Mostafa Kadiri

Mohammed Louaked

Houari Mechkour

Hydrodynamic design optimization using non stationary porous media model

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

HYDRODYNAMIC DESIGN OPTIMIZATION USING NON STATIONARY POROUS MEDIA MODEL

wostf uehss iole gentrle d9iletronique @igiErisAD sxiig F eserh genter smmeule olluxD QU ui de qrenelleEg UISPHD USUPS ris gedex ISD prne nd niversité de gen xormndieD vortoire de wthémtiques xiols yresmeD gmpus ssD fdF wréhl tuinD fF SIVTD IRHQP genD prne mostfFkdiridunienFfr wohmmed vyeuih niversité de gen xormndieD vortoire de wthémtiques xiols yresmeD gmpus ssD fdF wréhl tuinD fF SIVTD IRHQP genD prne mohmmedFloukeddunienFfr rouri wigruy iole gentrle d9iletronique @igiErisAD sxiig F eserh genter smmeule olluxD QU ui de qrenelleEg UISPHD USUPS ris gedex ISD prne mehkourdeeFfr Abstract. In this paper, we focus on the penalty nite element method for the non stationary porous media model. We begin by showing the existence and uniqueness of the solution for the initial problem.

Error estimates for the velocity and the pressure are obtained via the energy method.

We introduce a time discretization by the use of a backward Euler scheme combined with fully discrete nite element method to approximate the penalized problem and establish an error estimate for the velocity and the pressure which will be used to show the convergence of the approximate solution to the solution of the initial problem.

The shape optimization problem is to nd the shape which is optimal in that it minimizes a cost functional related to a comfort sh populations. We derive the adjoint system associated to the penalized problem. We compute the gradient in terms of state and adjoint variables. The optimization procedure is implemented using the continuous adjoint method and the nite element method. Numerical simulations are presented to show the eciency and the robustness of the proposed method.

IF Introduction sn this workD we re interested in studying the nonEsttionry xvierEtokes porhheimer @xpA equE tions

             u t -ν∆u + (u.∇)u + a | u | α u + ∇p = f in Ω × [0, T ], ∇.u = 0 in Ω × [0, T ], u = 0 on ∂Ω × [0, T ], u| t=0 = u 0 on Ω,
where Ω is n open ounded set in R d @d = 2 or d = 3A with su0iently smooth oundry ∂ΩF he unknown funtions here re u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) nd p = p(x, t)D whih stnd for the veE loity (led nd the pressure of the )owD respetivelyF sn dmping termD 1 ≤ α ≤ 2 nd a > 0 re two onstntsF he given funtion u 0 = u 0 (x) is the initil veloity nd the onstnt ν > 0 represents the visosity oe0ient of the )owF sn the litertureD there re severl ppers devoted to the mthemtil study of xp equtions without the nonliner onvetion termsF eferenes PD IRD IU reported results onerning the struturl stility for the oe0ients in )owsF e refer to the work of PI for the symptoti ehvior of solutionsF sn IPD the nlysis of the pressure stiliztion method for the unstedy inompressile frinkmnEporhheimer equtions is proposedF irror estimtes for the veloity nd the pressure re estlishedF he existene of regulr dissiptive solutions with the nonlinerity of n ritrry polynomil growth rte is investigted in IHF emong the pplitions of these wter )ow model re the prolems ssoited to the optimiztion of (shwys strutureF sn ftD some types of (sh omplish their dily or nnul migrtions nd for long or short distnes to omplete their life ylesF hese migrtory (sh speies re thretened y urniztionD griultureD or dmsF hms re uilt through enturies y men for griultureD nvigtionD eletriity or eonomi resonsF hey represent impssle rriers for migrtory speies for their reprodution or feeding res during the rise of the riversF o meet this hllengeD mn thought to uilt (shwys for permitting (sh to pss with omfortle onditionsF pishwys @(sh ldderD or (sh pssgeA re hydruli strutures pled ner from dms or weirs to llow migrtory (shes to ross to upstrem riverF everl types of (sh pssge re known nowdysF he est generl referene gly S for the pool nd weir typeD the stndrd work of utopodis et lF II for the henil (shwys nd the lssil work for the vertil slot type jrtnm et lF IVF e re involved in the third type of (shwysX vertil slot (shwy @pAF p hs n re of low veloity for (sh to rest during upstrem pssgeD good energy dissiptionF his mkes them suitle to sites where (sh pssge is needed t di'erent times of the yer nd for mximum numer of smll speiesF sn tul engineeringD in order to redue the )ow veloity in the (shwy poolD snd grvels re usully lid t the ottom whih n inrese the roughness of the (shwy nd ring the )ow informtion in the (shwy muh loser to nturl river nd indue fmilir )ow ptterns for (shF he )ow ehvior n only modeled urtely if the struture of porous medi is properly desriedF rnsition from the hry )ow regime to the porhheimer regime n e oserved y the investigtions of )ow in pore speF wny reserhers hve ttempted to lrify the physil mening of the linerity in the porhheimer eqution y reovering the porhheimer eqution from the xvierEtokes eqution y using the si priniples underlying the theory of hydrodynmisF hespite the diverse opinions on the origin of the nonlinerityD it is now generlly greed tht the qudrti term involves the seprtion of oundry lyers nd wke formtion ehind solid ostlesF hetiled disussions n e found in QD WD IQF yur purpose is to investigte these strutures using the non sttionry porous medi model nd to ssess the possiility to ompute the )ow pttern in vertil slot (sh ldder nd dedue n optiml struture llowing (sh to ross the ostle in onvenient onditionsF he resolution of the prolem @IFIA ould e di0ult numerillyD used y the inompressiility onE strint ∇.u = 0. e populr strtegy to overome this di0ulty is to relx the inompressiility term with the penlty pprohF yur purpose is to use penlty methodD introdued y gournt T nd used y emm PH for the pproximtion of the xvier tokes equtionsD to tret the xvier tokes porhheimer systemF he penlty method is to estlish n pproximtion of the ouple (u, p) solution of @IFIA y (u ε , p ε ) solution of the following penlized prolem

               u ε t -ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f in Ω × [0, T ], ∇.u ε + εp ε = 0 in Ω × [0, T ],
u ε = 0 on ∂Ω × [0, T ],

u ε | t=0 = u 0 in Ω.
@IFPA e prove the existene nd uniqueness of the initil prolem solutionF e nlyze the error ehvior for the linerized xp modelF hen we tret the penlty method for the fully nonliner xp systemF he penlized model is onsidered to study n optimiztion prolem relted to (sh pss strutureD whih onsist to (nd n optiml shpe of (shwy y minimizing ost funtion onneted to (sh omfortF e derive n djoint eqution nd ompute the shpe grdient of the ojetive funtionF e propose nd nlyze fully disrete (nite element method for the penlized prolemF he numeril pproh omines (nite element method to solve the nonEsttionry porous medi equtions nd grdient type lgorithm lled spetrl projeted grdient to solve the optimiztion prolemF he reminder of this pper is orgnized s followsF sn the next setionD we introdue nottions nd rell some preliminry resultsF e use lso the lssil pedo!qlerkin method nd the pourier trnsform tehnique to prove the existene of wek solutionF he etion Q dels with the shpe designF e de(ne shpe optimiztion prolemF e then derive the djoint system nd ompute the shpe grdient of the ost funtionF he etion R trets time disretiztion y the use of kwrd iuler sheme omined with (nite element method to pproximte the penlized prolem nd otin n error estimte for the veloity nd the pressureF pinllyD the lst setion is devoted to the grdient method to solve the shpe optimiztion prolem nd the presenttion of some numeril results to prove the e0ieny of the suggested tehniqueF PF Mathematical model PFIF Preliminaries. he xvier tokes porhheimer equtions governing the motion of n inompressile visous )uid inside Ω redX

u t -ν∆u + (u.∇)u + a|u| α u + ∇p = f in Ω × [0, T ] @PFIA ∇.u = 0 in Ω × [0, T ], u| ∂Ω = 0, u| t=0 = u 0
@PFPA e im to desrie some of the nottion nd results whih will e frequently used in this pperF por the mthemtil setting of the xvier tokes porhheimer systemD we introdue the rilert spes 1

H = v ∈ L 2 (Ω), : ∇.v = 0, v.n| ∂Ω = 0 V = v ∈ H 1 0 (Ω) : ∇.v = 0
1 The vector functions and vector spaces will be indicated by boldface type.

X = (H 1 0 (Ω)) d (d = 2, 3), Y = L 2 (Ω) , Ŷ = q ∈ H 1 (Ω) , M = q ∈ L 2 (Ω); Ω qdx = 0 V 0 (Ω) = u ∈ (H 2 (Ω)) d |u = 0 on Γ 1 D where ∂Ω = Γ 1 ∪ Γ 2 X = v ∈ (H 1 @PFTA vet introdue the pertured A ε u = -ν∆u - 1 ε ∇(∇.u)D
whih is the opertor relted to the penlty methodF st is positive selfEdjoint opertor from H 2 (Ω) ∩ H 1 0 (Ω) onto L 2 (Ω) nd the powers A β of e @β ∈ RA re well de(ned nd we hve

(A 1/2 u, A 1/2 v) = (∇u, ∇v)D (A 1/2 ε u, A 1/2 ε v) = (A 1/2 u, A 1/2 v) + 1 ε (divu, divv) for ll uDv ∈ XF por ε su0iently smllD ||∆u|| ≤ C||A ε u|| ∀u ∈ H 2 (Ω) ∩ X, ||∇u|| ≤ C||A 1 2 ε u|| ∀u ∈ X, ||A -1 ε u|| ≤ C||u|| -2 ∀u ∈ H -2 (Ω).

@PFUA

C is the generi onstnt tht n tke di'erent vlues in di'erent plesF sf the oundry ∂Ω is smooth enoughD for ny

v ∈ V ∩ H 2 (Ω) v 2 ≤ ∆v . @PFVA
st is well known tht the following qglirdoExirenerg inequlities holdX Lemma 2.1.

||v|| L 4 ≤ c||v|| 1/2 ||A 1/2 v|| 1/2 , ||v|| ≤ c||A 1/2 v||, ∀v ∈ X @PFWA ||∇v|| L 4 ≤ c||A 1/2 v|| 1/2 ||Av|| 1/2 , ||A 1/2 v|| ≤ c||Av||, ∀v ∈ D(A) @PFIHA ||v|| L ∞ ≤ c||v|| 1/2 ||Av|| 1/2 , ∀v ∈ D(A).
@PFIIA purthermoreD we hve the following properties for the tokes opertorF Lemma 2.2. here exists onstnt c 0 > 0 depending on

Ω nd suh tht if cε ≤ 1D ||Av|| ≤ c 0 ||A ε v||, ||A 1/2 v|| ≤ c 0 ||A 1/2 ε v|| @PFIPA
elsoD we remind two qronwll9s lemms type whih will e often usedF Lemma 2.3. @qronwll lemmA vet y(t), h(t), g(t) nd f (t) e non negtive funtions verifying

y(t) + t 0 h(s)ds ≤ y(0) + t 0 (g(s)y(s) + f (s))ds ∀0 ≤ t ≤ T, with T 0 g(t)dt ≤ M, then y(t) + t 0 h(s)ds ≤ exp(M )(y(0) + t 0 f (s)ds) ∀0 ≤ t ≤ T.
Lemma 2.4. @disrete qronwll lemmAF vet y n D h n D g n D f n e nonnegtive series verifying

y m + k m n=0 h n ≤ k m n=0 (g n y n + f n ), with k T /k n=0 g n ≤ M ∀0 ≤ m ≤ T /k. uppose kg n < 1 ∀n nd let σ = max 0≤n≤T /R (1 -kg n ) -1 D then y m + k m n=1 h n ≤ exp(σM )(B + k m n=0 f n ), ∀m ≤ T /k st is worth pointing out tht the mpping F X x → |x| α x is monotoneD thenX ∀u, v ∈ X, (|u| α u -|v | α v,u -v) ≥ 0.
@PFIQA he following oolev inequlity is useful to del with the nonliner form pF

p ≤ ∇v , 1 ≤ p ≤ 6 @PFIRA
st is onvenient to onsider the funtionl G de(ned y G(u) = P F (u) ∀u ∈ XF sn the next lemmD we provide the estimte for the veloity9s seond derivtiveD the veloity9s derivtive with respet to time nd the pressureF Lemma 2.5. e ssume tht the given dt u 0 nd f stisfy the following regulrity

u 0 ∈ V ∩ H 2 (Ω), f, f t ∈ L 2 (Ω) @IA
hen there exists T 1 ≤ T suh tht the solution of the system @PFIAE@PFPA stis(esX

u(t) 2 + u t (t) + p(t) 1 C, ∀t ∈ [0, T 1 ].
@PFISA roofF ke the inner produt of the eqution @PFIA with ∆u nd integrte ll to get

