Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions

Volume 2009, Article ID 207873,22pages doi:10.1155/2009/207873

Research Article

Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions

Dong-Gun Park,

Jin-Mun Jeong,

and Sun Hye Park

1Mathematics and Materials Physics, Dong-A University, Saha-Gu, Busan 604-714, South Korea

2Division of Mathematical Sciences, Pukyong National University, Busan 608-737, South Korea

3Department of Mathematics, Pusan National University, Busan 609-735, South Korea

Correspondence should be addressed to Jin-Mun Jeong,jmjeong@pknu.ac.kr Received 28 August 2008; Revised 4 December 2008; Accepted 1 January 2009 Recommended by Donal O’Regan

We prove the regularity for solutions of parabolic hemivariational inequalities of dynamic elasticity in the strong sense and investigate the continuity of the solution mapping from initial data and forcing term to trajectories.

Copyrightq2009 Dong-Gun Park et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we deal with the existence and a variational of constant formula for solutions of a parabolic hemivariational inequality of the form:

ux, t Δux, t˙ −divC

εux, t

Ξx, t fx, t inΩ×0,∞, 1.1

ux, t 0 onΓ1×0,∞, 1.2

C ε

ux, t

ν−β·νux, t onΓ0×0,∞, 1.3

Ξx, t∈ϕ ux, t

a.e.x, t∈Ω×0,∞, 1.4

ux,0 u0x inΩ, 1.5

where Ωis a bounded domain in R^N with sufficiently smooth boundaryΓ.Let x⁰ ∈ R^N, βx x−x⁰,Rmax_x∈Ω|x−x⁰|.The boundaryΓis composed of two piecesΓ0andΓ1, which

are nonempty sets and defined by Γ0:

x∈Γ:βx·ν≥α >0

, Γ1:

x∈Γ:βx·ν≤0

, 1.6

whereν is the unit outward normal vector to Γ. Here ˙u ∂u/∂t,u u1, . . . , uN^T is the displacement,εu 1/2{∇u ∇u^T} 1/2∂ui/∂xj ∂uj/∂xiis the strain tensor, ϕu ϕ1u1, . . . , ϕNuN^T,ϕi is a multi-valued mapping by filling in jumps of a locally bounded functionbi,i1, . . . , N. A continuous mapCfrom the spaceSofN×Nsymmetric matrices into itself is defined by

Cε atrεIbε, forε∈S, 1.7

whereI is the identity ofS, trεdenotes the trace ofε, and a > 0, b > 0. For example, in the caseN 2,Cε E/d1−μ²μtrεI 1−με, whereE >0 is Young’s modulus, 0< μ <1/2 is Poisson’s ratio anddis the density of the plate.

LetH andV be two complex Hilbert spaces. Assume thatV is a dense subspace in Hand the injection ofVintoHis continuous. LetAbe a continuous linear operator fromV intoV^∗which is assumed to satisfy G˚arding’s inequality. Namely, we formulated the problem 1.1as

u˙Au−divC εu

Ξ f inΩ×0,∞. 1.8

The existence of global weak solutions for a class of hemivariational inequalities has been studied by many authors, for example, parabolic type problems in1–4, and hyperbolic types in5–7. Rauch8and Miettinen and Panagiotopoulos1,2proved the existence of weak solutions for elliptic one. The background of these variational problems are physics, especially in solid mechanics, where nonconvex and multi-valued constitutive laws lead to differential inclusions. We refer to3,4to see the applications of differential inclusions. Most of them considered the existence of weak solutions for differential inclusions of various forms by using the Faedo-Galerkin approximation method.

In this paper, we prove the existence and a variational of constant formula for strong solutions of parabolic hemivariational inequalities. The plan of this paper is as follows. In Section 2, the main results besides notations and assumptions are stated. In order to prove the solvability of the linear case withΞx, t 0 we establish necessary estimates applying the result of Di Blasio et al.9to1.1–1.5considered as an equation inHas well asV^∗. The existence and regularity for the nondegenerate nonlinear systems has been developed as seen in10, Theorem 4.1or11, Theorem 2.6, and the references therein. InSection 3, we will obtain the existence for solutions of1.1–1.5by converting the problem into the contraction mapping principle and the norm estimate of a solution of the above nonlinear equation on L²0, T;V∩W^1,20, T;V^∗⊂C0, T;H. Consequently, ifuis a solution asociated withu0, andf, in view of the monotonicity ofA, we show that the mapping

H×L²

0, T;V^∗

u0, f

−→u∈L²0, T;V∩C

0, T;H

, 1.9

is continuous.

