≤C(kJ₀k²_|α|+1+ku₀k²_|α|+2). (4.3.5) Hence (4.3.4) and (4.3.5) yield

kuk²_|α|+2+kJk²_|α|≤C_δT(kJ₀k²_|α|+1+ku₀k²_|α|+2)e^−β2t, (4.3.6) hereC_δT depends upon δT. Thus

kn− C₀k²₅+kJk²₄+kVk²₄ ≤C(kJ₀k²₅+ku₀k²₆)e^−c0t, (4.3.7) the constantsC andc₀ are independent oft. Integrate (4.2.12) from 0 toT we deduce

kuk²_L2(0,T;H⁶(T^d))+kuk²_H1(0,T;H⁵(T^d))

≤CδTkJk²_L2(0,T;H⁵(T^d))+CY(0)

(4.3.8) Integrate (4.2.3) from 0 toT we deduce

kJk²_L2(0,T;H⁵(T^d))≤C

kuk²_L2(0,T;H⁶(T^d))+kuk²_H1(0,T;H⁵(T^d))

+kJ₀k²_H4(T^d).

(4.3.9) From (4.3.8) and (4.3.9) we obtain

kuk²_L2(0,T;H⁶(T^d))+kuk²_H1(0,T;H⁵(T^d))

≤CδT

kuk²_L2(0,T;H⁶(T^d))+kuk²_H1(0,T;H⁵(T^d))

+CδTkJ₀k²_H4(T^d)+CY(0).

For sufficiently smallδ_T

kuk²_L2(0,T;H⁶(T^d))+kuk²_H1(0,T;H⁵(T^d))≤Cδ_TkJ₀k²_H4(T^d)+CY(0). (4.3.10) Go back to (4.3.9) we obtain

kJk²_L2(0,T;H⁵(T^d))≤CδTkJ₀k²_H4(T^d)+CY(0) +kJ₀k²_H4(T^d). (4.3.11) By Theorem 2.4 in [17] there exists a local-in-time solution (n, J, V) of the IVP (3.0.1). Assume the initial data (n0, J0, V0) is sufficiently close to (C₀,0,0) then there exists a time interval [0, T^∗] such that (3.0.1) admits a local solution (n, J, V) in [0, T^∗] and δ_T^∗ is so small that (n− C₀, J, V) in [0, T^∗] satisfies (4.3.7), (4.3.10) and (4.3.11) whereT, δT is replaced by T^∗, δT^∗ respectively.

Further choosing the initial data kJ₀k²₅+ku₀k²₆ so small that C(kJ₀k²₅+ku₀k²₆)< δT^∗.

By Sobolev embedding theorem and (4.3.7) we conclude firstn >0 in [0, T^∗]× T^d; and second by the usual continuity argument (n, J, V) exists globally in time and satisfies (4.3.7).

Local Solutions to the

Non-Isothermal Viscous QHD

5.1 Introduction and the Main Result

Compared to the situation of constant temperature the non-isothermal system is on equations of n, J, V, ne which stands for particle density, the current density, the electrostatic potential and energy density respectively.

In this chapter we investigate the local existence of solutions to the following non-isothermal viscous model of QHD











































∂_tn−divJ =ν₀4n,

∂_tJ−div

J ⊗J n

− ∇(nT) +n∇V +²

2n∇

4√

√ n n

=ν₀4J−J

τ +µ∇n,

∂_t(ne)−div

((ne)E_d+P)J n

+J∇V =−2

τ(ne) + 3 τn +ν04(ne) +µdivJ, λ²4V =n− C(x),

(5.1.1)

whereµ is the interaction constant; the (scaled) stress tensor P and the tem-peratureT satisfy











P =nT E_d−²

4n(∇ ⊗ ∇) lnn, ne= |J|²

2n +3

2nT − ²

8n4lnn.

(5.1.2)

The (scaled) stress tensor consists of the classical pressure and a quantum

”pressure” term. The energy density is the sum of kinetic energy, thermal energy, and quantum energy.

115

The initial data and boundary conditions are

(n, J, ne)^T(0) = (n₀, J₀,(ne)₀), (5.1.3) (n, J, ne, V)^T|_∂Ω = (nΓ, JΓ,(ne)Γ, VΓ)^T (5.1.4) with

x∈Ωinf n₀(x)>0, n₀ ∈H³(Ω), J₀∈H²(Ω), (ne)₀ ∈H²(Ω), (5.1.5) n_Γ∈H^5/2(Ω), J_Γ∈H^3/2(Ω),(ne)_Γ ∈H^3/2(Ω), , V_Γ∈H^3/2(Ω). (5.1.6) We also require the following compatibility conditions.

( (n₀, J₀,(ne)₀)|_∂Ω= (n_Γ, J_Γ,(ne)_Γ), (ν04n₀+ divJ0)|_∂Ω= 0.

(5.1.7)

Furthermore the given data J0,∇(ne)₀,∇n₀, VΓ are supposed to be sufficiently small in

(H²(Ω))^d×(L²(Ω))^d×(H²(Ω))^d×H^3/2(Ω))

and C(x) is close to n₀ inL²(Ω) norm, which means physically that the initial data of the current density, the spatial change of the initial data of the particle density and the energy density, the boundary function of the electrostatic po-tential are small; and the doping profile of background charges is close to the initial data of the particle density. Under these assumptions we conclude Theorem 5.1.1. There exist T_∗ >0, C^∗>0 such that the system (5.1.1) with the initial data (5.1.3) and the boundary conditions (5.1.4) under the assump-tions (5.1.5), (5.1.6), (5.1.7) and

k∇n₀k_H2(Ω)+kJ₀k_H2(Ω)

+k∇(ne)₀k_L2(Ω)+kn₀− C(x)k_L2(Ω)+kV_Γk_H3/2(Ω)≤ C^∗,

(5.1.8)

has a unique local-in-time solution (n, J, ne, V) in [0,T_∗) 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),¯ ne∈L^∞(0,T_∗;H²(Ω)), V ∈C(0,T_∗;H²(Ω)), ∂_t(ne)∈L²(0,T_∗;H¹(Ω)),

∂_tV ∈L²(0,T∗;H¹(Ω)).

5.2 Proof of Theorem 5.1.1

We also study the equations for the perturbation (n−n₀, J−J₀,(ne)−(ne)₀).

The main steps to derive a local solution to (5.1.1) are first to construct ap-proximate solutions, second to derive a uniform time interval and their uniform bounds, finally to analysis the limit of the approximate solutions. For uniform bounds we refer to the a-priori estimates of solutions to the isothermal equations in previous chapters.

hold. From (5.1.2), we have the representions of the (scaled) stress tensor P andnT:

After an equivalent reformulation of (5.1.1) we obtain the following system.



SetP := (Pⁿ,P^J,P^ne) := (n−n0, J−J0, ne−(ne)0), then from (5.2.1) we obtain the following IBVP with respect to P.































∂tPⁿ−ν04Pⁿ−divP^J =F₀,

∂_tP^J −ν₀4P^J+P^J τ +²

6∇4Pⁿ−µ∇Pⁿ=F_d(Pⁿ,P^J,P^ne),

∂_tP^ne−ν₀4P^ne+ 2

τP^ne=F(Pⁿ,P^J,P^ne), λ²4V =Pⁿ+n0− C(x), P(0,·) = 0, P|_∂Ω = 0, V|_∂Ω =V_Γ,

(5.2.2)

where

F₀ :=ν04n₀+ divJ0, F_d(Pⁿ,P^J,P^ne) := 2

3∇P^ne+ 2

3∇(ne)₀+ div

(P^J +J0)⊗(P^J+J0) Pⁿ+n0

−(Pⁿ+n0)∇V +²div

(∇p

Pⁿ+n0)⊗(∇p

Pⁿ+n0)

−1

3∇|P^J+J₀|² Pⁿ+n₀ − ²

12∇|∇(Pⁿ+n₀)|² Pⁿ+n₀ +ν₀4J₀−1

τJ₀+µ∇n₀−²

6∇4n₀, F(Pⁿ,P^J,P^ne) := div

P^J +J₀ Pⁿ+n0

(P^ne+ (ne)₀+P(Pⁿ,P^J,P^ne))

+ν₀4(ne)₀− 2

τ(ne)₀+3

τ(Pⁿ+n₀) +µdiv(P^J+J₀)

−(P^J +J₀)∇V.