1 2 d dt ∇u 2 + γ ∆u 2 = -b(u, u, ∆u) -a(|u| α u, ∆u) + (f , ∆u).
@PFITA gomining the inequlities @PFUAE@PFIIA with oung inequlityD we hve b(u, u, ∆u) ≤ γ α+1) . @PFIVA ustituting @PFIUA nd @PFIVA into @PFITA nd using the ft tht

6 ∆u 2 + C ∇u 6 , @PFIUA a(|u| α u, ∆u) ≤ γ 6 ∆u 2 + C ∇u 2(
(f, ∆u) ≤ γ 6 ∆u 2 + 3 2γ f 2 ,
we otin 1 2

d dt ∇u 2 + γ 2 ∆u 2 ≤ C ∇u 6 + 3 2γ f 2 .
@PFIWA sn prtiulrD it is equivlent to solve the following eqution @y ssuming C ≥ 1AX

d dt y(t) ≤ C y(t) 3 , with y(t) = ∇u(t) 2 + C 1 , y(0) = ∇u(0) 2 + C 1 nd C 1 = 3 2γ sup 0≤t≤T f 2 . he eqution ove hs solution in [0, T 1 ] where T 1 ≤ 1 2C y(0) 2 = 1 C 0 nd y(t) ≤ √ 2y(0), t ∈ [0, T 1 ]. oD for ll t ∈ [0, T 1 ] with T 1 = min{T 1 , 1 C0 }D we hve ∇u(t) 2 ≤ √ 2( ∇u(0) 2 + C 1 ) := C 3 , @PFPHA
gonsequentlyD using @PFIWA nd @PFPHAD to infer tht

sup 0≤t≤T1 ∇u(t) 2 + γ T1 0 ∆u(t) 2 dt ≤ C, ∀t ∈ [0, T 1 ]. @PFPIA
heriving one time the eqution @PFIA with respet to tD we otin u tt -γ∆u t + aF (u)u t + B(u t , u) + B(u, u t ) + ∇p t = f t @PFPPA wultiplying the reltion @PFPPA with u t D we get

1 2 d dt u t 2 + γ ∇u t 2 = -a(F (u)u t , u t ) -b(u t , u,u t ) + (f t , u t ). ine (F (u)u t ) • u t is
positive de(niteD using the inequlities @PFUAD rölder9s nd oolev9s inequlities yields to

1 2 d dt u t 2 + γ ∇u t 2 c u t u 1 2 1 u 1 2 2 ∇u © t + 1 2 f t 2 , γ 2 ∇u t 2 + C( u 1 u 2 ) u t 2 + 1 2 f t 2 .
gomining guhyEhwrz inequlityD estimtes @PFVA nd @PFPIAD to hveX

T1 0 u(s) 1 u(s) 2 ds ≤ T1 0 u(s) 2 1 ds 1 2 T1 0 u(s) 2 2 ds 1 2 ≤ C. @PFPQA
epplying qronwll vemm to @PFPQAD we otinX

u t 2 + γ T1 0 ∇u t (s) 2 ds ≤ C, ∀t ∈ [0, T 1 ]. @PFPRA
wultiplying the equtions @PFIA y ∆uD we get

γ ∆u 2 = -(u t , ∆u) -a(|u| α u, ∆u) -b(u, u, ∆u) -(f , ∆u) ≤ γ 8 ∆u 2 + 2γ u t 2 + a|(|u| α u, ∆u)| + |b(u, u, ∆u)| + γ 8 ∆u 2 + 2γ f 2 .
imilrly s in @PFIVA nd @PFIUA we ontrol the nonliner termsF sing @PFPHAD @PFPRA nd @PFVAD we infer tht

u 2 ≤ C, ∀t ∈ [0, T 1 ]. @PFPSA
por the pressure estimteD we use the eqution @PFIA to hve ∇p ≤ u t + γ ∇u + a |u| α+1 + B(u, u) . @PFPTA sing the rölder9s nd the inequlities @PFUAD the estimtes @PFPHA nd @PFPSAD sine H 1 (Ω) → L 4 (Ω)D we oserve thtX

B(u, u) 2 C u α+2 ∇u 2(α+2) α C u α+2 ∆u ( 10-α α+8 ) 2 u 2( α-1 α+8 ) α+2 C ∆u 2( 10-α α+8 ) 2 u 3( α+2 α+8 ) α+2 ≤ C.
ustituting the inequlities @PFPHAD @PFPRA nd @PFPSA in @PFPTAD we otin the pressure estimteF Lemma 2.6. nder the sme ssumptions of the lemm PFSD we suppose in ddition tht

f t , f tt ∈ C([0, T ], v © 2 (Ω)) @QA
henD the solution of the system @PFIA stis(es

u t (t) 2 2 + ∇p t + t t0 ( u tt (s) 2 2 + p tt (s) 2 1 ) d s C, ∀t ∈ [t 0 , T 1 ]. @PFPUA
roofF heriving one time the eqution @PFIA with respet to tD to hve

u tt -γ ∆u t + aG (u)u t + B(u t , u) + B(u, u t ) = f t @PFPVA
wultiplying the eqution @PFPVA y ∆u t D we otinX

1 2 d dt ∇u t 2 + γ ∆u t 2 = -(f t , ∆u t ) + b(u t , u, ∆u t ) + b(u, u t , ∆u t ) +a(F (u)u t , ∆u t ) ≤ γ 8 ∆u t 2 + 2 f t 2 + |b(u t , u, ∆u t )| + |b(u, u t , ∆u t )| +a|(F (u)u t , ∆u t )|
ith similr mnner s in lemm PFS we write

|b(u t , u, ∆u t )| γ 8 ∆u t 2 + C u 2 2 ∇u t 2 @PFPWA |b(u, u t , ∆u t )| γ 8 ∆u t 2 + C u 2 2 ∇u t 2 . @PFQHA xextD a|(F (u)u t , ∆u t )| ≤ C(α + 1) u α ∞ u t ∆u t ≤ γ 8 ∆u t 2 + C u 2α ∞ ∇u t 2 @PFQIA
gomining the previous estimtes together nd using the inequlity @PFIRAD @PFPHA nd @PFPSAD to otin

d dt ∇u t 2 + γ ∆u t 2 ≤ 4 f t 2 + C ∇u t 2 .
@PFQPA hnks to qronwll9s inequlityD we infer tht

sup t0≤t≤T1 ∇u t (t) 2 + γ t t0 ∆u t (s) 2 ds C, ∀t ∈ [t 0 , T 1 ]. @PFQQA
xowD di'erentiting the eqution @PFPPA with respet to tD we hve u ttt -γ∆u tt + aF (u)|u t | 2 + aF (u)u tt + 2B(u t , u t ) + B(u tt , u) + B(u, u tt ) + ∇p tt = f tt . @PFQRA king the inner produt of the reltion @PFQRA with u tt D to get

1 2 d dt u tt 2 + γ ∇u tt 2 = -a(F (u)|u t | 2 , u tt ) -a(F (u)u tt , u tt ) -2b(u t , u t , u tt ) -b(u tt , u,u tt ) + (f tt , u tt ).
ine (F (u)u tt ) • u tt is positive de(niteD using oung9s inequlityD we hve

1 2 d dt u tt 2 + γ ∇u tt 2 ≤ 1 2 f tt 2 + |β(F (u)|u t | 2 , u tt )| + 2|b(u t , u t , u tt )| +|b(u tt , u,u tt )|. hnks to @PFIRAD @PFPHA nd @PFQQAD sine H 1 (Ω) → L 4 (Ω)D |(F (u)|u t | 2 , u tt )| α(α + 1)(|u| α-1 |u t | 2 , |u tt |) α(α + 1) u α-1 ∞ u t 2 4 u tt C u tt 2 + C u 2(α-1) ∞ ∇u t 4 ≤ C u tt 2 + C.
@PFQSA sing oungD rölder inequlitiesD the inequlity @PFUAE @PFIIA nd estimte @PFISAD we hve

|b(u tt , u,u tt )| ≤ u tt 1 u 2 u tt γ 2 ∇u tt 2 + C u tt 2 . @PFQTA |b(u t , u t , u tt )| u t 2 u t 1 u tt ≤ C ∇u © t 2 + u t 2 2 u tt 2 .
@PFQUA epling @PFQSAE@PFQUA in @PFQSAD using @PFQQA nd pplying qronwll vemm yield

u tt 2 + γ T1 t0 ∇u tt (s) 2 ds ≤ C. @PFQVA
wultiplying the equtions @PFIA with ∆u t D we otinX

γ ∆u t 2 = (u tt , ∆u t ) -(f t , ∆u t ) -a(F (u)u t , ∆u t ) -b(u t , u, ∆u t ) -b(u, u t , ∆u t ) ≤ 5γ 2 u tt 2 + γ 5 ∆u t 2 + 5γ 2 f t 2 + a|(F (u)u t , ∆u t )| +|b(u t , u, ∆u t )| + |b(u, u t , ∆u t )|
por ounding the nonliner terms we pply similr estimtions s in @PFPWAE@PFQIAD thnks to @PFVAD @PFQVA nd @PFQQAD we dedut

u t 2 ≤ C, ∀t ∈ [t 0 , T 1 ].
@PFQWA e n otin the remining estimtes in @PFPUA y following sme rguments of vemm PFSF xotie tht @PFQQA is n nlogue of t t0 ∆u tt (s) 2 ds for u tt insted of u t F woreoverD if we ssume tht

tf t ∈ L 2 (0, T 1 ; H 1 (Ω)),
@PA e otin the pressure estimte

tp t ∈ L 2 (0, T 1 ; H 1 (Ω))
@PFRHA Lemma 2.7. essume tht 1 ≤ α ≤ 2 nd u 0 ∈ HD then the solution u of the prolem @PFIAE@PFPA stis(es

sup t∈[0,T ] u ≤ C @PFRIA
roofF e multiply the eqution @PFIA y u t + u in L 2 (Ω)

2 u t (t) 2 + d dt ν ∇u(t) 2 + 2a α + 2 Ω |u(x, t)| α+2 dx + 2ν ∇u(t) 2 + 2a Ω |u(x, t)| α+2 dx +b(u, u, u t ) = (f , u t ) + (f , u) ≤ c f 2 + 1 2 u t 2 + ν 2 u 2 @PFRPA he eqution @PFRPA implies u t (t) 2 + d dt ν ∇u(t) 2 + 2a α + 2 Ω |u(x, t)| α+2 dx + ν ∇u(t) 2 + 2a Ω |u(x, t)| α+2 dx ≤ c f 2
@PFRQA e de(ne the funtion

ψ(t) = ν ∇u(t) 2 + 2a α + 2 Ω |u(x,
F (u) -F (v) ≤ C u -v @PFRSA
yur next point is the existene nd uniqueness of wek solutions to system@PFIAE@PFPAF Theorem 2.1. nder the ssumption @IA nd u 0 ∈ HD the system@PFIAE@PFPA hs t lest one solution

u ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ L α+2 (0, T ; L α+2 (Ω)) ∩ L ∞ (0, T ; L 2 (Ω))F
roofF e onsider n orthonorml sis (w k ) in nd de(ne the sequene (u m ) ∈ X stisfying

( ∂u m (t) ∂t , w k )+ν(∇u m (t), ∇w k )+b(u m (t), u m (t), w k )+a(|u m (t)| α u m (t), w k ) = (f (t), w k ), f or k = 1, ..., m @PFRTA with initil onditions u m (0) = u 0m @PFRUA
where for eh m ∈ N

u m (t) = m k=0
ξ km (t)w k @PFRVA nd u 0m is the projetion of u 0 from the spe X to the spe span{w 1 , ..., w m }F he guhy prolem @PFRTAE@PFRUA is (rst order nonliner ordinry di'erentil eqution in t for the funtions ξ 1m (t), ..., ξ mm (t) with the initil onditions ξ km (0) = the kth omponent of u 0m , k = 1, ..., m epplying the properties of n ordinry di'erentil equtionsD there exist solution of the guhy prolem

u m in the time intervl [0, t n )D 0 < t n ≤ T nd we hve if t n < T D then lim t→t - n u m = +∞F ytherwise t n =
T F e multiply the eqution @PFRTA y ξ km nd sum with respet to k from I to mD to otin

( ∂u m ∂t , u m ) + ν ∇u m 2 + a u m α+2 L α+2 = (f , u m ) @PFRWA
sing oinré inequlity nd rölder inequlity yields to

d dt u m 2 + 2ν ∇u m 2 + 2a u m α+2 L α+2 ≤ 2(f , u m ) ≤ 2 f u m ≤ 1 νλ f 2 + νλ u m 2 ≤ 1 νλ f 2 + ν ∇u m 2 @PFSHA herefore d dt u m 2 + ν ∇u m 2 + 2a u m α+2 L α+2 ≤ 1 νλ f 2
@PFSIA e integrte the inequlity @PFSIA with respet to t from H to s (≤ t n ) to hve

u m 2 ≤ u 0m 2 + 1 νλ s 0 f (t) 2 dt ≤ u 0 2 + 1 νλ T 0 f (t) 2 dt < +∞ @PFSPA hen t n = T we otin sup t∈[0,T ] u m 2 ≤ u 0 2 + 1 νλ T 0 f (t) 2 dt
@PFSQA e integrte @PFSIA with respet to t form H to D to get