2. Preliminaries and Linear Hemivariational Inequalities

We denoteξ·ζ_N

i1ξiζiforξ ξ1, . . . , ξN,ζ ζ1, . . . , ζN∈R^Nandε·ε_N

i,j1εijεij for ε,ε∈S. Throughout this paper, we consider

V

u∈

H¹ΩN

:u0 onΓ1

, H

L²Ω_N , u, v

Ωux·vxdx, u, v_Γ0

Γ0

ux·vxdΓ. 2.1

We denoteV^∗the dual space ofV,·,·the dual pairing betweenV andV^∗.

The norms onV,H, andV^∗will be denoted by · ,| · |and · ∗, respectively. For the sake of simplicity, we may consider

u_∗≤ |u| ≤ u, u∈V. 2.2

We denote · _L2Γ0^N by · _Γ0. Let Abe the operator associated with a sesquilinear form au, vwhich is defined G˚arding’s inequality

Reau, u≥ω1u²−ω2|u|², ω1>0, ω2≥0, foru∈V, 2.3

that is,

Au, v au, v, u, v∈V. 2.4

ThenAis a symmetric bounded linear operator fromV intoV^∗which satisfies

Au, u≥ω1u²−ω2u² 2.5

and its realization inHwhich is the restriction ofAto

DA {u∈V :Au∈H} 2.6

is also denoted byA. Here, we note thatDAis dense inV. Hence, it is also dense inH. We endow the domainDAofAwith graph norm, that is, foru∈ DA, we defineuDA

|u||Au|. So, for the brevity, we may regard that|u| ≤ u ≤ uDAfor allu∈V. It is known that—Agenerates an analytic semigroupSt t≥0in bothHandV^∗.

From the following inequalities

ω1u²≤Reau, u ω2u²≤C|Au||u|ω2|u|²

≤

C|Au|ω2|u|

|u| ≤max C, ω2

uDA|u|, 2.7

it follows that there exists a constantC0>0 such that

u ≤C0u^1/2_DA|u|^1/2. 2.8

So, we may regard asV DA, H_1/2,2whereDA, H_1/2,2is the real interpolation space betweenDAandH.

Consider the following initial value problem for the abstract linear parabolic type equation:

ut ˙ Aut−divC

εut

ft, t >0, u0 onΓ1×0,∞,

C εu

x, t

ν−β·νux, t onΓ0×0,∞, ux,0 u0x, x∈Ω.

LE

A continuous mapCfrom the spaceSofN×Nsymmetric matrices into itself is defined by Cε atrεIbε, fora >0, b >0, ε∈S. 2.9

It is easily known that

divCεw, v −Cεwν, v_Γ0 Cεw, εv, v, w∈V, 2.10 C

ε w1

−C ε

w2

C ε

w1−w2

, w1, w2 ∈V. 2.11

Note that the mapCis linear and symmetric and it can be easily verified that the tensorC satisfies the condition

λ0ε²≤Cε·ε≤λ1ε², ε∈Sfor someλ0, λ1>0. 2.12

Letλbe the smallest positive constant such that

v²≤λ∇v² ∀v∈V. 2.13

Simple calculations and Korn’s inequality yield that

λ2|∇u|²≤εu²≤λ3|∇u|², 2.14 and hence|εu|is equivalent to theH¹Ω^Nnorm onV.Then by virtue of9, Theorem 3.3, we have the following result on the linear parabolic type equationLE.

Proposition 2.1. Suppose that the assumptions stated above are satisfied. Then the following properties hold.

1For anyu0 ∈ V DA, H_1/2,2 andf ∈ L²0, T;H T > 0, there exists a unique solutionuofLEbelonging to

L²

0, T;DA

∩W^1,20, T;H⊂C

0, T;V

2.15

and satisfying

uL²0,T;DA∩W^1,20,T;H≤C1u0fL²0,T;H

, 2.16

whereC1is a constant depending onT.