5.2.2 Construction of Approximate Solutions and their Uni-form Bounds

Fix ϕ1, ϕ2 >0 define

c^∗: =p

kF(0,0,0)k²+ϕ₁+ϕ₂. (5.2.3) Lemma 5.2.1. Let T > 0, q^ne ∈ L^∞(0, T;H₀¹(Ω)), ∂tq^ne ∈ L^∞(0, T;L²(Ω)), with

kq^nek_L∞(0,T;H₀¹(Ω)) ≤c^∗, k∂_tq^nek_L^∞_(0,T;L2(Ω)) ≤c^∗, (5.2.4)

then for any γ > 0 there exist T ∈ (0, T], Cµ,ν0,,τ > 0 depending only on µ, ν₀, , τ; C^∗, C⁰ >0 depending only upon C₀ := max(A₁,A₂) with

A₁ :=

v u u u u u t

µkF₀k²+²

6k∇F₀k²+kF_d(0,0,0)k²

+γ min

µ,²

6,1

,

A₂:=

v u u u t

µkF₀k²+ ²

6k∇F₀k²+kF_d(0,0,0)k²

+γ C_µ,ν0,,τ

c^∗, the initial data and all physical constants such that the IBVP



















∂tqⁿ−ν04qⁿ−divq^J =ν04n₀+divJ0,

∂_tq^J−ν₀4q^J +q^J τ +²

6∇4qⁿ−µ∇qⁿ=F_d(qⁿ, q^J, q^ne), λ²4q^v=qⁿ+n0− C(x), (qⁿ, q^J)(0,·) = 0, (qⁿ, q^J)|_∂Ω= 0, q^v|_∂Ω =V_Γ,

(5.2.5)

has a unique local solution(qⁿ, q^J, q^v)^T in [0,T) with

qⁿ∈L^∞(0,T;H³(Ω)), q^J ∈L^∞(0,T;H²(Ω)),

∂tqⁿ∈L²(0,T;H²(Ω)), ∂tq^J ∈L²(0,T;H¹(Ω)), (qⁿ,∇qⁿ, q^J)∈C([0,T)×Ω),¯

q^v ∈C(0,T;H²(Ω)),

∂tq^v ∈L²(0,T;H¹(Ω)), and the following estimates



























kqⁿk_L^∞_(0,T;H3(Ω))≤C⁰, kq^Jk_L^∞_(0,T;H2(Ω))≤C⁰, kqⁿk_L^∞_(0,T;H2(Ω))≤C^∗, kq^Jk_L∞(0,T;H₀¹(Ω))≤C^∗, kqⁿk_L∞(0,T;H₀¹(Ω))≤C₀, kq^Jk_L∞(0,T;L²(Ω))≤C₀, kq˙ⁿk_L∞(0,T;H₀¹(Ω))≤C₀, kq˙^Jk_L∞(0,T;L²(Ω))≤C₀, kq˙ⁿk_L2(0,T;H²(Ω))≤C0, kq˙^Jk_L2(0,T;H₀¹(Ω))≤C0,

(5.2.6)

and



















[0,Tinf]inf

x∈Ω

(qⁿ+n0)≥δ0, sup

[0,T]

max

k∇(qⁿ+n0)k_L^∞_(Ω),kqⁿ+n0k_L^∞_(Ω), kq^J +J0k_L^∞_(Ω)

≤δ₀⁻¹,

(5.2.7)

hold, where δ₀ is defined in (3.3.17).

Proof. This is a direct consequence by making minor modifications of the proof of Theorem3.3.1.

Additionally we have the following conclusion which is useful for the remainder of the proof.

Lemma 5.2.2. Let T >0, f₁, f₂, f₃ be functions satisfying

f1 ∈L^∞(0, T;H³(Ω))∩C([0, t⁰];C¹(Ω))∩C([0, T];H²(Ω)), f2 ∈L^∞(0, T;H²(Ω))∩C([0, T];C(Ω))∩C([0, T];H₀¹(Ω)),

∂_tf₁ ∈L^∞(0, T;H₀¹(Ω))∩C([0, T];L²(Ω))∩L²(0, T;H²(Ω)),

∂_tf₂ ∈L^∞(0, T;L²(Ω))∩C([0, T];L²(Ω))∩L²(0, T;H₀¹(Ω)), f3 ∈L^∞(0, T;H¹(Ω)), ∂tf3 ∈L^∞(0, T;L²(Ω)),

and

inf

[0,T]inf

x∈Ω

(f₁+n₀)>0, (f₁, f₂, f₃)(0) = 0.

Then 









F(f₁, f₂, f₃)∈L^∞(0, T, L²(Ω)),

∂tF(f1, f2, f3)∈L²(0, T, H⁻¹(Ω)), F(f1, f2, f3)(0,·)∈L²(Ω).

Proof. At first

F(f1, f2, f3) :=Y₁+Y₂+Y₃+Y₄, where

Y₁ : = div

((f3+ (ne)0)Ed)f2+J0

f1+n0

, Y₂ := div

P(f1, f2, f3)f2+J0

f1+n0

Y₃ : =−(f₂+J₀)∇g,

Y₄ : =ν₀4(ne)₀−2τ⁻¹(ne)₀+ 3τ⁻¹(f₁+n₀) +µdiv(f₂+J₀),

λ²4g=f1+n0− C(x), g|_∂Ω=VΓ.

From our assumptions it is easy to verify that Y₁,Y₄ ∈L^∞(0, T, L²(Ω)) and

∂tY₁, ∂tY₄ ∈L²(0, T, H⁻¹(Ω)).

The item P(f1, f2, f3) has the representation P(f₁, f₂, f₃) =2

3(f₃+ (ne)₀)− |f₂+J₀|² 3(f1+n0) + ²

124(f₁+n₀)

− ² 12

|∇(f₁+n0)|² f₁+n₀

E_d−²

4(∇ ⊗ ∇)(f₁+n₀) +²

4

∇(f₁+n0)⊗ ∇(f₁+n0) f₁+n₀ ,

from which we obtain kP(f₁, f₂, f₃)k_L∞(0,T;H¹(Ω)) < ∞, which implies Y₂ ∈ L^∞(0, T, L²(Ω)).Furthermore it is easy to seek∂_tP(f₁, f₂, f₃)k_L2(0,T;L²(Ω)) <∞ according to our assumptions. Then k∂_tY₂k_L2(0,T;H⁻¹(Ω))<∞.

Now it remains to estimateY₃.

kY₃k_L^∞_(0,T;L2(Ω))≤Ckf₂+J0k_L^∞_(0,T;H1(Ω))kgk_L^∞_(0,T;H2(Ω))

≤Ckf₂+J0k_L^∞_(0,T;H1(Ω))

kf₁+n0− C(x)k_L^∞_(0,T;L2(Ω))+kV_Γk_H3/2(Ω)

, thenY₃∈L^∞(0, T;L²(Ω)).Next for a.e. 0≤t≤T

k∂_tY₃k_L2(Ω)≤Ck∂_tf₂k_H1(Ω)kgk_H2(Ω)+Ckf₂+J₀k_H1(Ω)k∂_tgk_H2(Ω)

≤C

kf₁+n₀− C(x)k_L∞(0,T;L²(Ω))+kV_Γk_H3/2(Ω)

k∂_tf₂k_H1(Ω)

+Ck∂_tf1k_L^∞_(0,T;L2(Ω))kf₂+J0k_L^∞_(0,T;H1(Ω)). Obviously,∂_tY₃∈L²(0, T;H⁻¹(Ω)).