T 0 ∇u m 2 dt ≤ 1 ν u 0 2 + 1 ν 2 λ T 0 f (t) 2 dt @PFSRA T 0 u m α+2 L α+2 dt ≤ 1 2a u 0 2 + 1 2aνλ T 0 f (t) 2 dt @PFSSA
sn order to estimte the derivtive term with respet to t of u m (t) for pssing to the limit in the nonliner term in @PFRTAD we de(ne

v m (t) = f (t) -ν∆u m (t) -(u m (t).∇)u m (t) -a|u m (t)| α u m (t)
e rewrite the eqution @PFRTA s

( ∂u m (t) ∂t , w k ) = (v m (t), w k ), f or k = 1, 2, ..., m @PFSTA e de(ne
ũm (t) = u m (t), t ∈ [0, T ] 0, t ∈ R\[0, T ]
ũm (t) is disontinuous t t = 0 nd t = T F por ll the set RD we hve

d ∂t (ũ m (t), w k ) = (ṽ m (t), w k ) + (u 0m , w k )δ (0) -(u m (T ), w k )δ (T ) , k = 1, 2, ..., m @PFSUA
with δ (0) nd δ (T ) re the hir distriutions t H nd D nd

ṽm (t) = v m (t), t ∈ [0, T ] 0, t ∈ R\[0, T ]
epplying the pourier trnsform on the oth sides in eqution @PFSTA yields to

2iπτ (û m (τ ), w k ) = (v m (τ ), w k ) + (u 0m , w k ) -(u m (T ), w k )exp(-2iπτ T ), k = 1, ..., m @PFSVA
ith similr mnnerD we de(ne ξkm nd we denote y ξkm the pourier trnsform of ξkm F e multiply the eqution @PFSTA y ξkm nd sum with respet to kD to otin

2iπτ ûm (τ ) 2 = (v m (τ ), ûm (τ )) + (u 0m , ûm (τ )) -(u m (T ), ûm (τ ))exp(-2iπτ T ) @PFSWA
sing the inequlities @PFSPAE@PFSSAD yields to

2πτ ûm (τ ) 2 ≤ |(v m (τ ), ûm (τ ))| + u 0m ûm (τ ) + u m (T ) ûm (τ ) ≤ |(v m (τ ), ûm (τ ))| + 2 √ K ûm (τ ) @PFTHA
where

K = u 0 2 + 1 νλ T 0 f (t) 2 dtF he term |(v m (τ ), ûm (τ ))| n e rised s |(v m (τ ), ûm (τ ))| ≤ |( f (τ ), ûm (τ ))| + ν ∇û m (τ ) 2 + b(û m (τ ), ûm (τ ), ûm (τ )) + a||û m (τ )|| α+2 L α+2 ≤ f (τ ) 2 + ûm (τ ) 2 + ν ûm (τ ) 2 + a ûm (τ ) α+2 L α+2
@PFTIA gomining the inequlities @PFTHAE@PFTIA yields to

2πτ ûm (τ ) 2 ≤ f (τ ) 2 + ûm (τ ) 2 + ν ûm (τ ) 2 + a ûm (τ ) α+2 L α+2 + 2 √ K ûm (τ ) @PFTPA gonsider γ ∈ [0, 1/4]D we hve the inequlityX |τ | 2γ ≤ (2γ + 1) 1 + |τ | 1 + |τ | 1-2γ D ∀τ ∈ RF herefore +∞ -∞ |τ | 2γ ûm (τ ) 2 dτ ≤ (2γ + 1) +∞ -∞ ûm (τ ) 2 1 + |τ | 1-2γ dτ + (2γ + 1) +∞ -∞ |τ | ûm (τ ) 2 1 + |τ | 1-2γ dτ ≤ (2γ + 1) +∞ -∞ ûm (τ ) 2 dτ + (2γ + 1) +∞ -∞ |τ | ûm (τ ) 2 1 + |τ | 1-2γ dτ @PFTQA
por rising the (rst term on the right hnd side in @PFTQAD we use the erevl equlityD the oinré inequlity nd the inequlity @PFSRA to hve

(2γ + 1) +∞ -∞ ûm (τ ) 2 dτ = (2γ + 1) +∞ -∞ ûm (t) 2 dt = (2γ + 1) +∞ -∞ u m (t) 2 dt ≤ c(γ, d) @PFTRA
st remins the seond term on the right hnd side of the inequlity @PFTQAF e use the inequlities @PFSQAE @PFSSA nd @PFTPA to get

(2γ + 1) +∞ -∞ |τ | ûm (τ ) 2 1 + |τ | 1-2γ dτ ≤ (2γ + 1) 2π +∞ -∞ f 2 dτ + (2γ + 1)a π +∞ -∞ ûm (τ ) α L α dτ + (2γ + 1)ν 2π +∞ -∞ ∇û m (τ ) 2 dτ + (2γ + 1) √ K 2π +∞ -∞ ∇û m (τ ) 1 + |τ | 1-2γ dτ ≤ c +∞ -∞ f 2 + ûm (t) α L α + ∇û m (t) 2 dt +c +∞ -∞ dτ 1 + |τ | 1-2γ 1/2 +∞ -∞ ûm (t) 2 dt 1/2 ≤ c T 0 f 2 + ûm (t) α L α + ∇û m (t) 2 dt +c +∞ -∞ dτ 1 + |τ | 1-2γ 1/2 T 0 ûm (t) 2 dt 1/2 ≤ c(π, γ, a, ν, d, T )
@PFTSA sing the inequlities @PFTQAE@PFTSAD there exists susequene denoted without lose of generlity y u m nd u suh tht

u m * u in L ∞ (0, T ; L 2 (Ω)) @PFTTA u m u in L 2 (0, T ; H 1 0 (Ω)) @PFTUA u m -→ u in L 2 (0, T ; L 2 (Ω)) @PFTVA u m u in L α+2 (0, T ; L α+2 (Ω)) @PFTWA gonsider φ(t) ∈ C ∞ 0 (0, T )F
wultiplying the eqution @PFRTA y φ(t)D integrting from H to nd integrting the (rst term of the result y prts to otin

- T 0 (u m (t), w k )φ (t)dt + ν T 0 (∇u m (t), φ(t)∇w k )dt + T 0 b(u m (t), u m (t), w k φ(t))dt a T 0 (|u m (t)| α u m (t), w k φ(t))dt = T 0 (f , w k φ(t))dt, k = 1, 2, ..., m
@PFUHA xowD we study the onvergene of the nonliner terms in @PFUHAF e egin y the triliner form

T 0 b(u m , u m , w k φ(t))dt - T 0 b(u, u, w k φ(t))dt ≤ 1 2 T 0 b(u m , u m , w k φ(t))dt - T 0 b(u, u, w k φ(t))dt + 1 2 T 0 b(u m , w k φ, u m ))dt - T 0 b(u, w k φ(t), u)dt ≤ 1 2 T 0 b(u m -u, u m , w k φ(t))dt + 1 2 ' T 0 b(u, u m -u, w k φ(t))dt + 1 2 T 0 b(u m -u, w k φ(t), u m )dt + 1 2 T 0 b(u, w k φ(t), u m -u)dt b 1 + b 2 + b 3 + b 4 @PFUIA where b 1 = 1 2 T 0 b(u m -u, u m , w k φ(t))dt ≤ 1 2 sup t∈[0,T ] |φ(t)| sup x∈Ω |w k | T 0 u m -u ∇u dt ≤ c u m -u L 2 (0,T,L 2 (Ω)) u m L 2 (0,T,H 1 0 (Ω)) -→ 0 s m → ∞ @PFUPA b 2 = 1 2 ' T 0 b(u, u m -u, w k φ(t))dt = 1 2 T 0 3 i,j=1 Ω (u) i ∂(u m -u ∂x j (φ(t)w k ) j dxdt ≤ 1 2 T 0 3 i,j=1 Ω (u m -u) j ∂(u) i ∂x j (φ(t)w k ) j dxdt + 1 2 T 0 3 i,j=1 Ω (u m -u) j (u) i φ(t) ∂(w k )j ∂w j dxdt ≤ 1 2 sup t∈[0,T ] |φ(t)| sup x∈Ω |w k | T 0 u m -u ∇u dt + 1 2 sup t∈[0,T ] |φ(t)| sup x∈Ω |∇w k | T 0 u m -u u dt ≤ c u m -u L 2 (0,T ;L 2 (Ω)) ( u L 2 (0,T ;H 1 0 (Ω)) + u L 2 (0,T ;L 2 (Ω)) ) -→ 0 s m → ∞ @PFUQA b 3 = 1 2 T 0 b(u m -u, w k φ(t), u m )dt ≤ sup t∈[0,T ] |φ(t)| sup x∈Ω |∇w k | T 0 u m -u ∇u m dt ≤ c u m -u L 2 (0,T ;L 2 (Ω) u m L 2 (0,T ;H 1 0 (Ω)) -→ 0 s m → ∞ @PFURA b 4 = 1 2 T 0 b(u, w k φ(t), u m -u)dt ≤ sup t∈[0,T ] |φ(t)| sup x∈Ω |∇w k | T 0 u m -u u dt ≤ c u m -u L 2 (0,T ;L 2 (Ω)) u L 2 (0,T ;L 2 (Ω)) -→ 0 s m → ∞ @PFUSA sing @PFUPAE@PFUSA to dedue tht T 0 b(u m , u m , w k φ(t))dt -→ T 0 b(u, u, w k φ(t))dt @PFUTA
st remins the porhheimer termF e write T 0

(|u m | α u m , w k φ(t))dt - T 0 (|u| α u, w k φ(t))dt ≤ T 0 ||u m | α u m -|u| α u| |w k ||φ(t)|dt ≤ c T 0 ∇(u m -u) dt ≤ c u m -u L 2 (0,T,H 1 0 (Ω)) -→ 0 s m → ∞ @PFUUA iqution @PFUUA implies a T 0 (|u m | α u m , w k φ(t))dt -→ a T 0 (|u| α u, w k φ(t))dt @PFUVA
xowD for (xed k = 1, 2, ..., mD we pss to the limit in @PFUHA nd using @PFTTAE@PFTWAD @PFUTA nd @PFUVA to infer tht

- T 0 (u, w k )φ (t)dt + ν T 0 (∇u, φ(t)∇w k )dt + T 0 b(u, u, w k φ(t))dt +a T 0 (|u| α u, w k φ(t))dt = T 0 (f (t), w k φ(t))dt, k = 1, 2, ... m @PFUWA
es it is known the nonliner terms in @PFUWAF fy ontinuity properties we hve ∀w ∈ H

- T 0 (u, w)φ (t)dt + ν T 0 (∇u, φ(t)∇w)dt + T 0 b(u, u, wφ(t))dt +a T 0 (|u| α u, wφ(t))dt = T 0 (f (t), wφ(t))dt @PFVHA iqution @PFVHA implies in the distriution sense d dt (u, w) + ν(∇u, ∇w) + b(u, u, w) + a(|u| α u, w) = (f , w), ∀w ∈ H @PFVIA
et lestD we verify if the veloity u stis(es the initil ondition u(0

) = u 0 F e hve u ∈ L 2 (0, T ; H 1 0 (Ω))∩ L α+2 (0, T ; L α+2 (Ω)) ∩ L ∞ (0, T ; L 2 (Ω))D nd we use the ft tht a T 0 |u(t)| α u(t) (α+2)/3 L (α+2)/3 dt ≤ c T 0 u α+2 L α+2 dt @PFVPA a|u| α u elongs in L (α+2)/3 (0, T ; L (α+2)/3 (Ω)) ⊂ L 4/3 (0, T ; H -1 (Ω)) e use the ft tht T 0 B(u(t), u(t)) 4/3 H -1 dt ≤ c T 0 u(t) 2/3 u(t) 2 H 1 0 ≤ c u(t) 2/3 L ∞ (0,T ;L 2 (Ω)) T 0 u(t) 2 1 dt @PFVQA B(u, u) ∈ L 4/3 (0, T, H -1 (Ω))F sing @PFVIA ∂u ∂t ∈ L 4/3 (0, T, H -1 (Ω)) @PFVRA fy the emedding theorem u ∈ C(0, T ; L 2 (Ω))F gonsider φ ∈ C 1 ([0, T ]) suh tht φ(0) = 0 nd φ(T ) = 0D then we hve - T 0 (u, w)φ (t)dt + ν T 0 (∇u, φ(t)∇w)dt + T 0 b(u, u, wφ(t))dt +a T 0 (|u| α u, wφ(t))dt = (u 0 , w)φ(0) + T 0 (f (t), wφ(t))dt ∀w ∈ H @PFVSA
wultiplying the eqution @PFVIA y φ(t) nd integrting yields to