2Letu0 ∈Handf ∈L²0, T;V^∗for anyT >0. Then there exists a unique solutionuof LEbelonging to

L²0, T;V∩W^1,20, T;V^∗⊂C

0, T;H

2.17

and satisfying

uL²0,T;V∩W^1,20,T;V^∗ ≤C1u0fL²0,T;V^∗

, 2.18

whereC1is a constant depending onT.

Proof. 1Letau, v be a bounded sesquilinear form defined inV×V by

au, v Au, v− divC

εu , v

, u, v∈V. 2.19

Noting that by2.10

− divC

εu , u

C

εu , εu

β·νu, u

Γ0, 2.20

and by2.12,2.14, and1.6, λ0λ2u²≤

C εu

, εu

, α|u|²≤

β·νu, u

Γ0, 2.21

it follows that there existω1>0 andω2≥0 such that

Reau, u ≥ω1u²−ω2|u|², foru∈V. 2.22 LetAbe the operator associated with this sesquilinear form:

Au, v au, v, u, v∈V. 2.23

ThenAis also a symmetric continuous linear operator fromV intoV^∗which satisfies Au, u ≥ω1u²−ω2|u|². 2.24

So we know that—Agenerates an analytic semigroupSt t ≥0in bothHandV^∗. Hence, by applying9, Theorem 3.3to the regularity for the solution of the equation:

ut ˙ Aut ft, t >0, u0 onΓ1×0,∞, C

ε

ux, t

ν−β·νux, t onΓ0×0,∞, ux,0 u0x, x∈Ω,

2.25

in the spaceH, we can obtain a unique solutionuofLEbelonging to L²

0, T;DA

∩W^1,20, T;H⊂C

0, T;V

2.26

and satisfying the norm estimate2.16.

2It is easily seen that

H

x∈V^∗: _T

0

Ae^tAx²

∗dt <∞

, 2.27

for the timeT >0. Therefore, in terms of the intermediate theory we can see that

V, V^∗_1/2,2 H 2.28

and follow the argument of1term by term to deduce the proof of2results.

3. Existence of Solutions in the Strong Sense

This Section is to investigate the regularity of solutions for the following parabolic hemivariational inequality of dynamic elasticity in the strong sense:

ut ˙ Aut−divC ε

ut

Ξx, t ft, t≥0, u0 onΓ1×0,∞,

C ε

ux, t

ν−β·νux, t onΓ0×0,∞ Ξx, t∈ϕ

ux, t

a.e.x, t∈Ω×0,∞, u0 u0.

HIE

Now, we formulate the following assumptions.

HbLetbii1, . . . , N:R → Rbe a locally bounded function verifying

bis≤μi|s| fors∈R, 3.1

whereμi>0.We denote

μmax

μ1, . . . , μN

. 3.2

The multi-valued functionϕi : R → 2^Ris obtained by filling in jumps of a function bi:R → Rby means of the functionsb_i, b_i, bi, bi:R → Ras follows.

b_is ess inf

|τ−s|≤biτ, b_is ess sup

|τ−s|≤biτ, bis lim

→0b_is, bis lim

→0bi

s,

ϕis

bis, bis .

3.3

We denotebξ: b1ξ1, . . . , bNξN,ϕξ: ϕ1ξ1, . . . , ϕNξNforξ ξ1, . . . , ξN∈R^N. We will need a regularization ofbidefined by

bⁿ_is n _∞

−∞bis−τρnτdτ, 3.4

whereρ∈C₀^∞−1,1, ρ≥0 and₁

−1ρτdτ 1.It is easy to show thatbⁿ_i is continuous for all n∈Nandb_i, b_i, bi, bi, bⁿ_i satisfy the same conditionHbwith possibly different constants ifbi satisfiesHb. It is also known thatb_iⁿsis locally Lipschitz continuous ins, that is for anyr >0, there exists a numberLir>0 such that

Hb-1

bⁿ_i s1

−bⁿ

s2≤Lirs1−s2 3.5

holds for alls1, s2∈Rwith|s1|< r, |s2|< r.We denote

Lr max{L1r, . . . , LNr}. 3.6

The following lemma is from12; Lemma A.5.