LetT >0,q₀^ne= 0. Then it is trivial that

kq₀^nek_L∞(0,T;H₀¹(Ω))≤c^∗, k∂_tq^ne₀ k_L∞(0,T;L²(Ω))≤c^∗.

By Lemma 5.2.1 there exists a time interval [0, t₀) ⊆ [0, T) such that the system

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∂_tqⁿ₀ −ν₀4qⁿ₀ −divq₀^J =ν₀4n₀+ divJ₀,

∂tq₀^J−ν04q₀^J+ q^J₀ τ +²

6∇4q₀ⁿ−µ∇qⁿ₀ =F_d(q₀ⁿ, q^J₀, q^ne₀ ), λ²4q^v₀ =q₀ⁿ+n0− C(x), (q₀ⁿ, q^J₀)(0,·) = 0, (qⁿ₀, q₀^J)|_∂Ω= 0, q₀^v|_∂Ω=VΓ,

(5.2.8)

admits a unique local-in-time solution (q₀ⁿ, q₀^J, q₀^v) with

qⁿ₀ ∈L^∞(0, t₀;H³(Ω)), q₀^J ∈L^∞(0, t₀;H²(Ω)),

∂tqⁿ₀ ∈L²(0, t0;H²(Ω)), ∂tq^J₀ ∈L²(0, t0;H¹(Ω)), (q₀ⁿ,∇q₀ⁿ, q^J₀)∈C([0, t₀)×Ω),¯

q^v₀ ∈C(0, t0;H²(Ω)),

∂tq^v₀ ∈L²(0, t0;H¹(Ω)),

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kq₀ⁿk_L^∞_(0,t0;H3(Ω))≤C⁰, kq^J₀k_L^∞_(0,t0;H2(Ω))≤C⁰, kq₀ⁿk_L^∞_(0,t0;H2(Ω))≤C^∗, kq^J₀k_L∞(0,t0;H₀¹(Ω))≤C^∗, kq₀ⁿk_L∞(0,t0;H¹₀(Ω))≤C₀, kq₀^Jk_L∞(0,t0;L²(Ω))≤C₀, kq˙₀ⁿk_L∞(0,t0;H¹₀(Ω))≤C0, kq˙₀^Jk_L∞(0,t0;L²(Ω))≤C0, kq˙ⁿ₀k_L2(0,t0;H²(Ω))≤C0, kq˙₀^Jk_L2(0,t0;H₀¹(Ω))≤C0,

(5.2.9)

and

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inf

[0,t0]inf

x∈Ω

(q₀ⁿ+n0)≥δ0, sup

[0,t0]

max

k∇(q₀ⁿ+n0)k_L^∞_(Ω),kqⁿ₀ +n0k_L^∞_(Ω), kq^J₀ +J0k_L^∞_(Ω)

≤δ⁻¹₀ .

(5.2.10)

Now we consider the third equation of (5.2.2). More precisely, we shall study the problem

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

∂_tq₁^ne−ν₀4q₁^ne+2

τq^ne₁ =F(q₀ⁿ, q₀^J, q^ne₀ ), q₁^ne(0,·) = 0, q₁^ne|_∂Ω= 0.

(5.2.11) This BVP admits a unique solution q₁^ne with q₁^ne ∈ L^∞(0, t₀, H₀¹(Ω)), ∂_tq₁^ne ∈ L^∞(0, t0, L²(Ω))∩L²(0, t0, H₀¹(Ω)) provided

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F(qⁿ₀, q₀^J, q₀^ne)∈L^∞(0, t₀, L²(Ω)),

∂_tF(qⁿ₀, q₀^J, q₀^ne)∈L²(0, t₀, H⁻¹(Ω)), F(q₀ⁿ, q^J₀, q^ne₀ )(0,·)∈L²(Ω).

(5.2.12)

Check that since (q₀ⁿ, q^J₀) satisfies (5.2.9) and it is obviously true thatF(q₀ⁿ, q₀^J, q₀^ne) satisfies (5.2.12) from Lemma 5.2.12.

Letk≥1,tk−1>0. Assume

q^ne_k ∈L^∞(0, tk−1;H₀¹(Ω)), ∂tq_k^ne∈L^∞(0, tk−1;L²(Ω))

with

kq^ne_k k_L∞(0,tk−1;H¹₀(Ω))≤c^∗, k∂_tq^ne_k k_L∞(0,tk−1;L²(Ω))≤c^∗, q^ne_k (0) = 0. (5.2.13) Then by Lemma 5.2.1 there exists a time interval [0, t_k)⊆[0, tk−1) such that the system

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∂tqⁿ_k−ν04qⁿ_k−divq_k^J =ν04n₀+ divJ0,

∂_tq^J_k −ν₀4q_k^J+ q^J_k τ +²

6∇4qⁿ_k−µ∇qⁿ_k =F_d(q_kⁿ, q^J_k, q^ne_k ), λ²4q^v_k=qⁿ_k +n0− C(x), (qⁿ_k, q_k^J)(0,·) = 0, (q_kⁿ, q^J_k)|_∂Ω = 0,q_k^v|_∂Ω=V_Γ,

(5.2.14)

admits a unique local-in-time solution (qⁿ_k, q_k^J, q_k^v) with

q_kⁿ∈L^∞(0, t_k;H³(Ω)), q_k^J ∈L^∞(0, t_k;H²(Ω)),

∂_tq_kⁿ∈L²(0, t_k;H²(Ω)), ∂_tq^J_k ∈L²(0, t_k;H¹(Ω)), (q_kⁿ,∇qⁿ_k, q_k^J)∈C([0, t_k)×Ω),¯

q_k^v ∈C(0, t_k;H²(Ω)),

∂_tq_k^v ∈L²(0, t_k;H¹(Ω)),

(5.2.15)

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kq_kⁿk_L∞(0,t_k;H³(Ω))≤C⁰, kq^J_kk_L∞(0,t_k;H²(Ω))≤C⁰, kq_kⁿk_L∞(0,tk;H²(Ω))≤C^∗, kq^J_kk_L∞(0,tk;H₀¹(Ω))≤C^∗, kq_kⁿk_L∞(0,tk;H₀¹(Ω))≤C0, kq^J_kk_L^∞_(0,tk;L2(Ω))≤C0, kq˙_kⁿk_L∞(0,tk;H₀¹(Ω))≤C₀, kq˙^J_kk_L∞(0,tk;L²(Ω))≤C₀, kq˙ⁿ_kk_L2(0,tk;H²(Ω))≤C₀, kq˙_k^Jk_L2(0,tk;H₀¹(Ω))≤C₀,

(5.2.16)

and

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inf

[0,tk]inf

x∈Ω

(qⁿ_k +n₀)≥δ₀, sup

[0,tk]

max

k∇(qⁿ_k +n0)k_L^∞_(Ω),kq_kⁿ+n0k_L^∞_(Ω), kq_k^J +J0k_L^∞_(Ω)

≤δ₀⁻¹,

(5.2.17)

Next consider the problem

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

∂_tq_k+1^ne −ν₀4q_k+1^ne + 2

τq^ne_k+1=F(q_kⁿ, q^J_k, q^ne_k ), q^ne_k+1(0,·) = 0, q^ne_k+1|_∂Ω= 0.

(5.2.18)

This BVP admits a unique solutionq^ne_k+1withq_k+1^ne ∈L^∞(0, tk, H₀¹(Ω)),∂tq_k+1^ne ∈

It is easy to verify that (5.2.19) is satisfied fromLemma 5.2.2.

Using (5.2.11), we take the inner product with −4q^ne_k+1 for a.e. t ∈ [0, t_k]. by which we use H¨older’s inequality and Cauchy’s inequality then obtain

1

Using (5.2.13), (5.2.9) and (5.2.10)

F(qⁿ_k, q_k^J, q_k^ne) we will use the same notations, i.e., letC_i, i= 1,2 denote the generic constants which may change from line to line.