- T 0 (u, w)φ (t)dt + ν T 0 (∇u, φ(t)∇w)dt + T 0 b(u, u, wφ(t))dt +a T 0 (|u| α u, wφ(t))dt = (u(0), w)φ(0) + T 0 (f (t), wφ(t))dt ∀w ∈ H
@PFVTA e look t the nlogy etween the equtions @PFVSA nd @PFVTA to infer tht

(u 0 -u(0), w) = 0 ∀w ∈ H @PFVUA herefore u(0) = u 0 F
he existene of the pressure follows from the existene of the veloity nd for eh t

∈ [0, T ]D p(t) ∈ L 2 (Ω)F woreover ∇p ∈ C(0, T ; H -1 (Ω)) nd we hve p ∈ C(0, T ; L 2 (Ω)) @PFVVA
nd stis(es in the distriution sense tht ∂u ∂t

-ν∆u + (u.∇)u + a|u| α u + ∇p = f @PFVWA
whih omplete the proofF Lemma 2.9. nder the ssumption @IAD there exists T 2 < T nd positive onstnt C independent of ε suh tht

||u ε || 2 + ν 2 t 0 ||u ε || 2 1 ds + a t 0 ||u ε || α+2 L α+2 ds + ε t 0 ||p ε || 2 ds ≤ C, ∀t ∈ [0, T 2 ]. @PFWHA
roofF e use the inner produt of @QFIA with u ε nd @QFPA with p ε to otin

1 2 d dt ||u ε || 2 + ν||∇u ε || 2 + a||u ε || α+2 L α+2 + ε||p ε || 2 = (f, u ε ) ≤ 1 2ν ||f|| 2 + ν 2 ||∇u ε || 2
fy integrting with respet to tD we get the inequlity @PFWHAF QF Study of the perturbed problem sn this setion we fous on the existeneD the uniqueness nd the regulrity for the penlized prolem ssoited to porous medi model @PFIAE@PFPAF he ssoited penlized prolem to the equtions @PFIAE@PFPA writes

u ε t -ν∆u ε + B(u ε , u ε ) + a|u ε | α u ε + ∇p ε = f, in Ω × [0, T ], @QFIA ∇.u ε + εp = 0, in Ω × [0, T ], u ε | ∂Ω = 0, u ε | t=0 = u 0 , @QFPA where B(u, v) = (u.∇)v + 1 2 (∇.u)v
is the modi(ed iliner termF sing the opertor A ε , we n reformulte the pertured system @PFIAE@PFPA s

u ε t + A ε u ε + B(u ε , u ε ) + a|u ε | α u ε = f, @QFQA u ε | ∂Ω = 0, u ε | t=0 = u 0 @QFRA
Proposition 3.1. nder the ssumption @IAD there exists T 2 < T nd positive onstnt C 0 nd C 1 suh tht the solution u the system @QFIAE@QFPA hs one solution

(u ε , p ε ) whih elongs in ∈ L 2 (0; T 0; H 2 (Ω)) ∩ L ∞ (0; T 0 ; H 1 (Ω)) × L ∞ (0; T 0 ; H 1 (Ω)) nd we hve ||u ε || 2 + ν 2 t 0 ||u ε || 2 1 ds + a t 0 ||u ε || α+2 L α+2 ds + ε||∇p ε || 2 ≤ C 0 , ∀t ∈ [0, T 2 ]. @QFSA nd sup 0≤t≤T0 ||∇u ε || 2 + ν 2 t 0 ||∆u ε || 2 ds ≤ C 1 , ∀t ∈ [0, T 2 ]. @QFTA
roofF ith similr mnner s done for the system @PFIAE@PFPAD we estlish the existene of solution vi pedoEqlerkin methodF o derive the regulrity of the veloity u ε nd the pressure p ε Y pirstD we use the inner produt of @QFIA with u ε nd the eqution @QFPA y p ε to otin

1 2 d dt ||u ε || 2 + ν||∇u ε || 2 + a||u ε || α+2 L α+2 + ε||∇p ε || 2 = (f, u ε ) ≤ 1 2ν ||f|| 2 + ν 2 ||∇u ε || 2 @QFUA
fy integrting with respet to tD we get the inequlity @QFSA with

C 0 = 1 2ν ||f|| 2 + ||u ε 0 || 2 F xextD
to otin more regulrity for the veloity uD we multiply the system @PFIA y -∆u to infer

1 2 d dt ||∇u ε || 2 + ν||∆u ε || 2 = a(|u ε | α u ε , ∆u ε ) + b(u ε , u ε , ∆u ε ) -(f -∇p ε , ∆u ε ) ≤ ν 6 ||∆u ε || 2 + 3 2ν (||f|| 2 + ||∇p ε || 2 )
@QFVA fy pplying guhyEhwrtz inequlity nd oung9s inequlity to otin

a(|u| α u ε , ∆u ε ) ≤ a||u ε || α+1 ||∆u ε ν 6 ||∆u ε || 2 + a 3 3 2ν ||∇u ε || 2(α+1) king 1 ≤ α ≤ 2 to dedue a(|u| α u ε , ∆u ε ) ≤ ν 6 ||∆u ε || 2 + C||∇u ε || 6 @QFWA sing egmon9s inequlity to ound b(u ε , u ε , ∆u ε ) s follow b(u ε , u ε , ∆u ε ) ≤ C||u ε || ∞ ||u ε || 1 ||u ε || 2 ≤ ν 6 ||∆u ε || 2 + C ν ||∇u ε || 2 @QFIHA
ustituting @QFWA nd@QFIHA in @QFUA to hve

d dt ||∇u ε || 2 + ν||∆u ε || 2 ≤ C 2 + C 3 ||∇u ε || 6 @QFIIA vet y(t) = ||∇u ε || 2 + K 1 F xote tht for K 2 ≥ 1 d dt ≤ K 2 y(t) 3 , where y(t) = ||∇u ε || 2 + K 1 and y(0) = ||∇u ε || 2 + K 1 @QFIPA
prom @QFIHA we derive solution de(ned on t @t ≥ 0A stis(es

t ≤ 1 2K 2 y(0) 2 = 1 K 2 @QFIQA his leds to y(t) ≤ √ 2y(0) @QFIRA et T 2 = min(T, 1 K 2 )D for t ∈ [0, T 2 ]D we dedue ||∇u ε (t)|| 2 ≤ √ 2||∇u ε (0)|| 2 + K 1 := K3 @QFISA
o summrizeD we omine the equtions @QFUAnd @QFIHA to onlude

sup 0≤t≤T0 ||∇u ε || 2 + ν 2 t 0 ||∆u ε || 2 ds ≤ C 1 , ∀t ∈ [0, T 2 ]. @QFITA vet us onsider T 0 = min(T 1 , T 2 )F por simpliity we denote y ||v|| L p (X) the norm T0 0 ||v|| p X dt 1/p in L p (0, T 0 ; X)F RF Error Analysis
he min result of this setion is stted in heorem RFPF he proof is split into two stepsX the error of the liner se is given in the (rst step nd the seond one is relted to the error ehvior for the fully nonliner prolemF RFIF Error estimates for a linearly perturbed problem. rimrilyD we will fous on the nlysis of the error for the liner seF e onsider the linerized xvier tokes equtions t u = 0 :

u t -ν∆u + ∇p = f in Ω × [0, T ] ∇.u = 0, in Ω × [0, T ] u| ∂Ω = 0, u(0) = u 0 @RFIA
he ssoited pertured prolem to the system @RFIA tkes the form

u ε t -ν∆u ε + ∇p ε = f in Ω × [0, T ] ∇.u ε + εp ε = 0, in Ω × [0, T ] u ε | ∂Ω = 0, u ε (0) = u 0 @RFPA
vet e = uu ε nd q = p -p ε F utrting @RFPA from @RFIA to get e t -ν∆e + ∇q = 0 @RFQA ∇.e + εq = εp, e(0) = 0 @RFRA he following result summrizes the error relting to the linerized prolem Lemma 4.1. vet ssume @IAD we hve

||e|| L ∞ (L 2 ) + √ ν||e|| L 2 (H 1 ) + √ ε||q|| L 2 (L 2 ) ≤ C √ ε @RFSA ||e|| L 2 (L 2 ) ≤ Cε @RFTA
roofF wultiplying @RFQA y e nd @RFRA y q nd summing upD we otin

1 2 d dt ||e|| 2 + ν||∇e|| 2 + ε||q|| 2 = ε(p, q) ≤ ε 2 ||q|| 2 + ε 2 ||p|| 2
e integrte the previous inequlity from H to t ≤ T 0 D nd we use the ft tht e(0) = 0 nd lemm PFS to hve

||e|| 2 L ∞ (L 2 ) + ν||e|| 2 L 2 (H 1 ) + ε||q|| 2 L 2 (L 2 ) ≤ Cε
whih is equivlent to @RFSAF o dedue the ssertion @RFTAD we need to introdue the uxiliry prolemX for 0 < t ≤ T 0 we de(ne (w, φ)

w s + ν∆w + ∇φ = e(s) ∀ 0 < s ≤ t, ∇.w = 0, w(t) = 0.
@RFUA pirstlyD we derive the following inequlity

ν||w|| L 2 (H 2 ) + ||∇φ|| L 2 (L 2 ) ≤ C||e|| L 2 (L 2 ) @RFVA
por this imD we use the inner produt of @RFUA with AwD nd we integrte from 0 to tD to otin

ν||w|| L 2 (H 2 ) + ||∇w(0)|| L 2 (L 2 ) ≤ C||e|| L 2 (L 2 )
sing the projetion opertor P H on @RFUA to get

||w s || L 2 (L 2 ) ≤ C||e|| L 2 (L 2 )
e use the eqution @RFUA gin to get

||∇φ|| L 2 (L 2 ) ≤ C||e|| L 2 (L 2 )
his ltter inequlity ompletes the demonstrtion of @RFVAF e now multiply the eqution @RFUA y e(s), using @RFQA nd ft ∇.w = 0 we otin

||e|| 2 = (
t 0 ε(φ, p ε )ds ≤ δ||φ|| 2 L 2 (L 2 ) + C δ ε 2 ||p ε || 2 L 2 (L 2 )
with C δ is positive onstnt depending only of δF epplying @RFVA nd hoosing δ smll enough to hve

t 0 ||e|| 2 ds ≤ Cε 2 ||p ε || L 2 (L 2 ) ≤ Cε 2 ∀ t ∈ [0, T 0 ].
he following result is neessry for the susequent investigtionsF Lemma 4.2. rovided @IA nd @PA re stis(edD we hve

t 0 s 2 ||p ε t || 2 ds ≤ C ∀ t ∈ [0, T 0 ].
roofF e derive one time with respet to t on @RFQA nd @RFRAD we get e tt -ν∆e t + ∇q t = 0 @RFWA ∇.e t + εq t = εp t , e(0) = 0 @RFIHA wultiplying @RFQA y te t nd @RFIHA y tq to get

t||e|| 2 + ν 2 d dt t||∇e|| 2 + ε 2 d dt t||q|| 2 = ν 2 ||∇e|| 2 + ε 2 ||q|| 2 + εt(p t , q) ≤ ν 2 ||∇e|| 2 + ε||q|| 2 + εt 2 2 ||p t || 2
sntegrting from 0 to tD using vemm RFTD @PFRHA nd qronwll lemmD we derive

t 0 s||e t || 2 ds + t||e|| 2 1 + εt||q|| 2 ≤ Cε @RFIIA
e wultiply @RFWA y t 2 e t nd @RFIHA y t 2 q t nd we sum up we otin

1 2 d dt t 2 ||e t || 2 + νt 2 ||∇e t || 2 + εt 2 ||q t || 2 = t||e t || 2 + εt 2 (p t , q t ) ≤ t||e t || 2 + εt 2 2 ||q t || 2 + εt 2 2 ||p t || 2
sntegrting over [0, t], using @RFIIA nd @PFRHA we get

ε t 0 s 2 ||q t || 2 ds ≤ C t 0 s||e t || 2 ds + Cε t 0 s 2 ||p t || 2 ds ≤ Cε
he results follows from the previous inequlity nd @PFRHAF Lemma 4.3. e suppose tht @IA nd @PA re stis(ed then

t||e(t)|| 2 + ν t 0 s||∇e(s)|| 2 ds + ε t 0 s||q(s)|| 2 ds ≤ Cε 2 ∀ t ∈ [0, T 0 ], @RFIPA t 2 ||∇e(t)|| 2 + t 0 s 2 ||q(s)|| 2 ds ≤ Cε 2 ∀ t ∈ [0, T 0 ]. @RFIQA
roofF e onsider the deomposition