Lemma 3.1. Letm∈L¹0, T;Rsatisfyingmt≥0 for allt∈0, Tanda≥0 be a constant. Letd be a continuous function on0, T⊂Rsatisfying the following inequality:

1

2d²t≤ 1 2a²

_t

0

msdsds, t∈0, T. 3.7

Then,

dt≤a _t

0

msds, t∈0, T. 3.8

Proof. Let

βt 1

2a² _t

0

msdsds, >0. 3.9

Then

dβt

dt mtdt, t∈0, T, 3.10

and

1

2d²t≤β0t≤βt, t∈0, T. 3.11

Hence, we have

dβt

dt ≤mt√ 2

βt. 3.12

Sincet → βtis absolutely continuous and d

dt

βt 1

2 βt

dβt

dt 3.13

for allt∈0, T, it holds

d dt

βt≤ 1

√2mt, 3.14

that is,

βt≤

β0 1

√2 _t

0

msds, t∈0, T. 3.15

Therefore, combining this with3.11, we conclude that

|dt| ≤√ 2

βt≤√ 2

β0

_t

0

msds

a _t

0

msds, t∈0, T

3.16

for arbitrary >0.

From now on, we establish the following results on the local solvability of the following equation,

ut ˙ Aut−divC ε

ut

−bⁿ ut

ft, t≥0, n∈N, u0 onΓ1×0,∞,

C εu

x, t

ν−β·νux, t onΓ0×0,∞, u0 u0.

HIE-1

Lemma 3.2. Letube a solution of HIE-1andu ∈Br {v∈L²0, T;V :||v|| ≤ r}. Then, the following inequality holds, for any 0< t≤T,

ut²u²_L20,t;Γ0u²_L20,t;V≤c⁻¹₁ 1

2u0²f²_L20,t;H

e^ω2Lr1t, 3.17

wherec1min{1/2, α, ω1c0}.

Proof. We remark that from2.11,2.12, it follows that there is a constantc0>0 such that

c0u1t−u2t²≤ C

ε u1t

−C ε

u2t , ε

u1t

−ε u2t

. 3.18

Consider the following equation:

ut ˙ Aut−divC ε

ut

−bⁿ ut

ft, t >0, n∈N. 3.19

Multipying on both sides ofut, we get ut, ut˙

Aut, ut

C ε

ut

, ε

ut

β·νut, ut

bⁿ ut

, ut

ft, ut

, 3.20

and integrating this over0, t, by1.6,2.5,3.18andHb-1, we have 1

2ut²α _t

0

uτ²

Γ0dτ ω1c0

^t

0

uτ²dτ

≤ 1

2u0²

ω2Lr^t

0

uτ²dτ _t

0

fτ²uτ² dτ,

3.21

that is,

c1ut²u²_L20,t;Γ0u²_L20,t;V

≤ 1

2u0²f²_L20,t;H

ω2Lr 1^t

0

uτ²dτ.

3.22

Applying Gronwall lemma, the proof of the lemma is complete.

Theorem 3.3. Assume thatu0∈H,f ∈L²0, T;V^∗and (Hb). Then, there exists a timeT0>0 such thatHIE-1admits a unique solution

u∈L²0, T0;V∩W^1,20, T0;V^∗∩C

0, T0;H

, 0< T0≤T. 3.23 Proof. Assume that2.5holds forω2/0. Let the constantrsatisfy the following inequality:

c⁻¹₁ 1

2u0²f²_L20,T;H

e^ω2Lr1T < r. 3.24

Let us fixT ≥T0>0 such that

max

μ, Lr 4ω2

ω1c0

e^2ω2T0−1

<1, 3.25

whereμis given byHb.

InvokingProposition 2.1, for a givenw∈Br{v∈L²0, T0;V:v ≤r}, the problem ut ˙ Aut−divC

ε

ut

−bⁿ

wt

ft, t≥0, n∈N, u0 onΓ1×0,∞,

C ε

ux, t

ν−β·νux, t onΓ0×0,∞, u0 u0.

HIE-2

has a unique solutionu∈ L²0, T;V∩C0, T;H. To prove the existence and uniqueness of solutions of semilinear typeHIE-1, by virtue ofLemma 3.2, we are going to show that the mapping defined byw→umaps is strictly contractive fromBrinto itself if the condition 3.25is satisfied.