Differentiate formally (5.2.11) with respect to 0≤t≤tk, then take theL²(Ω) (5.2.9) and (5.2.10) we estimate

kP(qⁿ_k, q_k^J, q_k^ne)k_L4(Ω)≤C2.

Thus

kS₂k_L2(Ω) ≤C₂kq˙_k^Jk_H1(Ω)+C₂. (5.2.25)

• Estimates of S₃. It is easy to verify that

kS₃k_L2(Ω)≤C₂. (5.2.26)

• Estimates of S₄.

kS₄k_L2(Ω) ≤δ₀⁻²

P(q˙ _kⁿ, q^J_k, q^ne_k ) L²(Ω). By (5.2.9) and (5.2.10) we obtain

kS₄k_L2(Ω)≤C2kq˙ⁿ_kk_H2(Ω)+C2. (5.2.27) Combining (5.2.23)-(5.2.27) we deduce

1

2∂_tkq˙^ne_k+1k²+ν₀k∇q˙_k+1^ne k²+ 2

τkq˙^ne_k+1k²

≤C₂+C₂kq˙_kⁿk²_H2(Ω)+C₂kq˙_k^Jk²_H1(Ω),

(5.2.28)

from which via integration from 0 to t∈(0, t_k] we conclude from (5.2.9) kq˙^ne_k+1k_L^∞_(0,tk;L2(Ω))

≤q

kq˙^ne_k+1(0)k²+C2t_k+C2kq˙_kⁿk²_L2(0,t

k;H²(Ω))+C2kq˙^J_kk²_L2(0,t

k;H¹(Ω))

≤ q

kq˙^ne_k+1(0)k²+C2t_k+C2C₀².

(5.2.29)

It is obvious that

˙

q_k+1^ne (0) =F(0,0,0),

then (5.2.29) yields that there exit C, t_C > 0 independent of k such that if infx∈Ωn₀ > δ₀,t_k ≤tC and

k∇n₀k_H2(Ω)+kJ₀k_H2(Ω)

+k∇(ne)₀k_L2(Ω)+γ+kn₀− C(x)k_L2(Ω)+kV_Γk_H3/2(Ω) ≤ C, (5.2.30) then after recalling (5.2.13)

kq˙_k+1^ne k_L∞(0,tk;L²(Ω))≤p

kF(0,0,0)k²+ϕ₁+ϕ₂ =c^∗, kq_k+1^ne k_L∞(0,tk;H₀¹(Ω))≤c^∗.

(5.2.31)

Recursively using Lemma 5.2.1 once more we obtain [0, tk+1) which guarantees thatγ satisfying (5.2.30) and t_k+1 ≤tC, and functions (q_k+1ⁿ , q_k+1^J ) in [0, t_k+1) satisfying

q_k+1ⁿ ∈L^∞(0, t_k+1;H³(Ω)), q^J_k+1 ∈L^∞(0, t_k+1;H²(Ω)),

∂tq_k+1ⁿ ∈L²(0, t_k+1;H²(Ω)), ∂tq^J_k+1∈L²(0, t_k+1;H¹(Ω)), (q_k+1ⁿ ,∇q_k+1ⁿ , q^J_k+1)∈C([0, t_k+1)×Ω),¯

q_k+1^v ∈C(0, tk+1;H²(Ω)),

∂_tq_k+1^v ∈L²(0, t_k+1;H¹(Ω)),

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kq_k+1ⁿ k_L∞(0,tk+1;H³(Ω))≤C⁰, kq_k+1^J k_L∞(0,tk+1;H²(Ω))≤C⁰, kq_k+1ⁿ k_L^∞_(0,tk+1;H2(Ω))≤C^∗, kq_k+1^J k_L∞(0,tk+1;H¹₀(Ω))≤C^∗, kq_k+1ⁿ k_L∞(0,t_k+1;H₀¹(Ω))≤C₀, kq^J_k+1k_L∞(0,tk+1;L²(Ω))≤C₀, kq˙_k+1ⁿ k_L∞(0,tk+1;H₀¹(Ω))≤C0, kq˙^J_k+1k_L∞(0,t_k+1;L²(Ω))≤C0, kq˙_k+1ⁿ k_L2(0,tk+1;H²(Ω))≤C₀, kq˙_k+1^J k_L2(0,tk+1;H¹₀(Ω))≤C₀,

(5.2.32)

and

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inf

[0,tk+1]inf

x∈Ω

(qⁿ_k+1+n₀)≥δ₀, sup

[0,tk+1]

max

k∇(q_k+1ⁿ +n0)k_L^∞_(Ω),kq_k+1ⁿ +n0k_L^∞_(Ω), kq^J_k+1+J0k_L∞(Ω)

≤δ₀⁻¹.

(5.2.33)

In the next lemma we will show thatt_k tends not to zero ask→ ∞.

Lemma 5.2.3. There exits an in k uniform time interval [0, tu) with tu ≤tC

such that forγ satisfying (5.2.30)and all k= 0,1,· · ·

q_kⁿ∈L^∞(0, tu;H³(Ω)), q_k^J ∈L^∞(0, tu;H²(Ω)),

∂_tq_kⁿ∈L²(0, t_u;H²(Ω)), ∂_tq^J_k ∈L²(0, t_u;H¹(Ω)), (q_kⁿ,∇qⁿ_k, q_k^J)∈C([0, tu)×Ω),¯ q^ne_k ∈L^∞(0, tu;H₀¹(Ω)), q_k^v ∈C(0, t_u;H²(Ω)), ∂_tq^ne_k ∈L^∞(0, t_u;L²(Ω)),

∂_tq_k^v ∈L²(0, t_u;H¹(Ω)),

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kq_kⁿk_L∞(0,tu;H³(Ω))≤C⁰, kq_k^Jk_L∞(0,tu;H²(Ω))≤C⁰, kq_kⁿk_L∞(0,tu;H²(Ω))≤C^∗, kq_k^Jk_L∞(0,tu;H₀¹(Ω))≤C^∗, kq_kⁿk_L∞(0,tu;H₀¹(Ω))≤C₀, kq^J_kk_L∞(0,tu;L²(Ω))≤C₀, kq˙_kⁿk_L∞(0,tu;H₀¹(Ω))≤C0, kq˙^J_kk_L∞(0,tu;L²(Ω))≤C0, kq˙_kⁿk_L2(0,tu;H²(Ω))≤C₀, kq˙_k^Jk_L2(0,tu;H₀¹(Ω))≤C₀, kq^ne_k k_L∞(0,tu;H₀¹(Ω))≤c^∗, kq˙_k^nek_L^∞_(0,tu;L2(Ω))≤c^∗,

(5.2.34)

and

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inf

[0,tu]inf

x∈Ω

(qⁿ_k+n0)≥δ0, sup

[0,tu]

max

k∇(qⁿ_k+n0)k_L^∞_(Ω),kq_kⁿ+n0k_L^∞_(Ω), kq_k^J+J0k_L^∞_(Ω)

≤δ₀⁻¹,

(5.2.35)

hold.