H 1 0 (Ω) = V ⊕ V ⊥ , where V ⊥ = (-∆) -1 ∇q, q ∈ L 2 (Ω) nd v = (-∆) -1 ∇q if nd only if -∆v = ∇q nd v| ∂Ω = 0F e note tht for p(t) ∈ L 2 (Ω)/R there exists unique ϕ(t) ∈ V ⊥ suh tht ∇.ϕ(t) = p(t) with ϕ(t) 1 ≤ C||p(t)|| ∀ t ∈ [0, T 0 ]. @RFIRA woreoverD if p t (t) ∈ L 2 (Ω)/RD we then hve ∇.ϕ t (t) = p t (t) suh tht ϕ t (t) 1 ≤ C||p t (t)|| ∀ t ∈ [0, T 0 ]. @RFISA
wultiplying @RFQA y te nd @RFRA y tq, we sum up nd use @RFQAD we infer

1 2 d dt t||e|| 2 +tν||∇e|| 2 + εt||q|| 2 = 1 2 ||e|| 2 + εt(p, q) = 1 2 ||e|| 2 + εt(∇.ϕ, q) = 1 2 ||e|| 2 -εt(∇q, ϕ) = 1 2 ||e|| 2 + εt(e t , ϕ) + ενt(∇e, ∇ϕ) = 1 2 ||e|| 2 + ε d dt t(e, ϕ) -ε(e, ϕ) -εt(e, ϕ t ) + ενt(∇e, ∇ϕ) @RFITA fy the ssumption @IAD we hve √ tp ∈ L ∞ (0, T 0 ; H 1 (Ω)/R)F sing @RFIRA we otin εt(e(t), ϕ(t)) ≤ t 4 ||e(t)|| 2 + ε 2 t||ϕ|| 2 ≤ t 4 ||e(t)|| 2 + Cε 2
fy integrting @RFITA from 0 to tD using the previous inequlityD the guhyEhwrtz inequlityD vemm RFTD @PFRHA nd @RFISA we dedue

t||e(t)|| 2 + t 0 (νs||∇e(s)|| 2 + εs||q(s)|| 2 )ds ≤ Cε 2 + C t 0 ||e(s)|| 2 ds + cε 2 t 0 ||ϕ(s)|| 2 ds + Cε 2 t 0 s 2 ||ϕ t (s)|| 2 ds ≤ Cε 2 + Cε 2 t 0 (||p|| 2 + s 2 ||p t || 2 )ds ≤ Cε 2
king the inner produt of @RFQA with t 2 e t D @RFIHA with t 2 q we dedut

t 2 ||e t || 2 + 1 2 d dt t 2 ||∇e|| 2 + ε 2 d dt t 2 ||q|| 2 = t||e|| 2 1 + εt||q|| 2 + εt 2 (p t , q) @RFIUA
sing eqution @RFQA nd the inequlity @RFISA to otin

εt 2 (p t , q) = εt 2 (∇.ϕ t , q) = -εt 2 (ϕ t , ∇q) = εt 2 (e t , ϕ t ) + εt 2 (∇e, ∇ϕ t ) ≤ t 2 2 ||e t || 2 + cε 2 t 2 ||ϕ t || 2 + t 2 ||∇e|| 2 + ε 2 t 2 ||∇ϕ t || 2 ≤ t 2 2 ||e t || 2 + t 2 ||∇e|| 2 + Cε 2 t 2 ||p t || 2
e integrte @RFIUA nd using the qronwll lemm to hve

t 0 s 2 ||e t (s)|| 2 ds + t 2 ||∇e|| 2 + εt 2 ||q(t)|| 2 ≤ Cε 2 ∀ t ∈ [0, T 0 ]
egin from the eqution @RFQA

||q|| 2 ≤ C||∇q|| 2 -1 ≤ C(||∆e|| 2 -1 + ||e t || 2 -1 ) ≤ C(||e|| 2 1 + ||e t || 2 ) gonsequently T0 0 s 2 ||q|| 2 ds ≤ C T0 0 s 2 (||e|| 2 1 + ||e t || 2 )ds ≤ Cε 2
he result elow desries the ehvior of the liner error prtF Theorem 4.1. essume the ssumption @IAF henD there exists onstnt g depending on the given dt suh tht

||e(t)|| + t 0 ||e(s)|| 2 1 ds 1 2 ≤ C √ ε ∀ t ∈ [0, T 0 ].
sf we ssume moreover the hypothesis @PAD then

√ t||e(t)|| + t||e(t)|| 1 + t 0 s 2 ||q|| 2 ds 1 2 ≤ Cε ∀ t ∈ [0, T 0 ].
holdsF he following susetion trets the trnsfer of the results tht hve een derived for the liner se to the non liner oneF RFPF Error estimates for the nonlinear perturbed problem. vet (u; p) e the solution of the sysE tem @IFIA nd let (v; γ) e the solution of the pertured non linerly prolemX

v t -ν∆v + ∇γ = f -B(u, u) -a|u| α v in Ω × [0, T ] @RFIVA ∇.v + εγ = 0, in Ω × [0, T ] u| ∂Ω = 0, v(0) = v 0 , @RFIWA
king ξ = v -uD φ = γ -pD nd sutrting @RFIVAE@RFIWA from @IFIAD we get

ξ t -ν∆ξ + ∇φ = 0 ∇.ξ + εφ = -εp, ξ(0) = 0.
@RFPHA he next vemm desries the ontrol of the errors through the nonElineritiesF Lemma 4.4. e suppose tht @IA nd @PA re vlidF hen we hve

t 0 ||ξ(s)|| 2 ds 1 2 + √ t||ξ(t)|| + t||ξ(t)|| 1 + t 0 s 2 ||φ(s)|| 2 ds 1 2 ≤ Cε ∀ t ∈ [0, T 0 ].
roofF he lemm is onsequene of vemmFRFI nd heoremFRFI pplied to the prolem @RFPHAF e now tke η = u ε -vD q = p ε -γD nd sutrting @RFIVAE@RFIWA from @QFIAE@QFPA to otin

η t -ν∆η + B(u ε , u ε ) -B(u, u) + a(|u ε | α u ε -|u| α u) + ∇q = 0 @RFPIA ∇.η + εq = 0, η(0) = 0 @RFPPA yr B(u ε , u ε ) -B(u, u) = B(u ε , u ε -u) + B(u ε -u, u ε ) = B(u ε , ξ + η) + B(ξ + η, u ε )
@RFPQA e rewrite the eqution @RFPIA s

η t + A ε η + B(u ε , ξ + η) + B(ξ + η, u ε ) + a(|u ε | α u ε -|u| α u) = 0 @RFPRA
he following theorem sttes generl priori estimte for the errors e = uu ε nd q = p -p ε Y with (u, p) re solutions of the stte prolem @IFIAD nd (u ε , p ε ) re solutions of the penlized prolem @QFIAF Theorem 4.2. essume tht @IA nd @PA holdF hen we hve

√ t||u(t) -u ε (t)|| + √ νt||u(t) -u ε (t)|| 1 + t 0 s 2 ||p(t) -p ε (t)|| 2 d 1 2 ≤ Cε ∀ t ∈]0, T 0 ].
roofF wultiplying @RFPRA y A -1 ε ηD we otin

1 2 d dt ||A 1 2 ε η|| 2 + ν||η|| 2 = -b(u ε , ξ + η, A -1 ε η) -b(ξ + η, u, A -1 ε η) -a(|u ε | α u ε -|u| α u, A -1 ε η) = I 1 + I 2 + I 3
sing @PFTAD @PFUAD nd guhy hwrtz9s inequlity we derive tht

I 1 ≤ C||u ε || 2 ||ξ + η||||A -1 ε η|| 1 ≤ C||u ε || 2 ||ξ + η||||∇A -1 ε η|| ≤ C||u ε || 2 (||ξ|| + ||η||)||A -1 2 ε η|| ≤ ν 4 ||η|| 2 + C||ξ|| 2 + C||u ε || 2 2 ||A -1 2 ε η|| 2
ith the sme rgumentD we hve

I 2 ≤ ν 4 ||η|| 2 + C||ξ|| 2 + C||u|| 2 2 ||A -1 2 ε η|| 2
por the third term we proeed similrly

I 3 ≤ C||u ε -u|| 2 ||A -1 2 ε η|| ≤ C(||u ε || 2 2 + ||u|| 2 2 ) + C||A -1 2 ε η|| 2
edding the ove inequlities we get

d dt ||A -1 2 ε η|| 2 ν||η|| 2 ≤ C||ξ|| 2 + C(1 + ||u ε || 2 2 + ||u|| 2 2 )||A -1 2 ε η|| 2 + C(||u ε || 2 2 + ||u|| 2 2 ) @RFPSA hnks to lemm PFSD T0 0 (||u ε || 2 2 + ||u|| 2
2 )dt ≤ CD we n pply the qronwll lemm to @RFPSAD using lemm PFV to otin

||A -1 2 ε η(t)|| 2 + ν t 0 ||η(s)|| 2 ds ≤ C t 0 ||ξ(s)|| 2 ds ≤ Cε 2 ∀ t ∈ [0, T 0 ].
@RFPTA e now multiply the eqution @RFPIA y tη nd @RFPPA y tq nd we sum them upD then using @PFTA nd guhy hwrtz9s inequlity one gets

1 2 d dt t||η|| 2 + νt||∇η|| 2 + εt||q|| 2 = 1 2 ||η|| 2 -t b(u ε , ξ + η, η) -t b(ξ + η, u, η) -at(|u ε | α u ε -|u| α u, η) ≤ 1 2 ||η|| 2 + Ct||u ε || 2 ||ξ + η|| 1 ||η|| + Ct||u|| 2 ||ξ + η|| 1 ||η|| + Ct||u ε -u|| 1 ||η|| ≤ 1 2 ||η|| 2 + Ct||u ε || 2 ||ξ + η|| 1 ||η|| + Ct||u|| 2 ||ξ + η|| 1 ||η|| + Ct||(u ε -v) -(u -v)|| 1 ||η|| ≤ 1 2 ||η|| 2 + Ct||u ε || 2 ||ξ + η|| 1 ||η|| + Ct||u|| 2 ||ξ + η|| 1 ||η|| + Ct||η -ξ|| 1 ||η|| ≤ 1 2 ||η|| 2 + νt 2 ||∇η|| 2 + Ct||ξ|| 2 1 + Ct(||u ε || 2 + ||u|| 2 )||η|| 2
e integrte from H to t nd use @RFPTAD vemm PFV nd the qronwll lemm we get

t||η(t)|| 2 + ν t 0 s||∇η(s)|| 2 ds + ε t 0 s||q(s)|| 2 ds ≤ Cε 2 @RFPUA
e now derive one time with respet to the eqution @RFPPA ∇.η t + εq t = 0 @RFPVA wultiplying the eqution @RFPIA y t 2 η t D nd @RFPVA y t 2 qD summing them up to get

t 2 ||η t || 2 + ν 2 d dt t 2 ||∇η|| 2 + ε 2 d dt t 2 ||q|| 2 = νt||∇η|| 2 + εt||q|| 2 -t 2b (u ε , ξ + η, η t ) -t 2b (ξ + η, u, η t ) -at 2 (|u ε | α u ε -|u| α u, η t ) @RFPWA
por the (rst nonliner term in the right hnd sideD we proeed s follow

t 2 b(u ε , ξ + η, η t ) ≤ t 2 ||u ε || 2 ||ξ + η|| 1 ||η t || ≤ t 2 6 ||η t || 2 + Ct 2 ||u ε || 2 2 (||ξ|| 2 1 + ||η|| 2 1 ) ≤ t 2 6 ||η t || 2 + Cε 2 ||u ε || 2 2 + Ct 2 ||u ε || 2 2 ||∇η|| 2
ith the sme mnner we otin

t 2 b(ξ + η, u, η t ) ≤ t 2 6 ||η t || 2 + Cε 2 ||u|| 2 2 + Ct 2 ||u|| 2 2 ||∇η|| 2
por the lst non liner term we hve

at 2 (|u ε | α u ε -|u| α u, η t ) ≤ at 2 ||η t ||||u ε -u|| 1 ≤ t 2 6 ||η t || 2 + Ct 2 η -ξ 2 1
umming the ove inequlities into @RFPWA to get

t 2 ||η t || 2 + ν d dt t 2 ||∇η|| 2 + ε d dt t 2 ||q|| 2 ≤ νt||∇η|| 2 + εt||q|| 2 + C(ε 2 + t 2 ||∇η|| 2 )(||u ε || 2 2 + ||u|| 2 2 ) + Ct 2 η -ξ 2 1
e integrte from H to tD using @RFPUA nd the qronwll lemm we otin

t 0 s 2 ||η t || 2 ds + νt 2 ||∇η(t)|| 2 + εt 2 ||q(t)|| 2 ≤ Cε 2 @RFQHA
sing @PFTAD we hve

|| B(u ε , ξ + η)|| -1 ≤ C(||u ε || 1 ||ξ + η|| 1 ) ≤ C(||u ε || 1 ||ξ|| 1 + ||η||) || B(ξ + η, u)|| -1 ≤ C(||u|| 1 ||ξ + η|| 1 ) ≤ C(||u|| 1 ||ξ|| 1 + ||η|| 1 ) a(|u ε | α u ε -|u| α u) ≤ C(||ξ|| 1 + ||η|| 1 )
prom @RFPIA nd @RFPQA we otin