Lemma 3.4. Letu1, u2be the solutions of HIE-2withwreplaced byw1, w2 ∈Br whereBris the ball of radiusrcentered at zero ofL²0, T0;V, respectively. Then the following inequality holds:

u1t−u2t≤ _t

0

e^ω2t−sGsds, 3.26

where

Gt Lrw1t−w2t. 3.27

Proof. Let u1, u2 be the solutions of HIE-2 with w replaced by w1, w2 ∈ L²0, T0;V, respectively. Then, we have that

d dt

u1t−u2t A

u1t−u2t

− divC

εu1t

−divC ε

u2t −

bⁿ w1t

−bⁿ w2t

, t >0, n∈N.

3.28

Multiplying on both sides ofu1t−u2tand by2.8, we get 1

2 d

dtu1t−u2t²a

u1t−u2t, u1t−u2t

C ε

u1t

−C ε

u2t , ε

u1t

−ε u2t

βx·ν

u1t−u2t

, u1t−u2t

Γ0

− bⁿ

w1t

−bⁿ w2t

, u1t−u2t ,

3.29

and so, by3.18,2.5,Hb, we obtain 1

2 d

dtu1t−u2t²

ω1c0u1t−u2t²

≤ω2u1t−u2t|²Lrw1t−w2tu1t−u2t.

3.30

Putting

Gt Lrw1t−w2t, Ht Gtu1t−u2t 3.31

and integrating3.30over0, t, this yields 1

2|u1t−ut|² ω1c0

^t

0

u1s−u2s²ds≤ω2

_t

0

u1s−u2s²ds _t

0

Hsds.

3.32

From3.32it follows that d dt

e^−2ω2t

_t

0

u1s−u2s²ds

2e^−2ω2t 1

2u1t−u2t²−ω2

_t

0

u1s−u2s²ds

≤2e^−2ω2t _t

0

Hsds.

3.33

Integrating3.33over0, twe have

e^−2ω2t _t

0

u1s−u2s²ds≤2 _t

0

e^−2ω2τ _τ

0

Hsds dτ

2 _t

0

_t

s

e^−2ω2τdτHsds2 _t

0

e^−2ω2s−e^−2ω2t

2ω2 Hsds

1 ω2

_t

0

e^−2ω2s−e^−2ω2t

Hsds,

3.34

thus, we get

ω2

_t

0

u1s−u2s²ds≤ _t

0

e^2ω2t−s−1

Hsds. 3.35

From3.32and3.35it follows that 1

2u1t−u2t² ω1c0

^t

0

u1s−u2s²ds

≤ _t

0

e^2ω2t−sHsds

_t

0

e^2ω2t−sGsu1s−u2sds,

3.36

which implies 1 2

e^−2ω2tu1t−u2t² ω1c0

e^−2ω2t _t

0

u1s−u2s²ds

≤ _t

0

e^−ω2sGse^−ω2su1s−u2sds.

3.37

By usingLemma 3.1, we obtain that

e^−ω2tu1t−u2t≤ _t

0

e^−ω2sGsds. 3.38

The proof of lemma is complete.

From3.26and3.36it follows that 1

2u1t−u2t² ω1c0

^t

0

u1s−u2s²ds

≤ _t

0

e^2ω2t−sGs _s

0

e^ω2s−τGτdτ ds

e^2ω2t _t

0

e^−ω2sGs _s

0

e^−ω2τGτdτ ds

e^2ω2t _t

0

1 2

d ds

_s

0

e^−ω2τGτdτ 2

ds

1 2e^2ω2t

_t

0

e^−ω2τGτdτ 2

≤ 1 2e^2ω2t

_t

0

e^−2ω2τdτ _t

0

Gτ²dτ

1

2e^2ω2t1−e^−2ω2t 2ω2

_t

0

Gτ²dτ

Lr² 4ω2

e^2ω2t−1^t

0

w1s−w2s²ds.

3.39

Starting from the initial valueu0t u0, consider a sequence{un·}satisfying u˙_n1t Au_n1t−divC

ε

u_n1t −bⁿ

unt

ft, t≥0 un10 onΓ1×0,∞

Cε

un1x, t

ν−β·νu˙n1x, t, onΓ0×0,∞ u_n10 u0.

3.40

Then from3.39it follows that 1

2u_n1t−unt² ω1c0

^t

0

u_n1s−uns²ds

≤ Lr² 4ω2

e^2ω2t−1^t

0

uns−u_n−1s²ds.