Proof. Using reduction to absurdity, assume there exists no such time interval, then for any sufficiently small [0, t_s) witht_s≤tC we can find an earliest k∈N such that though q_k^ne∈L^∞(0, t_s;H₀¹(Ω)) and ∂_tq_k^ne∈L^∞(0, t_s;L²(Ω)) with

kq^ne_k k_L∞(0,ts;H₀¹(Ω))≤c^∗, k∂_tq_k^nek_L∞(0,ts;L²(Ω))≤c^∗, the system (5.2.14) has no slutions in [0, ts) satisfying

qⁿ_k ∈L^∞(0, t_s;H³(Ω)), q_k^J ∈L^∞(0, t_s;H²(Ω)),

∂tqⁿ_k ∈L²(0, ts;H²(Ω)), ∂tq^J_k ∈L²(0, ts;H¹(Ω)), (q_kⁿ,∇q_kⁿ, q_k^J)∈C([0, ts)×Ω),¯

q_k^v∈C(0, ts;H²(Ω)),

∂_tq_k^v∈L²(0, t_s;H¹(Ω)),

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kq_kⁿk_L^∞_(0,ts;H3(Ω))≤C⁰, kq_k^Jk_L^∞_(0,ts;H2(Ω))≤C⁰, kq_kⁿk_L^∞_(0,ts;H2(Ω))≤C^∗, kq_k^Jk_L∞(0,ts;H₀¹(Ω))≤C^∗, kq_kⁿk_L∞(0,ts;H₀¹(Ω))≤C₀, kq_k^Jk_L∞(0,ts;L²(Ω))≤C₀, kq˙_kⁿk_L∞(0,ts;H₀¹(Ω))≤C₀, kq˙_k^Jk_L∞(0,ts;L²(Ω))≤C₀, kq˙_kⁿk_L2(0,ts;H²(Ω))≤C0, kq˙^J_kk_L2(0,ts;H₀¹(Ω))≤C0,

(5.2.36)

and

Now we go back to the proof of Theorem3.3.1. In order to solve (5.2.14) with respect to (qⁿ_k, q_k^J, q_k^v) we first construct approximate solutions{(qⁿ_k,l, q_k,l^J , q_k,l^v )}^∞_l=1. From Lemma 3.3.2 and 3.3.3 we claim there existsC_c >0 independent ofksuch that if ts ≤ C_c (in the sequal we will always use this assumption of ts), then

From this the local-in-time existence of solutions to (5.2.14) in [0, ts) can be obtained where the solution satisfies (5.2.36) and (5.2.37).

Recall our premise that the system (5.2.14) has no slutions satisfying (5.2.36) and (5.2.37) in [0, ts), there must be an earliestlsuch that (q_k,lⁿ , q^J_k,l, q^v_k,l) violate (5.2.36) or (5.2.38). From the proof of Theorem 3.3.1 there exists t_γ > 0 independent of k such that if ts ≤ tγ (in the sequal we will always use this assumption ofts) and (qⁿ_k,l−1, q^J_k,l−1) satisfies (5.2.36) and

kq^J_k,l(t₁,·)−q^J_k,l(t₂,·)k_C(Ω)

≤2^(2−β)/2CβCinterk∂_tq^J_k,lk^β/2_L∞(0,tk;L²(Ω))kq_k,l^J k^(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. By a similar reasoning

k∇q_k,lⁿ (t₁,·)− ∇q_k,lⁿ (t₂,·)k_C(Ω)

≤2^(2−β)/2C_βC_interk∇∂_tqⁿ_k,lk^β/2_L∞(0,tk;L²(Ω))kq_k,lⁿ k^(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.

Sincetsis sufficiently small then from the interpolations mentioned above, that (qⁿ_k,l, q^J_k,l) violate (5.2.38) produces a contradiction.

Remark 5.2.1. Recall (5.2.18),

kq^ne_k+1k_L∞(0,tu;H²(Ω))≤C k∂_tq_k+1^ne k_L∞(0,tu;L²(Ω))+kF(q_kⁿ, q_k^J, q_k^ne)k_L∞(0,tu;L²(Ω))

. Integrate (5.2.28)from 0 to t_u then recall (5.2.29) we infer

k∂_tq_k+1^ne k_L2(0,tu;H¹(Ω))

≤q

kq˙_k+1^ne (0)k²+C₂t_k+C₂kq˙ⁿ_kk²_L2(0,t

k;H²(Ω))+C₂kq˙_k^Jk²_L2(0,t

k;H¹(Ω))

≤q

kq˙_k+1^ne (0)k²+C₂t_k+C₂C₀².

From (5.2.18), (5.2.23), (5.2.29), (5.2.34) and (5.2.35) we conclude that there exists C˜independent of k such that for all k= 0,1,· · ·

kq^ne_k k_L∞(0,tu;H²(Ω))+k∂_tq_k^nek_L2(0,tu;H¹(Ω))≤C.˜ (5.2.40) 5.2.3 Analysis of the Limit of the Approximate Solutions After having obtained the uniform bounds we deduce the following

Lemma 5.2.4. There exits C_∗ > 0 independent of k such that if t_u ≤ C_∗, {q^ne_k }^∞_k=0 is a Cauchy sequence inL^∞(0, tu;L²(Ω). Namely there exists aq_∗^ne∈ L^∞(0, t_u;L²(Ω) such thatq_k^ne converges to q_∗^ne in L^∞(0, t_u;L²(Ω).

Proof. Since (qⁿ_k, q_k^J) solves



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









∂_tq_kⁿ−ν₀4q_kⁿ−divq^J_k =ν₀4n₀+ divJ₀,

∂tq_k^J −ν04q_k^J+q_k^J τ +²

6∇4q_kⁿ−µ∇q_kⁿ=F_d(qⁿ_k, q_k^J, q_k^ne), λ²4q_k^v =q_kⁿ+n0− C(x), (q_kⁿ, q_k^J)(0,·) = 0,(qⁿ_k, q_k^J)|_∂Ω= 0,q^v_k|_∂Ω=VΓ,

(5.2.41)

and (q_k−1ⁿ , q^J_k−1) solves



















∂tq_k−1ⁿ −ν04qⁿ_k−1−divq_k−1^J =ν04n₀+ divJ0,

∂_tq_k−1^J −ν₀4q_k−1^J +q_k−1^J τ + ²

6∇4qⁿ_k−1−µ∇q_k−1ⁿ =F_d(qⁿ_k−1, q_k−1^J , q^ne_k−1), λ²4q_k−1^v =q_k−1ⁿ +n0− C(x), (qⁿ_k−1, q_k−1^J )(0,·) = 0,(q_k−1ⁿ , q^J_k−1)|_∂Ω= 0,q^v_k−1|_∂Ω =VΓ.

(5.2.42) (5.2.41)-(5.2.42) yields



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





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





∂t(q_k−q_k−1)^T +A(∂_x)(q_k−q_k−1)^T = 0,F_d(q_k, q^ne_k )−F_d(q_k−1, q^ne_k−1)T

, λ²4(q^v_k−q_k−1^v ) =q_kⁿ−qⁿ_k−1,

(q_k−q_k−1)^T(0,·) = 0, (q_k−q_k−1, q_k^v−q_k−1^v )^T|_∂Ω = 0,

(5.2.43) whereq_i:= (q_iⁿ, q^J_i)^T, i=k, k−1,

A(∂_x) :=





−ν₀4 −div

−µ∇+ ²

6∇4 −ν₀4+τ⁻¹



. By a similar calculations as (3.3.28)-(3.3.33) we find

1 2∂t

µkqⁿ_k−qⁿ_k−1k²+²

6k∇(qⁿ_k −q_k−1ⁿ )k²+kq_k^J−q_k−1^J k²

+ν0µk∇(qⁿ_k−q_k−1ⁿ )k²+²

6ν0k4(q_kⁿ−qⁿ_k−1)k² +ν₀k∇(q^J_k −q_k−1^J )k²+1

τkq_k^J−q^J_k−1k²

=(F_d(q_k, q_k^ne)−F_d(q_k−1, q_k−1^ne ), q_k^J−q^J_k−1).

By H¨older’s inequality and Cauchy’s inequlity we obtain

∂t

µkq_kⁿ−qⁿ_k−1k²+²

6k∇(qⁿ_k−q_k−1ⁿ )k²+kq_k^J−q^J_k−1k²

+C k∇(qⁿ_k−qⁿ_k−1)k²+k4(qⁿ_k −q_k−1ⁿ )k² +k∇(q_k^J−q_k−1^J )k²+kq_k^J−q^J_k−1k²

≤CkF_d(q_k, q_k^ne)−F_d(q_k−1, q_k−1^ne )k²_H−1(Ω).