∇q = -η t + ν∆η -B(u ε , ξ + η) -B(ξ + η, u) -a(|u ε | α u ε -|u| α u)
onsequentlyD y pplying the ove estimtes on the previous eqution we get

T0 0 s 2 ||q|| 2 ds ≤ T0 0 s 2 ||∇q|| 2 -1 ds ≤ Cε 2
whih end the proof of the theorem RFPFPF SF Optimal shape design he gol of shpe design prolem is to (nd n optiml shpe domin in set of dmissile domins when ost funtionl hieves minimum on this dominF sn grdientEsed optimiztion tehniquesD the ojetive is to minimize suitle ost funtion with respet to set of design vrilesF yne of the key ingredients to ompute e0iently the ost funtion grdient is the use of the djoint system tehniques to simplify the omputtion of grdientsF SFIF Adjoint equation. sn this setion we derive n djoint eqution relted to the )uid )ow system

               u ε t -ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f in Ω × [0, T ], ∇.u ε + εp ε = 0 in Ω × [0, T ], u ε = 0 on ∂Ω × [0, T ], u ε | t=0 = u 0 in Ω.
@SFIA he im is to otin design in whih the veloity is lose to typil suitle veloity nd )ow turulene s low s possileF eording to thisD we de(ne the ojetive funtion s

J(Ω) = J 1 (Ω) + J 2 (Ω), @SFPA
where

J 1 (Ω) = 1 2 T 0 Ω |u ε -u d | 2 dtdxD nd J 2 (Ω) = σ 2 T 0 Ω |curl(u ε )| 2 dtdx.
he generi optimiztion prolem n e suintly stted s follows

min Ω∈Q ad J(Ω) = 1 2 T 0 Ω |u ε -u d | 2 dtdx + σ 2 T 0 Ω |curl(u ε )| 2 dtdx such that (u ε , p ε ) is solution of @SFIA @SFQA
nd the orresponding djoint stte system stisfy the following theoremF Theorem 5.1. vet (u ε , p ε ) ∈ H 2 (0, T ; Ω) × Y e solution of the penlized prolem @SFIAF hen the djoint eqution ssoited to the eqution @SFIA tkes the form

                             -v t -ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u ε -u d ) -σ curl(curl(u ε )) in Ω × [0, T ] ∇.v + εq = 0 in Ω × [0, T ] ν ∂v ∂n -nq = σcurl(u ε ).τ on ∂Ω × [0, T ] v(T ) = 0 @SFRA
roofF e de(ne lso the following funtion

F (Ω, u ε , p ε , v, q) = T 0 Ω u ε t + ν∇u ε : ∇v + (u ε .∇)u ε .v + 1 2 (∇.u ε )u ε .v + a|u ε | α u ε .v dtdx - T 0 Ω f.v dtdx - T 0 Ω p ε ∇.v -q∇.u ε -εqp ε dtdx.
vet L(Ω, u ε , p ε , v, q) e vgrngin funtionl de(ned y

L(Ω, u ε , p ε , v, q) = J(Ω) -F (Ω, u ε , p ε , v, q), @SFSA
we (rst derive L with respet to the stte vrile p in ny diretion p ∈ M (Ω), we otin

∂L ∂p (Ω, u ε , p ε , v, q).p = T 0 Ω p∇.v + εpq dtdx,
the vrition p is ritrryD we get ∇.v + εq = 0 on Ω. @SFTA xextD we derive L with respet of the stte vrile u ε in the ritrry diretion ũ ∈ V 0 (Ω) we hve

0 = ∂L ∂u ε (Ω, u ε , p ε , v, q).ũ, = T 0 Ω (u ε -u d ).ũ -σcurl(u ε ).curl(ũ) dtdx - T 0 Ω -v t ũ + ν∇ũ : ∇v + (ũ.∇)u ε .v + (u ε .∇)ũ.v + 1 2 (∇.ũ)u ε .v + (∇.u ε )ũ.v dtdx + T 0 Ω q∇.ũ -a|u ε | α ũ + aα|u ε | α-2 (u ε .ũ)u ε .v dtdx, = T 0 Ω (u ε -u d ) -σ curl(curl(u ε )) + v t + ν∆u ε -(∇u ε ) T .v + (u ε .∇)v + 1 2 ∇(u ε .v) - 1 2 (∇.u ε )v -(a|u ε | α v + aα|u ε | α-2 (u ε .v)u ε -∇q .ũ dtdx + T 0 ∂Ω σcurl(u ε )τ -ν ∂v ∂n -nq .ũdtdx + Ω [v.ũ] T 0 dx.
gonsidering n ritrry diretion ũ whih vnishes in neighorhood of the oundry ∂ΩD we otin

-v t -ν∆v+(∇u) T .v-(u ε .∇)v- 1 2 ∇(u ε .v)+ 1 2 (∇.u ε )v+a|u| α-2 (|u ε | 2 v+α(u ε .v)u ε )+∇q = (u ε -u d )-σ curl(curl(u ε )),
nd n ritrry ũ in ∂Ω gives ν ∂v ∂n -nq = σcurl(u ε ).τ on ∂Ω pinlly we otin the djoint eqution

                             -v t -ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u ε -u d ) -σ curl(curl(u ε )) in Ω × [0, T ] ∇.v + εq = 0 in Ω × [0, T ] ν ∂v ∂n -nq = σcurl(u ε ).τ on ∂Ω × [0, T ] v(T ) = 0
@SFUA SFPF Shape gradient. sn the followingD we derive the strutures of shpe grdients for timeEdependent ost funtionls y using the the veloity method @see gé IA nd its ssoited djoint stteF vet Ω e referene domin in R 2 D the perturtion of Ω y the veloity method is desried s the veloity (eld

V(t)(x) = V(t, x), ∀x ∈ Ω, t ≥ 0.
st n generte trnsformtions

T t (V)X = x(t, X), t ≥ 0, X ∈ Ω,
the )ow de(ned y the initil vlue prolem

   dχ dt (t, X) = V(t, χ(t)), χ(0, X) = X, @SFVA
with the initil vlue X givenF vet Ω t e perturtion domin of Ω nd J(Ω) e funtionl ssoited to Ω t F he shpe derivtive of the funtionl J(Ω t ) t ∂Ω in the diretion of the deformtion (eld V is written s

dJ(Ω; V) = lim t→0 J(Ω t ) -J(Ω) t .
sf dJ(Ω; V) exists for ll V ∈ C([0, T ]; (D k (R 2 )) 2 )D for smll positive onstnt T, the funtionl J is lled shpe di'erentile t Ω nd its shpe grdient veri(es

dJ(Ω; V) = (∇J, V) ((D k (R 2 )) 2 ) ×(D k (R 2 )) 2
e de(ne the veloity (eld dmissile domin s follow

V ad = V ∈ C 0 (0, τ ; (C 2 (R 2 )) 2 )|V = 0 on ∂Ω .
he use of the veloity V for t ≥ 0D implies the trnsformtion of the domin Ω into Ω t = T t (Ω) y the veloity method with formultion @SFVAF vet us (nd n expression of the derivtive of the sddle point prolem j@tA with respet to t where

j(t) = min (u ε t ,py)∈Vg(Ωt)× Ŷ (Ωt) max (vt,qt)∈V0(Ωt)× Ŷ (Ωt) L(Ω t , u ε t , p ε t , v t , q t )
with (u, p) nd (v, q) re solutions of @IFIA nd @SFRA in the pertured domin Ω t D respetivelyF e onsider the rilert spes whih depend on the prmeter t de(ned y

V 0 (Ω t ) = u ε • T -1 t : u ε ∈ V 0 (Ω) Ŷ (Ω t ) = p ε • T -1 t : p ε ∈ Ŷ (Ω) sine T t nd T -1
t re di'eomorphismsD the prmetriztion do not in)uene j@tAF end we hve

j(t) = min (u ε ,p ε )∈Vg(Ω)× Ŷ (Ω) max (v,q)∈V0(Ω)× Ŷ (Ω) L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t )
we de(ne the following funtions whih depend on the prmeter t

l 1 (t) = 1 2 T 0 Ωt |u ε • T -1 t -u d | 2 dx + σ 2 T 0 Ωt |curl(u ε • T -1 t )| 2 dtdx, l 2 (t) = T 0 Ωt [ν∇(u ε • T -1 t ) : ∇(v • T -1 t ) + ((u ε • T -1 t ).∇)(u ε • T -1 t ).(v • T -1 t ) + a|u ε • T -1 t | α (u ε • T -1 t ).(v • T -1 t ) -(p ε • T -1 t )∇.(v • T -1 t ) -(f • T -1 t ).(v • T -1 t )]dtdx - T 0 Ωt (q • T -1 t )∇.(u ε • T -1 t )dtdx -ε T 0 Ωt (q • T -1 t )(p ε • T -1 t )dtdx
he vgrngin funtionl writes

L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t ) = l 1 (t) -l 2 (t) sf Φ : [0, τ ] × R 2 →
R is su0iently smoothD we hve the following rdmrd formul

d dt T 0 Ωt Φ(t, x)dx| t=0 = T 0 Ω ∂Φ ∂t (0, x)dx + T 0 ∂Ω Φ(0, x)v(0, X).nds @SFWA
e onsider V(0, X) ∈ V ad D nd oserve tht V(0, X) = VF herefore we n derive the shpe grdient using the formul @SFVA

d dt L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t )| t=0 = l 1 (0) -l 2 (0) @SFIHA
where

l 1 (0) = T 0 Ω (u ε -u d ).(-∇u ε .V)dx + σ T 0 Ω curl(u ε )curl(-∇u ε .V)dx + 1 2 T 0 ∂Ω (|u ε -u d | 2 V.n)ds + σ 2 T 0 ∂Ω (|curl(u ε )| 2 V.n)ds @SFIIA l 2 (0) = T 0 Ω [(-∇u ε .V) t v + (-∇v.V) t u ε + ν∇(-∇u ε .V) : ∇v + ν∇u ε : ∇(-∇v.V) + ((-∇u ε .V).∇u ε ).v + (u ε .∇(-∇u ε .V).v + (u ε .∇u ε ).(-∇v.V) 1 2 ∇.(-∇u ε .V)u ε .v + 1 2 ∇.u ε (-∇u ε .V).v + 1 2 ∇.u ε u ε .(-∇v.V) + aα|u ε | α-2 (-∇u ε .V.u ε )(u ε .v) + a|u ε | α (-∇u ε .V).v + a|u ε | α u ε .(-∇v.V) -(f.V).v -f.((-∇v).V) -(∇p ε .V)∇.v -p ε ∇.(-∇v.V) -q∇.(-∇u ε .V) -(∇.u ε )(-∇q.V)]dx
sing qreen9s formul nd the ondition u ε = 0 on ∂ΩD we otin

l 2 (0) = - T 0 Ω [(u ε t -ν∆u ε + (u ε .∇)u ε + 1 2 ∇.u ε u ε + a|u ε | α u ε + ∇p ε -f ).(∇v.V)]dtdx + T 0 Ω (∇.u ε )(∇q.V)dtdx + T 0 Ω (∇.v)(∇p ε .V)dtdx + T 0 Ω [-v t -ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) -∇q].(∇u ε .V)dtdx - T 0 ∂Ω [ν ∂v ∂n -nq].(∇u ε .V)dtds - T 0 ∂Ω [ ∂u ε ∂n -nq].(∇v.V)dtds + T 0 ∂Ω [ν∇u ε : ∇v + (u ε .∇u ε ).v + a|u ε | α u ε .v -p∇.v -q∇.u ε ]V.n dtds
@SFIPA epling @SFIIA nd @SFIPA in @SFIHA nd using the ft tht (u ε , p ε ) is solution of @IFPA nd (v, q) is solution of @SFRA respetively yields

dJ(Ω; V) = d dt L| t=0 = 1 2 T 0 ∂Ω ((|u ε -u d | 2 + σ|curl(u ε )| 2 )V.n)ds + T 0 ∂Ω [ν ∂v ∂n -nq].(∇u ε .V)dtds + T 0 ∂Ω [ν ∂u ε ∂n -np ε ].(∇v.V)dtds - T 0 ∂Ω [(ν∇u ε : ∇v)V.n dtds -σ T 0 ∂Ω ( ∂u ε ∂n .(curl(u ε ) ∧ n))V.n dtds @SFIQA xote tht u ε = 0 nd v = 0 in ∂Ω we hve n.(∇u ε .V) = ∇u ε .(n ⊗ n).V.n = ∇u ε .n.n(V.n) = (∇.u ε )(V.n) = 0, ∀x ∈ ∂Ω @SFIRA ∂v ∂n .(∇u ε .V) = ∇u ε .(n ⊗ n).V. ∂v ∂n = ∂u ε ∂n . ∂v ∂n (V.n) = (∇u ε : ∇v)V.n @SFISA
imilrly we otin n.(∇v.V) = 0,