3.41

So by virtue of the condition3.25 the contraction principle gives that there exists u·∈L²0, T0;Vsuch that

un·−→u· inL²0, T0;V, 3.42

and hence, from3.41there existsu·∈C0, T0;Hsuch that un·−→u· inC

0, T0;H

. 3.43

Now, we give a norm estimation of the solution HIE and establish the global existence of solutions with the aid of norm estimations.

Theorem 3.5. Let the assumption (Hb) be satisfied. Assume thatu0 ∈ Handf ∈L²0, T;V^∗for anyT >0. Then, the solutionuofHIEexists and is unique in

u∈L²0, T;V∩W^1,20, T;V^∗⊂C

0, T;H

. 3.44

Furthermore, there exists a constantC2depending onTsuch that

uL²∩W^1,2 ≤C2

1|u0|fL²0,T;V^∗

. 3.45

Proof. Letw∈Br be the solution of

wt ˙ Awt−divC ε

wt

ft, t≥0, w0 onΓ1×0,∞,

C ε

wx, t

ν−β·νwx, t onΓ0×0,∞, w0 u0.

3.46

Then, since d

dt

ut−wt A

ut−wt

−divC ε

ut

divC ε

wt

−bⁿ ut

, 3.47

by multiplying byut−wt, fromHb,3.18and the monotonicity ofA, we obtain 1

2 d

dtut−wt²

ω1c0ut−wt²≤ω2ut−wt²μutut−wt. 3.48

By integrating on3.48over0, twe have 1

2ut−wt² ω1c0

^t

0

us−ws²ds

≤ω2

_t

0

us−ws²dsμ _t

0

usus−wsds.

3.49

By the procedure similar to3.39we have 1

2ut−wt ω1c0

^t

0

us−ws²ds≤ μ² 4ω2

e^2ω2t−1^t

0

us²ds. 3.50

Put

M μ²

4ω2

ω1c0

e^2ω2t−1

. 3.51

Then it holds

u−wL²0,T0;V≤M^1/2uL²0,T0;V 3.52

and hence, from2.16inProposition 2.1, we have that

uL²0,T0;V≤ 1

1−M^1/2wL²0,T0;V

≤ C0

1−M^1/2

1u0fL²0,T0;V^∗

≤C2

1u0fL²0,T0;V^∗

3.53

for some positive constantC2. Noting that byHb bⁿu

L²0,T;H≤const·uL²0,T;V 3.54 and byProposition 2.1

u_W^1,2_0,T;V^∗ ≤C1

1u0bⁿu f

L²0,T;V^∗

, 3.55

it is easy to obtain the norm estimate ofuinW^1,20, T0;V^∗satisfying3.45.

Now fromTheorem 3.3it follows that u

T0≤ u_C

0,T0,H ≤C2

1u0fL²0,T0;V^∗

. 3.56

So, we can solve the equation inT0,2T0and obtain an analogous estimate to3.53. Since the condition3.25is independent of initial values, the solution ofHIE-1can be extended the internal0, nT0for a natural numbern, that is, for the initialunT0in the intervalnT0,n 1T0, as analogous estimate3.53holds for the solution in0,n1T0. Furthermore, the estimate3.45is easily obtained from3.53and3.56.

We show thatu,Ξis a solution of the problemHIE.Lemma 3.4andHbgive that bⁿ

ut≤μut≤c, 3.57

and foru0∈H, there exists a unique solutionuofHIEbelonging to L²0, T;V∩W^1,20, T;V^∗⊂C

0, T;H

3.58

and satisfying3.44.

From3.44and3.57, we can extract a subsequence from{un}, still denoted by{un}, such that

un−→uweakly inL²0, T;V∩W^1,20, T;V^∗ 3.59 bⁿ

un

−→Ξweakly inL²0, T;H. 3.60

Here, we remark that if V is compactly embedded in H and u ∈ L²0, T;V, the following embedding

L²0, T;V∩W^1,20, T;V^∗⊂L²0, T;H 3.61

is compact in view13, Theorem 2. Hence, the mapping

un−→ustrongly inL²0, T;H. 3.62

By a solution ofHIE-1, we understand a mild solution that has a form

unt Stu0 _t

0

St−s divC

ε unt

−bⁿ uns

fs

ds, t≥0, 3.63

so lettingn → ∞and using the convergence results above, we obtain ut ˙ Aut−divC

ε

ut

Ξt ft, 0≤t≤T. 3.64

Now, we show thatΞx, t∈ϕux, ta.e. inQ: Ω×0, T0. Indeed, from3.62we haveuni → ui strongly inL²0, T;L²Ωand henceunix, t → uix, ta.e. inQ for each i1,2, . . . , N.Leti∈ {1,2, . . . , N}andη >0.Using the theorem of Lusin and Egoroff, we can choose a subsetω ⊂Qsuch that|ω|< η, ui ∈L²Q\ωanduni → uiuniformly onQ\ω.