(5.2.44)

Using (5.2.9), (5.2.10) we calculate similarly as (3.3.121)-(3.3.126) to obtain for a.e. t∈[0, t⁰)

kF_d(q_k, q_k^ne)−F_d(q_k−1, q_k−1^ne )k²_H−1(Ω)

≤C

kq_kⁿ−q_k−1ⁿ k²_H1(Ω)+kq_k^J−q_k−1^J k²_L2(Ω)+kq_k^ne−q_k−1^ne k²_L2(Ω)

.

(5.2.45) Then we have

kqⁿ_k −q_k−1ⁿ k²_L∞(0,tu;H₀¹(Ω))+kq^J_k −q_k−1^J k²_L∞(0,tu;L²(Ω))

≤Ctu

kq_kⁿ−q_k−1ⁿ k²_L∞(0,tu;H¹₀(Ω))+kq^J_k −q^J_k−1k²_L∞(0,tu;L²(Ω))

+kq_k^ne−q_k−1^ne k²_L∞(0,tu;L²(Ω))

.

Assume [0, t_u) is sufficiently small such thatCt_u <1, then kq_kⁿ−q_k−1ⁿ k²_L∞(0,tv;H₀¹(Ω))+kq_k^J−q^J_k−1k²_L∞(0,tu;L²(Ω))

≤ Ct_u

1−Ct_ukq_k^ne−q_k−1^ne k²_L∞(0,tu;L²(Ω)).

(5.2.46)

Integrate (5.2.44) from 0 to tu we obtain

kqⁿ_k −q_k−1ⁿ k²_L2(0,tu;H²(Ω))+kq_k^J−q_k−1^J k²_L2(0,tu;H¹(Ω))

≤Ct_u

kq_kⁿ−q_k−1ⁿ k²_L∞(0,tu;H¹₀(Ω))+kq^J_k −q^J_k−1k²_L∞(0,tu;L²(Ω))

+kq_k^ne−q_k−1^ne k²_L∞(0,tu;L²(Ω))

. Together with (5.2.46) we conclude

kqⁿ_k−q_k−1ⁿ k²_L2(0,tu;H²(Ω))+kq_k^J−q_k−1^J k²_L2(0,tu;H¹(Ω))

≤ Ct_u 1−Ctu

kq^ne_k −q_k−1^ne k²_L∞(0,tu;L²(Ω)).

(5.2.47)

From (5.2.11) we find



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









∂_t(q_k+1^ne −q_k^ne)−ν₀4(q_k+1^ne −q_k^ne) + 2

τ(q^ne_k+1−q_k^ne)

=F(qⁿ_k, q_k^J, q_k^ne)−F(q_k−1ⁿ , q^J_k−1, q_k−1^ne ), (q^ne_k+1−q_k^ne)(0,·) = 0,(q_k+1^ne −q_k^ne)|_∂Ω = 0.

Taking the inner product withq^ne_k+1−q_k^nefor a.e. t∈[0, t_u) yields

∂t

1

2kq_k+1^ne −q_k^nek²+ν0k∇(q_k+1^ne −q_k^ne)k²+ 2

τkq^ne_k+1−q_k^nek²

≤ kF(q_kⁿ, q^J_k, q^ne_k )−F(qⁿ_k−1, q_k−1^J , q_k−1^ne )k_H⁻¹_(Ω)kq_k+1^ne −q^ne_k k_H1 0(Ω).

Cauchy’s inequality provides By a similar reasoning as (3.3.122)-(3.3.123):

Further, from which we infer

kK₁k_L2(Ω)≤ Recall (5.2.50) we obtain

kK₂k_L2(Ω) ≤C

It remains to estimate

J_k∇q_k^v−Jk−1∇q_k−1^v

L²(Ω). We first have the reformu-lation:

J_k∇q^v_k−Jk−1∇q^v_k−1 =J_k∇4^k_V +∇q_k−1^v 4^k_J.

Since4^k_V is the unique solution of







λ²44^k_V =4^k_n, 4^k_V|_∂Ω= 0,

(5.2.52) thenk4^k_Vk_H2(Ω) ≤Ck4^k_nk_L2(Ω). Thus

Jk∇q_k^v−Jk−1∇q_k−1^v

L²(Ω)≤C

k4^k_Jk_H1(Ω)+k4^k_nk_L2(Ω)

. (5.2.53) Recalling (5.2.48), combining (5.2.49), (5.2.51) and (5.2.53) we obtain

∂tkq_k+1^ne −q_k^nek²+Ckq_k+1^ne −q_k^nek²_H1 0(Ω)

≤C

k4^k_nek²_L2(Ω)+k4^k_Jk²_H1(Ω)+k4^k_nk²_H2(Ω)

.

(5.2.54)

Integrating (5.2.54) from 0 to anyt∈[0, tu) yields k(q_k+1^ne −q^ne_k )(t,·)k²

≤Ctuk4^k_nek²_L∞(0,tu;L²(Ω))+C

k4^k_Jk²_L2(0,tu;H¹(Ω))+k4^k_nk²_L2(0,tu;H²(Ω))

. From (5.2.47) we decuce

kq^ne_k+1−q_k^nek²_L∞(0,tu;L²(Ω))≤

Ctu+ Ctu

1−Ct_u

k4^k_nek²_L∞(0,tu;L²(Ω)), which implies{q_k^ne}^∞_k=0 is a Cauchy sequence in L^∞(0, tu;L²(Ω)) assuming

Ctu+ Ct_u 1−Ctu

<1, thus there existsq_∗^ne∈L^∞(0, tu;L²(Ω)) such that

q^ne_k −→q_∗^ne inL^∞(0, t_u;L²(Ω)).

Now we are in a position to study the convergence behavior of the sequence {(q_kⁿ, q_k^J, q_k^v, q^ne_k )}^∞_k=0. Select 0 < T∗ < min (t_u,C∗). Since the embeddings H³(Ω),→H²(Ω),H²(Ω),→H¹(Ω) are compact,{(q_kⁿ, q^J_k, q^ne_k )}^∞_k=0are bounded inL^∞(0,T_∗;H³(Ω)×(H²(Ω))^d+1) and {(∂_tq_kⁿ, ∂_tq^J_k, ∂_tq^ne_k )}^∞_k=0 are bounded in L²(0,T∗;H¹(Ω)×(L²(Ω))^d+1) from Lemma 5.2.3 and Remark 5.2.1, then Aubin’s Lemma (Corollary 4 in [72]) yields{q_kⁿ}^∞_k=0,{q_k^J}^∞_k=0 and {q_k^ne}^∞_k=0 are relatively compact inC([0,T_∗], H²(Ω)),C([0,T_∗]; (H¹(Ω))^d) andC([0,T_∗];H¹(Ω)) respectively, i.e., there is a subsequence of (qⁿ_k, q_k^J, q_k^ne) and functions (Pⁿ,P^J, V,P^ne) which satisfy (maybe after extracting a subsequence)

(q_kⁿ, q_k^J)→(Pⁿ,P^J) inC([0,T∗];H²(Ω)×(H¹(Ω))^d), (5.2.55)

(q^v_k, q_k^ne)→(V,P^ne) in C([0,T_∗];H²(Ω))×C([0,T_∗];H¹(Ω)). (5.2.56) Furthermore we also obtain

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















∂_tq_kⁿ* ∂_tPⁿ in L²(0,T_∗;H²(Ω)),

∂tq_k^J * ∂tP^J in L²(0,T_∗;H¹(Ω)),

∂tq_k^ne* ∂tP^ne in L²(0,T_∗;H¹(Ω)), q_kⁿ*^∗ Pⁿ in L^∞(0,T_∗;H³(Ω)), q_k^J *^∗ P^J in L^∞(0,T_∗;H²(Ω)), q_k^ne*^∗ P^ne in L^∞(0,T_∗;H²(Ω)).