∂u ε ∂n .(∇v.V) = ∂u ε ∂n . ∂v ∂n (V.n) = (∇u ε : ∇v)V.n @SFITA
ustituting @SFIRAD @SFISA nd @SFITA in @SFIQA the shpe derivtive tkes the form

dJ(Ω; V) = T 0 ∂Ω 1 2 |u ε -u d | 2 + σ 2 |curl(u ε )| 2 + ∂u ε ∂n . ν ∂v ∂n -σcurl(u ε ) ∧ n V.n dtds @SFIUA
gonsequentlyD the shpe grdient writes

∇J = 1 2 |u ε -u d | 2 + σ 2 |curl(u ε )| 2 + ∂u ε ∂n . ν ∂v ∂n -σcurl(u ε ) ∧ n n @SFIVA
TF Discrete finite element method TFIF Preliminaries. sn this setion we write some results nd nottions whih will e used in this workF e de(ne the following forms

a(u, v) = ν(A 1/2 u, A 1/2 v), a ε (u, v) = ν(A 1/2 ε u, A 1/2 ε v), c α (u, v) = (a|u| α u, v) ∀u, v ∈ X d(v, q) = (∇.v, q), ∀v ∈ X, ∀q ∈ M
he following ssumptions on the dtum u 0 nd f will e needed throughout this setion he initil veloity u 0 ∈ D(A) with ∇.u 0 = 0

the foring funtion f ∈ H 1,∞ (0, T ; Y ) with ||Au 0 || + sup t∈[0,T ] {||f (t)|| + ||f t (t)||} ≤ C.
@eIA e rell the two wek formultions onerning the initil prolem nd its penlizedF

F ind (u, p) ∈ L ∞ (0, T ; Y ) ∩ L 2 (0, T ; X) × L 2 (0, T ; M ) such that (u t , v) + a(u, v) -d(v, p) + d(u, q) + b(u, u, v) + c α (u, v) = (f , v), ∀(v, q) ∈ X × M @TFIA nd F ind (u ε , p ε ) ∈ L ∞ (0, T ; Y ) ∩ L 2 (0, T ; X) × L 2 (0, T ; M ) such that (u εt , v) + a(u ε , v) -d(v, p ε ) + d(u ε , q) + ε(p ε , q) + b(u ε , u ε , v) + c α (u ε , v) = (f , v) f or all (v, q) ∈ X × M @TFPA
with the initil dt u(0) = u 0 nd u ε (0) = u 0 respetivelyF xowD we de(ne the time disretiztion of the penlized wek formultion @TFPA y the kwrd iuler sheme

(d t u n ε , v) + a(u n ε , v) -d(v, p n ε ) + d(u n ε , q) + ε(p n ε , q) + b(u n ε , u n ε , v) + c α (u n ε , v) = (f (t n ), v) @TFQA for ll (v, q) ∈ X × M nd 1 ≤ n ≤ N D where 0 < ∆t < 1 is the time step sizeD t n = n∆D t N = T D (u 0 ε , p 0 ε ) = (u 0 , 0)D d t u n ε = u n ε -u n-1 ε ∆t for 1 ≤ n ≤ N nd d t u 0 ε stis(es the following eqution (d t u 0 ε , v) = a(u 0 , v) + ((u 0 .∇)u 0 , v) + c α (u 0 , v) = (f (0), v)
, ∀v ∈ X with divv = 0 fy using the inequlity @PFIIAD we infer tht

||d t u 0 ε || ≤ ν||Au 0 || + ||(u 0 .∇)u 0 || + ||A 1/2 u 0 || + ||f (0)|| ≤ c ν ||Au 0 || + c||u 0 ||||A 1/2 u 0 || 2 + ||f (0)|| @TFRA
e egin with generl regulrity resultD on the pertured solutionD useful for the estlishment of n error ound ssoited with the (nite elements disretiztionF Theorem 6.1. nder the ssumptions @eIA nd Cε ≤ 1D there is onstnt k 0 > 0 suh tht if k 0 ∆t ≤ 1D then the following estimtes hold

||A 1/2 ε u m ε || 2 + ∆t m n=1 (||d t u n ε || 2 + ||A ε u n ε || 2 + ||p n ε || 2 + a||u n ε || α+2 L α+2 ) ≤ C, @TFSA ||d t u m ε || 2 + ||A ε u m ε || 2 + ||p m ε || 2 1 + ∆t m n=1 ||A 1/2 ε d t u n ε || 2 ≤ C, @TFTA
for ll 1 ≤ m ≤ N F roofF king (v, q) = 2(u n ε , p n ε )∆t in @TFQAD using @TFQA nd the reltion

2(u -v, u) = ||u|| 2 -||v|| 2 + ||u -v|| 2 , ∀u, v ∈ Y @TFUA to hve ||u n ε || 2 -||u n-1 ε || 2 + 2ν||A 1/2 u n ε || 2 ∆t + a∆t m n=1 u n ε α+2 L α+2 ≤ 2||f (t n )||||u n ε ||∆t
e sum this inequlity from 1 to m nd using @PFWA nd the oung9s inequlityD to otin

||u m ε || 2 + ν∆t m n=1 ||A 1/2 u n ε || 2 + a∆t m n=1 u n ε α+2 L α+2 ≤ ||u 0 || 2 + ν -1 c∆t m n=1 ||f (t n )|| 2 ≤ C @TFVA
e n estlish from @TFQA tht

d t u n ε + νA ε u n ε + B(u n ε , u n ε ) + a|u n ε | α u n ε = f (t n ) @TFWA
king the slr produt of @TFWA with @ν -1 d t u n ε + A ε u n ε )∆t in Y nd using the reltion

2(A 1/2 ε (u -v), A 1/2 ε u) = ||A 1/2 ε u|| 2 -||A 1/2 ε v|| 2 + ||A 1/2 ε (u -v)|| 2 , ∀u, v ∈ X @TFIHA we otin ||A 1/2 ε u n ε || 2 -||A 1/2 ε u n-1 ε || 2 + ν -1 ||d t u n ε || 2 ∆t + ν||A ε u n ε || 2 ∆t +c α (u n ε , ν -1 d t u n ε + A ε u n ε )∆t + b(u n ε , u n ε , ν -1 d t u n ε + A ε u n ε )∆t = (f (t n ), ν -1 d t u n ε + A ε u n ε )∆t @TFIIA
sing @PFIIAE@PFIPAD we get pirstly

| b(u n ε , u n ε , ν -1 d t u n ε + A ε u n ε )| ≤ c||A 1/2 u n ε ||||u n ε || L ∞ ||ν -1 d t u n ε + A ε u n ε || ≤ 1 6ν ||d t u n ε || 2 + ν 6 ||A ε u n ε || 2 + ν -1 c||u n ε || 2 ||A 1/2 u n ε || 2 ||A 1/2 ε u n ε || 2 eondly |c α (u n ε , ν -1 d t u n ε + A ε u n ε )| ≤ 1 6ν ||d t u n ε || 2 + ν 6 ||A ε u n ε || 2 +ν -1 c||A 1/2 u n ε || 2 ||A 1/2 ε u n ε || 2 pinlly |(f (t n ), ν -1 d t u n ε + A ε u n ε )| ≤ 1 6ν ||d t u n ε || 2 + ν 6 ||A ε u 2 ε || 2 + ν -1 c||f (t n )|| 2
gomining these inequlities with @TFIIA to otin

||A 1/2 ε u n ε || 2 -||A 1/2 ε u n-1 ε || 2 + 1 2 (ν -1 ||d t u n ε || 2 + ν||A ε u n ε || 2 )∆t +c α (u n ε , ν -1 d t u n ε + A ε u n ε ) ≤ d n ||A 1/2 ε u n ε || 2 ∆t + ν -1 c||f (t n )|| 2 ∆t with d n = ν -1 c(1 + ν -2 ||u n ε || 2 )||A 1/2 u n ε || 2 F umming this inequlity from I to m nd oserving tht A 1/2 ε u 0 = A 1/2 u 0 D we get ||A 1/2 ε u m ε || 2 + 1 2 ∆t m n=1 (ν -1 ||d t u n ε || 2 + ν||A ε u n ε || 2 ) + ∆t n n=1 c α (u n ε , ν -1 d t u n ε + A ε u n ε ) ≤ ||A 1/2 u 0 || 2 + ∆t m n=1 d n ||A 1/2 ε u n ε || 2 + ν -1 c∆t m n=1 ||f (t n )|| 2 ≤ C + ∆t m n=1 d n ||A 1/2 ε u n ε || 2
@TFIPA sf we tke ∆t suh tht d n ∆t ≤ 1 2 D y pplying vemm PFR to @TFIPA we infer

||A 1/2 ε u m ε || 2 + 1 2 ∆t m n=1 (ν -1 ||d t u n ε || 2 + ν||A ε u n ε || 2 ) + ∆t m n=1 c α (u n ε , ν -1 d t u n ε + A ε u n ε ) ≤ Cexp(2∆t m n=1 d n )
@TFIQA for ll 1 ≤ m ≤ N F fy using @PFIPAD @TFVA nd @TFIPA there exists onstnt C 0 > 0 suh tht

2∆t m n=1 d n ≤ C and d m = ν -1 c(1 + ν -2 ||u m ε || 2 )||A 1/2 u m ε || 2 ≤ 1 2 C 0 @TFIRA for ll 1 ≤ m ≤ N.
xextD using @TFQA to otin

(d tt u n ε , v) + a ε (d t u n ε , v) + b(d t u n ε , u n ε , v) + b(u n-1 ε , d t u n ε , v) +a (d t (|u n ε | α u n ε ) , v) = 1 ∆t tn tn-1 f t (t)dt, v , ∀v ∈ X
@TFISA e tke v = 2d t u n ε ∆t in @TFISA nd we use @PFRA nd @TFUA to otin

||d t u n ε || 2 -||d t u n-1 ε || 2 + 2ν||A 1/2 ε d t u n ε || 2 ∆t + 2 b(d t u n ε , u n ε , d t u n ε )∆t +2a (d t (|u n ε | α u n ε ) , d t u n ε ) ∆t ≤ 2 tn tn-1 f t (t)dt, d t u n ε @TFITA ine 2a(d t (|u n ε | α u n ε ), d t u n ε ) ≥ 0 thnks to @PFIQAF e use @PFWA nd @PFIPA to get pirstly 2| b(d t u n ε , u n ε , d t u n ε )| ≤ c||d t u n ε ||||A 1/2 d t u n ε ||0||A 1/2 u n ε || +c||d t u n ε || 1/2 ||A 1/2 d t u n ε || 3/2 ||A 1/2 u ε || 1/2 ||u n ε || 1/2 ≤ ν 3 ||A ε d t u n ε || 2 + d n ||d t u n ε || 2 eondly | tn tn-1 f t (t)dt, d t u n ε | ≤ ν 3 ||A ε d t u n ε || 2 + ν -1 c tn tn-1 ||f (t)|| 2 dt
edding these inequlities nd repling them in @TFITA we hve

||d t u n ε || 2 -||d t u n-1 ε || 2 + ν||A 1/2 ε d t u n ε || 2 ∆t ≤ (ν -1 c + d n )||d t u n ε || 2 ∆t + ν -1 c tn tn-1 ||f (t)|| 2 dt @TFIUA
umming this inequlity from I to m nd using @TFRAD @TFIQA nd @TFIRA we otin

||d t u n ε || 2 + ν∆t m n=1 ||A 1/2 ε d t u n ε || 2 ≤ C, ∀1 ≤ m ≤ N @TFIVA
pinllyD we n estlish from @TFQAD @TFISA nd the inf sup ondition IW tht

ν||A ε u n ε || ≤ ||d t u n ε || + || B(u n ε , u n ε )|| + |||u n ε | α u n ε || + ||f (t n )|| @TFIWA ||p n ε || 1 ≤ c||d t u n ε || + νc||Au n ε || + c|| B(u n ε , u n ε )|| + c|||u n ε | α u n ε || + c||f (t n )|| @TFPHA
epplying @PFIIAE@PFIPAD we get

|| B(u n ε , u n ε )|| ≤ c||A 1/2 u n ε ||||u n ε || L ∞ ≤ c||A 1/2 u n ε ||||A ε u n ε || 1/2 ||u n ε || 1/2 ≤ ν 4 ||A ε u n ε || + ν -1 c||u n ε ||||A 1/2 u n ε || 2 nd |||u n ε | α u n ε || ≤ c||A ε u n ε ||
epling theses inequlities in @TFIWA nd @TFPHA nd using @PFIPAD we otin