Thus, for each >0,there is anM >2/such that unix, t−uix, t<

2 forn > M andx, t∈Q\ω. 3.65 Then, if|unix, t−s|<1/n,we have|uix, t−s|< for alln > Mandx, t∈Q\ω.

Therefore we have b_i

uix, t

≤b_iⁿ

unix, t

≤b_i

uix, t

, ∀n > M, x, t∈Q\ω. 3.66

Letφ∈L²0, T;L²Ω, φ≥0.Then

Q\ωbi

uix, t

φx, tdx dt≤

Q\ωbⁿ_i

unix, t

φx, tdx dt

≤

Q\ωbi

uix, t

φx, tdx dt.

3.67

Lettingn → ∞in this inequality and using3.60, we obtain

Q\ωbi

uix, t

φx, tdx dt≤

Q\ωΞix, tφx, tdx dt

≤

Q\ωbi

uix, t

φx, tdx dt,

3.68

whereΞ Ξ1, . . . ,ΞN.Letting → 0in this inequality, we deduce that Ξix, t∈ϕi

uix, t

a.e.inQ\ω, 3.69

and lettingη → 0we get

Ξix, t∈ϕi

uix, t

a.e.inQ. 3.70

This implies thatΞx, t∈ϕux, ta.e. inQ.This completes the proof of theorem.

Remark 3.6. In terms ofProposition 2.1, we remark that ifu0 ∈V DA, H_1/2,2 andf ∈ L²0, T;Hfor anyT >0 then the solutionuofHIEexists and is unique in

x∈L²0, T;DA∩W^1,20, T;H⊂C

0, T;V

. 3.71

Futhermore, there exists a constantC2depending onT such that uL²∩W^1,2 ≤C2

1u0fL²0,T;H

. 3.72

Theorem 3.7. Let the assumption (Hb) be satisfied

1ifu0, f∈V×L²0, T;H, then the solutionuofHIEbelongs tou∈L²0, T;DA∩ C0, T;Vand the mapping

H×L²0, T;H u0, f

−→u∈L²

0, T;DA

∩C

0, T;V

3.73

is continuous.

2letu0, f∈H×L²0, T;V^∗. Then the solutionuof HIEbelongs tou∈L²0, T;V∩ C0, T;Hand the mapping

H×L²0, T;V^∗u0, f−→u∈L²0, T;V∩C

0, T;H

3.74

is continuous.

Proof. 1 It is easy to show that if u0 ∈ V and f ∈ L²0, T;H, then u belongs to L²0, T;DA∩W^1,20, T;H. letu0i, fi ∈ V ×L²0, T;Handui ∈ Br be the solution of HIEwithu0n, fiin place ofu0, ffori1,2.Then in view ofProposition 2.1, we have

u1−u2

L²0,T;DA∩W^1,20,T;H

≤C1u01−u02bⁿu1−bⁿu2

L²0,T;Hf1−f2

L²0,T;H

≤C1u01−u02Lru1−u2

L²0,T:Vf1−f2

L²0,T;H

.

3.75

Since

u1t−u2t u01−u02 _t

0

u˙1s−u˙2s

ds, 3.76

we get

u1−u2

L²0,T;H≤√

Tu01−u02 T

√2u1−u2

W^1,20,T;H. 3.77

Hence, arguing as in2.8, we get u1−u2

L²0,T;V≤C0u1−u2^1/2

L²0,T;DAu1−u2^1/2

L²0,T;H

≤C0u1−u2^1/2

L²0,T;DA

T^1/4u01−u02^1/2 T

√2 _1/2

u1−u2^1/2

W^1,20,T;H

≤C0T^1/4u01−u02^1/2u1−u2^1/2

L²0,T;DA0

C0

√T 2

_1/2

u1−u2

L²0,T;DA∩W^1,20,T;H

≤2^−7/4C0u01−u022C0

√T 2

1/2

u1−u2

L²0,T;DA∩W^1,20,T;H.