(5.2.57)

Fix now 0 < γ < 1

2 and any t ∈ (0,T_∗) we deduce the interpolation (after extracting a subsequence)

kq_kⁿ− Pⁿk_H^3−γ_(Ω)≤Cinkqⁿ_k− Pⁿk^(3−γ)/3_H3(Ω) kq_kⁿ− Pⁿk^γ/3_L2(Ω), then byLemma 5.2.3

kq_kⁿ− Pⁿk_H3−γ(Ω)≤Cin2^(3−γ)/3C^0(3−γ)/3kqⁿ_k − Pⁿk^γ/3_L2(Ω). (5.2.58) Similarly

kq^J_k − P^Jk_H2−γ(Ω) ≤Cin2^(2−γ)/2C^0(2−γ)/2kq^J_k − P^Jk^γ/2_L2(Ω), (5.2.59) kq_k^ne− P^nek_H^2−γ_(Ω) ≤Cin2^(2−γ)/2C˜^(2−γ)/2kq_k^ne− P^nek^γ/2_L2(Ω). (5.2.60) By Sobolev embedding theorem we deduce

(qⁿ_k,∇q_kⁿ, q^J_k, q^ne_k )→(Pⁿ,∇Pⁿ,P^J,P^ne) in C([0,T_∗]×Ω). (5.2.61) Notice that the functions (Pⁿ,P^J, V,P^ne) depend upon the selection of the subsequence.

Then we can find a subsequence {(qⁿ_k

j, q_k^J

j, q_k^v

j, q^ne_k

j)}^∞_j=0 ⊆ {(qⁿ_k, q_k^J, q_k^v, q_k^ne)}^∞_k=0 such that there are functions (P_aⁿ,P_a^J, Va,P_a^ne) and P_b^ne with (q_kⁿ

j, q_k^J

j, q^v_k

j, q^ne_k

j) converges to (P_aⁿ,P_a^J, Va,P_a^ne) in the sense of (5.2.55)-(5.2.61) asj→ ∞;q_k^ne

j+1

converges to P_b^ne in the sense of (5.2.55) and (5.2.61) as j → ∞. Then from Lemma 5.2.4P_a^ne coincides withP_b^ne.

We consider the system

According to (5.2.61)-(5.2.57) it is easy to verify that

Furthermore we have the calculus ZZ

Using (5.2.61)-(5.2.57) we obtain the convergence ZZ

converges to ZZ

Q∗

∇ϕ

∇(P_aⁿ+n₀)⊗ ∇(P_aⁿ+n₀) P_aⁿ+n₀

P_a^J +J₀ P_a^J+n₀

dxdt, asj→ ∞, which implies

ZZ

Q∗

ϕF(qⁿ_kj, q^J_kj, q_k^ne_j)dxdt−→

ZZ

Q∗

ϕF(P_aⁿ,P_a^J,P_a^ne)dxdt, as j→ ∞.

Thus we conclude (P_aⁿ,P_a^J, Va,P_a^ne) is the solution we seek.

5.2.4 Uniqueness

Let (n¹, J¹, V¹,(ne)¹) and (n², J², V²,(ne)²) be two solutions of (5.1.1)-(5.1.4).

Put

n4 =n¹−n², J4=J¹−J², V4 =V¹−V², (ne)4 = (ne)¹−(ne)², then we obtain the system



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

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





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







∂_tn4−ν₀4n₄= divJ4,

∂tJ4−ν04J₄+1

τJ4−µ∇n₄+²

6∇4n₄=4_Fd,

∂_t(ne)4−ν₀4(ne)4+ 2

τ(ne)4=4F, λ²4V₄=n4,

(n4, J4, V4,(ne)4)(t, x) = 0 on [0,T_∗)×∂Ω, (n4, J4,(ne)4)(0, x) = 0,

where

4_Fd : =F_d(n¹, J¹,(ne)¹)− F_d(n², J²,(ne)²), 4_F : =F(n¹, J¹,(ne)¹)− F(n², J²,(ne)²).

Similarly as (5.2.44) we obtain for a.e. t∈[0,T_∗] 1

2∂_t

µkn₄k²+²

6k∇n₄k²+kJ₄k²

+ν₀T₀k∇n₄k² +²

6ν0k4n₄k²+ν0k∇J₄k²+1 τkJ₄k²

=(4_Fd, J4)≤ k4_Fdk_H−1(Ω)kJ₄k_H1 0(Ω).

Using Young’s inequality 1

2∂t

µkn₄k²+²

6k∇n₄k²+kJ₄k² +C

ν0T0k∇n₄k² +²

6ν₀k4n₄k²+ν₀k∇J₄k²+ 1

τkJ₄k²

≤Ck4F_dk²_L∞(0,T∗;H⁻¹(Ω)),

(5.2.62)

which implies

kn₄k²_L∞(0,T∗;H₀¹(Ω))+kJ₄k²_L∞(0,T∗;L²(Ω))≤CT∗k4_Fdk²_L∞(0,T∗;H⁻¹(Ω)). (5.2.63) We go back to the proof of Lemma5.2.4. According to (5.2.45) we deduce for a.e. t∈[0,T∗]

k4F_dk²_H−1(Ω)≤C

kn4k²_H1(Ω)+kJ4k²_L2(Ω)+k(ne)4k²_L2(Ω)

. (5.2.64) Integrate (5.2.62) from 0 to t∈[0,T_∗) then we infer

kn₄k²_L∞(0,T∗;H₀¹(Ω))+kJ₄k²_L∞(0,T∗;L²(Ω))

≤ CT∗

1−CT_∗k(ne)₄k²_L∞(0,T∗;L²(Ω)),

(5.2.65)

which implies

kn₄k²_L2(0,T∗;H²(Ω))+kJ₄k²_L2(0,T∗;H¹(Ω))

≤ CT_∗ 1−CT∗

k(ne)₄k²_L∞(0,T∗;L²(Ω)).

(5.2.66)

Next we have

∂_t1

2k(ne)₄k²+ν₀k∇((ne)₄)k²+ 2

τk(ne)₄k²

≤ k4_Fk_H−1(Ω)k(ne)₄k_H1 0(Ω).

(5.2.67)

By a similar reasoning as (5.2.49)-(5.2.53) we obtain for a.e. t∈[0,T∗) k4_Fk²_H−1(Ω)≤C

kn₄k²_H2(Ω)+kJ₄k²_H1(Ω)+k(ne)₄k²_L2(Ω)

. (5.2.68) Integrate (5.2.67) from 0 to anyt∈[0,T∗) we see

k(ne)₄k²_L∞(0,T∗;L²(Ω))

≤CT_∗k(ne)₄k²_L∞(0,T∗;L²(Ω))

+C

kn₄k²_L2(0,T∗;H²(Ω))+kJ₄k²_L2(0,T∗;H¹(Ω))

.

(5.2.69)

Finally let us substitute (5.2.66) into (5.2.69), then we conclude k(ne)₄k²_L∞(0,T∗;L²(Ω)) ≤

CT∗+ CT∗

1−CT_∗

k(ne)₄k²_L∞(0,T∗;L²(Ω)). SinceT_∗ ≤t_u,

CT∗+ CT∗

1−CT_∗ <1,

from which it follows (ne)4 ≡0.Furthermore from (5.2.66) we have n4≡J4 ≡0.

Thus we have completed the proof of the uniqueness of local solutions.

Calculus Facts

A.1 Gauss-Green Theorem

Let Ω⊂Rⁿ be open and bounded, and ∂Ω isC¹.

• Gauss-Green Theorem Supposeu∈H¹(Ω). Then

Z

Ω

uxidx= Z

∂Ω

uνidS (i= 1,· · ·, n).

• Integration-by-parts formula Letu, v∈H¹(Ω). Then

Z

Ω

uxivdx=− Z

Ω

uvxidx+ Z

∂Ω

uvνidS (i= 1,· · ·, n).

• Green’s formulasLet u, v∈H²(Ω). Then 1.

Z

Ω

4udx= Z

∂Ω

∂u

∂νdS, 2.