3.78

Combining3.75with3.78, we obtain u1−u2

L²0,T;DA∩W^1,20,T;H

≤C1u01−u02f1−f2

L²0,T;H

2^−7/4C0C1μu01−u02 2C0C1

√T 2

_1/2

Lru1−u2

L²0,T;DA∩W^1,20,T;H.

3.79

Suppose thatu0n, fn → u0, finV ×L²0, T;Hand letunandube the solutionsHIE withu0n, fnandu0, f, respectively. Let 0< T1 ≤Tbe such that

2C0C1

T1/√ 21/2

Lr<1. 3.80

Then by virtue of3.79withT replaced byT1we see that

un−→u inL²

0, T1;D A0

∩W^1,20, T1;H. 3.81

This implies thatunT1→uT1inV. Hence the same argument shows thatun → uin

L²

T1,min 2T1, T

; D A0

∩W^1,2

T1,min 2T1, T

;H

. 3.82

Repeating this process we conclude thatun → uinL²0, T;DA0∩W^1,20, T;H.

2 If u0, f ∈ H × L²0, T;H then u belongs to L²0, T;V ∩ C0, T;H from Theorem 3.5. Letu0i, fi∈H×L²0, T;Handui ∈Br be the solution ofHIEwithu0i, fi in place ofu0, ffori1,2. MultiplyingHIEbyu1t−u2t, we have

1 2

d

dtu1t−u2t²

ω1c0u1t−u2t²

≤ω2u1t−u2t²bⁿ u1t

−bⁿ

u2tu1t−u2t f1t−f2tu1t−u2t.

3.83

Put

Gt Lru1t−u2tf1t−f2t. 3.84

Then, by the similar argument in3.32, we get 1

2u1t−u2t² ω1c0

^t

0

u1s−u2s²ds

≤ 1

2u01−u02²ω2

_t

0

u1s−u2s²ds _t

0

Gsu1s−u2sds

3.85

and we have that d

dt

e^−2ω2t _t

0

u1s−u2s²ds

≤2e^−2ω2t 1

2u01−u02² _t

0

Gsu1s−u2sds

, 3.86

thus, arguing as in3.35we have

ω2

_t

0

u1s−u2s²ds

≤ 1 2

e^2ω2t−1u01−u02² _t

0

e^2ω2t−s−1

Gsu1s−u2sds.

3.87

Combining this inequality with3.85it holds that 1

2u1t−x2t² ω1c0

^t

0

u1s−u2s²ds

≤ 1

2e^2ω2tu01−u02² _t

0

e^2ω2t−sGsu1s−u2sds.

3.88

ByLemma 3.1the following inequality 1

2

e^−2ω2tu1t−u2t² ω1c0

e^−2ω2t _t

0

u1s−u2s²ds

≤ 1

2u01−u02² _t

0

e^−ω2sGse^−ω2su1s−u2sds

3.89

implies that

e^−ω2tu1t−u2t≤u01−u02 _t

0

e^−ω2sGsds. 3.90

Hence, from3.88and3.90it follows that 1

2u1t−u2t² ω1c0

^t

0

u1s−u2s²ds≤ 1

2e^2ω2tu01−u02²

_t

0

e^2ω2t−sGse^ω2su01−u02 _s

0

e^ω2s−τGτds

≤ 1

2e^2ω2tu01−u02²u01−u02e^2ω2t _t

0

Gsds Lr² 4ω2

e^2ω2t−1^t

0

Gs²ds.

3.91

The last term of3.91is estimated as Lr²

e^2ω2t−1 4ω2

_t

0

2

L²x1s−x2s²k1s−k2s²

ds. 3.92

LetT2< T be such that

ω1c0−Lr² 2ω2

e^2ω2T2−1

>0. 3.93

Hence, from3.91and3.92it follows that there exists a constantC >0 such that u1

T2

−u2

T2² _T2

0

u1s−u2s²ds≤Cu01−u02² _T2

0

f1s−f2s²ds

. 3.94

Suppose u0n, un → u0, fin H×L²0, T2;V^∗, and let un and ube the solutions HIEwithu0n, fnandu0, f, respectively. Then, by virtue of3.94, we see thatun → u