Z

Ω

∇u· ∇vdx=− Z

Ω

u4vdx+ Z

∂Ω

u∂v

∂νdS, 3.

Z

Ω

(u4v−v4u)dx= Z

∂Ω

u∂v

∂ν −v∂u

∂ν

dS.

A.2 Convolution

We introduce tools which provide smooth approximations to given functions.

Let Ω⊂Rⁿ be open, write Ωε:={x∈Ω|dist(x, ∂Ω)> ε}.

143

Definition A.2.1. Define η∈C^∞(Rⁿ) by

η(x) :=





 Cexp

1

|x|²−1

if |x|<1, 0 if |x| ≥1, the constant C >0 is selected such that R

Rⁿηdx= 1.For each ε >0, set η_ε(x) := 1

εⁿηx ε

.

We call η the standard mollifier. Obviously, the functionsηε(x) are inC^∞(Rⁿ) and satisfy

Z

Rⁿ

ηε(x)dx= 1, suppηε⊂B(0, ε), where B(0, ε) is a closed ball with center 0, radius ε.

Definition A.2.2. Iff : Ω→Ris locally integrable, we define its mollification f^ε:=η_ε∗f in Ω_ε,

where

η_ε∗f = Z

Ω

η_ε(x−y)f(y)dy= Z

B(0,ε)

η_ε(y)f(x−y)dy for x∈Ω_ε. Theorem A.2.1. (Properties of mollifiers)

1. f^ε∈C^∞(Ωε).

2. f^ε→f a.e. asε→0.

3. Iff ∈C(Ω), then f^ε→f uniformly on compact subsets of Ω.

4. If1≤p <∞ andf ∈L^ploc(Ω), then f^ε→f in L^ploc(Ω).

5. Iff ∈C(Ω), then f^ε→f uniformly on Ω.

Convergence and Compactness

Theorem B.0.2. Any bounded set in a reflexive Banach space is weakly com-pact, i.e., any sequence in a bounded set has a weakly converging subsequence.

Theorem B.0.3. Any bounded set in L^p(Ω) with 1 < p ≤ ∞ is weakly star compact.

Theorem B.0.4. (J.P. Aubin [12])

Let T >0, X, B, Y be Banach spaces with continuous embeddingsX ⊂B ⊂Y, the first embedding being compact. Then we have

1. Let 1 ≤p≤ ∞, F be a bounded set of L^p(0, T;X) and be precompact in L^p(0, T;Y), then F is precompact in L^p(0, T;B).

2. Let 1 ≤p < ∞, F be a bounded set of L^p(0, T;X), ∂_tF:={∂_tf | f ∈F} be bounded inL¹(0, T;Y). Then F is precompact in L^p(0, T;B).

3. Let F be bounded of L^∞(0, T;X), ∂tF be bounded in L^r(0, T;Y) where r >1. Then F is precompact in C([0, T];B).

Theorem B.0.5. (almost everywhere Point-wise Convergence)

1. Suppose that Ω is a bounded or unbounded domain in Rⁿ, uk(x), u(x) are real functions in L^p(Ω),(1≤p ≤ ∞) such that u_k converges to u in L^p(Ω). Then if 1≤p <∞, u_k has a subsequence a.e. converging to u; if p=∞, then uk itself converges to u a.e. inΩ.

2. Suppose that Ω is a bounded domain in Rⁿ, {u_k(x)}^∞_k=1 is a bounded sequence in L^p(Ω),(1 ≤p < ∞) such that u_k converges to a function u a.e. inΩ. Then u also belongs toL^p(Ω), and u_k converges weakly tou in L^p(Ω). If p = ∞, the conclusion becomes that u_k converges to u weakly star in L^∞(Ω).

145

Definition B.0.3. A Banach space B is called uniformly conves if for any φ, ϕ ∈ B, ε > 0 such that kφk = kϕk = 1, kφ−ϕk ≥ ε, then there exists a constant δ >0 depending only upon εand kφ+ϕk ≤2(1−δ)<2.

Remark B.0.1. A Hilbert space is uniformly convex. A uniformly convex Banach space is reflexive. If 1 < p < ∞, then L^p(Ω) is uniformly convex.

L^p(Ω) is reflexive if and only if 1< p <∞. Thus in L^p(Ω) Reflexivity⇔Uniform Convexity⇔1< p <∞.

Theorem B.0.6. Let B be a uniformly convex Banach space. Suppose that uk, u ∈ B, ku_kk → kuk, and uk converges weakly to u. Then uk converges strongly to u, i.e., ku_k−uk →0 as k→ ∞.

Theorem B.0.7. Suppose 1 < p ≤ ∞, B is a Banach space, B^∗ is the dual space of B, and

(u_k*^∗u in L^p(0, T;B^∗), u⁰_k*^∗u⁰ in L^p(0, T;B^∗), then u_k(0)*^∗ u(0) in B^∗.

Furthermore suppose 1 < p <∞, B^∗ is a reflexive Banach space. Then the weakly star convergence above is equivalent to the weak convergence.

Inequalities

• Young’s inequalities. Let a,b and ε be positive numbers andp, q ≥1, 1

p +1

q = 1.Then

ab≤ ε^pa^p p + b^q

qε^q.

• H¨older’s inequality. Let 1≤p₁,· · ·, p_m ≤ ∞, with Pm j=1

1 pj

= 1, and assumeu_k ∈L^pk(Ω) fork= 1,· · ·, m. Then

Z

Ω

|u₁· · ·u_m|dx≤

m

Y

k=1

ku_ik_Lpk(Ω).

• Gronwall’s inequality

1. Suppose that a, b are nonnegative constants T > 0, and u(t) is a nonnegative integrable function. Suppose the inequality

u(t)≤a+b Z t

0

u(s)ds holds fort∈[0, T]. Then for 0≤t≤T,

u(t)≤ae^bt.

2. Lety(·) be a nonnegative, absolutely continuous function on [0, T], which satisfies for a.e. t∈[0, T] the differential inequality

y⁰(t)≤φ(t)y(t) +ψ(t),

whereφ(t) and ψ(t) are nonnegative, summable functions on [0, T].

Then

y(t)≤e

Rt 0φ(s)ds

y(0) +

Z _t

0

ψ(s)ds

.

147

• Interpolation inequalities.

1. Assume 1 ≤ s ≤ r ≤ t ≤ ∞ and 1 r = γ

s + 1−γ

t for γ ∈ [0,1].

Suppose alsou∈L^s(Ω)∩L^t(Ω),thenu∈L^r(Ω),and kuk_Lr(Ω)≤ kuk^γ_Ls(Ω)kuk^1−γ_Lt(Ω).

2. Suppose Ω ⊂ Rⁿ satisfies the cone condition and 0 ≤ j ≤ m,1 ≤ p < ∞, u ∈ W_p^m(Ω). Then there exists K > 0 depending only on n, m, p,Ω such that

kukWp^j(Ω)≤Kkuk^j/m_Wm

p (Ω)kuk^(m−j)/m_Lp(Ω) . (C.0.1) 3. Suppose Ω⊂Rⁿsatisfies the cone condition and m≥0,1≤p <∞, u ∈W_p^m(Ω). Ifmp > n, let q ∈[p,∞]; if mp=n, let q ∈[p,∞); if mp < n,let q ∈[p,_n−mp^np ]. Then there existsK >0 depending only on n, m, p, q,Ω such that

kuk_Lq(Ω)≤Kkuk^θ_Wm

p (Ω)kuk^1−θ_Lp(Ω) withθ= _mpⁿ −_mqⁿ .

4. Suppose Ω ⊂ Rⁿ satisfies the cone condition, p > 1, mp > n, u ∈ W_p^m(Ω). Suppose either 1≤q ≤p orq > p and mp−p < n. Then there existsK >0 depending only on n, m, p, q,Ω such that

kuk_L^∞_(Ω)≤Kkuk^θ_Wm

p (Ω)kuk^1−θ_Lq(Ω).

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