ν||A ε u n ε || 2 ≤ ν -1 ||d t u n ε || 2 + ν -3 c||u n ε || 2 ||A 1/2 ε u n ε || 4 + ν -1 c||f (t n )|| 2 @TFPIA ||p n ε || 2 1 ≤ c ||d t u n ε || 2 + ||A ε u n ε || 2 + cν -2 ||u n ε || 2 ||A 1/2 ε u n ε || 4 + ||f (t n )|| 2 @TFPPA ∆t m n=1 ||p n ε || 2 1 ≤ c∆t m n=1 ||d t u n ε || 2 + ||A ε u n ε || 2 +c∆t m n=1 (ν -2 ||u n ε || 2 ||A 1/2 ε u n ε || 4 + ||f (t n )|| 2 )
@TFPQA gomining the inequlities @TFPIAE@TFPQA with @TFIQAE@TFIRA nd @TFIVAD we end the proof of the heorem TFI Theorem 6.2. nder the ssumptions of heorem TFI we hve the following estimte

τ (t m )||A 1/2 d t u m ε || 2 + ∆t m n=1 τ (t n ) ||d tt u n ε || 2 + ||A ε d t u n ε || 2 + ||d t p n ε || 2 1 + 2ν -1 ∆t m n=1 a(d t (|u n ε | α u n ε ), d tt u n ε ) ≤ C, for ll 1 ≤ m ≤ N F roofF king v = 2ν -1 d tt u n ε ∆t in @TFISA yields 2 A 1/2 ε d t u n ε , A 1/2 ε (d t u n ε -d t u n-1 ε ) + 2ν -1 ||d tt u n ε || 2 ∆t + 2ν -1b (d t u n ε , u n ε , d tt u n ε )∆t +2ν -1b (u n ε , d t u n ε , d tt u n ε )∆t + 2ν -1 a(d t (|u n ε | α u n ε ), d tt u n ε )∆t = 2ν -1 tn tn-1 f t (t)dt, d tt u n ε @TFPRA e use @PFWAE@PFIPA to rise 2ν -1 | b(d t u n ε , u n ε , d tt u n ε )| +2ν -1 | b(u n ε , d t u n ε , d tt u n ε )| ≤ cν -1 ||A 1/2 d t u n ε ||||A ε u n ε ||||d tt u n ε || ≤ 1 6ν ||d tt u n ε || 2 + ν -1 c||A ε u n ε || 2 ||A 1/2 ε d t u n ε || 2 2ν -1 a|(d t (|u n ε | α u n ε ), d tt u n ε )| ≤ 1 6ν ||d tt u n ε || 2 + ν -1 c||A 1/2 ε d t u n ε || 2 2ν -1 | tn tn-1 f t (t)dt, d tt u n ε | ≤ 1 6ν ||d tt u n ε || 2 ∆t + ν -1 c tn tn-1 ||f t (t)|| 2 dt
epling these inequlities in @TFPRA yields

2τ (t n )(A 1/2 ε d t u n ε , A 1/2 ε (d t u n ε -d t u n-1 ε )) + ν -1 τ (t n )||d tt u n ε || 2 ∆t ≤ ν -1 c 1 + ||A ε u n ε || 2 ||A 1/2 ε d t u n ε || 2 + ν -1 c tn tn-1 ||f t (t)|| 2 dt @TFPSA e oserve tht 2τ (t n )(A 1/2 ε d t u n ε , A 1/2 ε (d t u n ε -d t u n-1 ε )) ≥ τ (t n )||A 1/2 ε d t u n ε || 2 -τ (t n-1 )||A 1/2 ε d t u n-1 ε || 2 -||A 1/2 ε d t u n-1 ε || 2 ∆t, 2 ≤ n ≤ N 2τ (t n )(A 1/2 ε d t u 1 ε , A 1/2 ε (d t u 1 ε -d t u 0 ε )) ≥ τ (t 1 )||A 1/2 ε d t u 1 ε || 2 -||A ε u 1 ε -Au 0 ||||d t u 0 ε || e sum @TFPSA in 1 ≤ n ≤ m to otin τ (t m )||A 1/2 ε d t u m ε || 2 +ν -1 ∆t m n=1 τ (t n )||d tt u n ε || 2 + 2ν -1 ∆t m n=1 a(d t (|u n ε | α u n ε ), d tt u n ε ) ≤ ||A ε u 1 ε -Au 0 ||||d t u 0 ε || + ∆t m n=1 ||A 1/2 ε d t u n ε || 2 +ν -1 c∆t m n=1 (1 + ||A ε u n ε || 2 )||A 1/2 ε d t u n ε || 2 + ν -1 c tn tn-1 ||f t (t)|| 2 dt @TFPTA
gomining @TFPTA with @TFTA yields

τ (t m )||A 1/2 ε d t u n ε || 2 + ν -1 ∆t m n=1 τ (t n )||d tt u n ε || 2 ≤ C, ∀1 ≤ m ≤ N @TFPUA
xextD we derive from @TFQA nd the infEsup ondition PH tht

ν||A ε d t u n ε || ≤ ||d tt u n ε || + || B(d t u n ε , u n ε )|| + || B(u n ε , d t u n ε )|| + ||d t (|u n ε | α u n ε )|| + ||f (t n )|| @TFPVA ||d t p n ε || 1 ≤ c||d tt u n ε || + νc||Ad t u n ε || +c|| B(d t u n ε , u n ε )|| + c|| B(u n ε , (d t u n ε )|| +c||d t (|u n ε | α u n ε )|| + c||f (t n )|| @TFPWA hue to @PFWAE@PFIPAD we get || B(d t u n ε , u n ε )|| + || B(u n ε , d t u n ε )|| ≤ c||A ε u n ε ||||A 1/2 ε d t u n ε ||
gomining this inequlity with @TFPVA nd @TFPWAD we otin

ν 2 ∆t m n=1 τ (t n )||A ε d t u n ε || 2 ≤ c∆t m n=1 τ (t n )||d tt u n ε || 2 +c∆t m n=1 (1 + ||A ε u n ε || 2 )||A 1/2 ε d t u n ε || 2 + c∆t m n=1 ||f t (t n )|| 2 @TFQHA ∆t m n=1 τ (t n )||d t p n ε || 2 1 ≤ c∆t m n=1 τ (t n ) ||d tt u n ε || 2 + ν 2 ||A ε d t u n ε || 2 +c∆t m n=1 (1 + ||A ε u n ε || 2 )||A 1/2 ε d t u n ε || 2 + ||f t (t n )|| 2 @TFQIA
gomining the inequlities @TFQHA nd @TFQIA with @TFPUAD this omplete the proof of the heorem TFP TFPF Finite element penalty method. vet us onsider rel positive prmeter h. e denote y (X h , M h ) the (nite element pir of (X, M ). st is disretiztion of these domin into tringles τ F e introdue the L 2 Eorthogonl projetionsX π h : M → M h de(ned y (π h q, q h ) = (q, q h ), ∀q ∈ M, q h ∈ M h e shll mke stnding the ssumption for the (nite element pir (X h , M h )

here exists mpping r h : D(A) ∩ X → X h suh tht @ePA (div(u -r h u), q h ) = 0, ∀q ∈ M, q h ∈ M h @TFQPA ||A 1/2 (u -r h u)|| ≤ ch||Au||, ||p -π h p|| ≤ ch||p|| 1 , ∀p ∈ H 1 (Ω) ∩ M @TFQQA
woreoverD we suppose the inverse inequlity

||A 1/2 v h || ≤ ch -1 ||v h ||, ∀v h ∈ X h @TFQRA nd the infEsup ondition ||q|| ≤ c sup v h ∈X h (div(v h ), q h ) ||A 1/2 v h || , ∀q h ∈ M h @TFQSA
vet us introdue the following (nite element disretiztion of the formultion @TFQA

(d t u n εh , v h ) + a(u n εh , v h ) -d(v h , p n εh ) + d(u n εh , q h ) + ε(p n εh , q h ) + b(u n εh , u n εh , v h ) +c α (u n εh , v h ) = (f (t n ), v h ), ∀(v h , q h ) ∈ X h × M h @TFQTA where u 0 εh = r h u 0 , p 0 εh = 0, {u n εh } N n=1 ⊂ X h nd {p n εh } N n=1 ⊂ M h re the (nite element pproximtions of {u n ε } N n=1 ⊂ X nd {p n ε } N n=1 ⊂ M respetivelyF
Theorem 6.3. uppose @eIAD @ePA nd εc 0 ≤ 1D ∆tk 0 ≤ 1 re vlidF hen the following proprieties holds

||u n εh || 2 + ∆t m n=1 ||A 1/2 u n εh || 2 + a∆t n n=1 u n εh α+2 L α+2 ≤ C, @TFQUA ||A 1/2 u n εh || 2 + ∆t m n=1 ||d t u n εh || 2 ≤ C + Ch -2 ∆t m n=1 ||A 1/2 (u n ε -u n εh )|| 2 ||u n εh || 2 , +Ch -2 ∆t m n=1 ||u n ε -u n εh || 2 ||A 1/2 u n εh || 2 , @TFQVA for ll 1 ≤ m ≤ N.
roofF sn similr mnner s the estimtion @TFVA in the heorem TFID we n derive @TFQUAF xext we n estlish from @TFQTA tht

(d t u n εh , v h ) + a(u n εh , v h ) -d(v h , p n εh ) + d(d t u n εh , q h ) + ε(d t p n εh , q h ) + b(u n εh , u n εh , v h ) +c α (u n εh , v h ) = (f (t n ), v h ), ∀(v h , q h ) ∈ X h × M h @TFQWA etting (v h , q h ) = 2(d t u n εh , p n εh )
∆t in @TFQWA nd using @TFIHAD we get

2||d t u n εh || 2 ∆t + ν||A 1/2 u n εh || 2 -ν||A 1/2 u n-1 εh || 2 + ε(||p n εh || 2 -||p n εh || 2 ) +2c α (u n εh , d t u n εh )∆t + 2 b(u n εh , u n εh , d t u n εh )∆t = 2(f (t n ), d t u n εh )∆t @TFRHA
epplying @PFWAE@PFIIAD @PFRA nd @TFQRAD we otin

2| b(u n εh , u n εh , d t u n εh )| ≤ 2| b(u n εh -u n ε , u n εh , d t u n εh )| + 2| b(u n ε , u n εh , d t u n εh )| 2| b(u n ε , u n εh , d t u n εh )| ≤ c||Au n ε ||||A 1/2 u n εh ||||d t u n εh || ≤ 1 4 ||d t u n εh || 2 + c||Au n ε || 2 ||A 1/2 u n εh || 2 2| b(u n εh -u n ε , u n εh , d t u n εh )| ≤ c||u n εh -u n ε || L 4 ||u n εh || L 4 ||A 1/2 d t u n εh || +c||A 1/2 (u n εh -u n ε )|| 2 ||u n εh || L 4 ||A 1/2 d t u n εh || L 4 ≤ 1 4 ||d t u n εh || 2 + ch -2 ||A 1/2 (u n ε -u n εh )|| 2 ||u n εh || 2 +ch -2 ||u n ε -u n εh || 2 ||A 1/2 u n εh || 2 2|c α (u n εh , d t u n εh )| ≤ 1 4 ||d t u n εh || 2 + c||A 1/2 u n εh || 2 2|(f (t n ), d t u n εh )| ≤ 1 4 ||d t u n εh || 2 + c||f (t n )|| 2 epling
these inequlities in @TFRHAF with some smple lultion we otin

ν||A 1/2 u n εh || 2 -ν||A 1/2 u n εh || 2 + ||d t u n εh || 2 ∆t + ε(||p n εh || 2 -||p n-1 εh || 2 ) ≤ c||Au n ε || 2 ||A 1/2 u n εh || 2 ∆t + ch -2 ||A 1/2 (u n ε -u n εh )|| 2 ||u n εh || 2 ∆t + ||u n ε -u n εh || 2 ||A 1/2 u n εh ||∆t + c||A 1/2 u n εh || 2 + c||f (t n )|| 2
@TFRIA umming these inequlity from I to m nd using heorem TFI nd @PFIPA we get @TFQVA sn order to estlish the error estimtes of the (nite element penlty methodD we introdue the following qlerkin projetion

R h ; (X, M ) → X h D Q h : (X, M ) → M h de(ned y a(u -R h (u, p), v h ) -d(v h , p -Q h (u, p)) + d(u -R h (u, p), q h ) + ε(p -Q h (u, p), q h ) = 0 ∀(v h , q h ) ∈ X h × M h , @TFRPA for ll (u, p) ∈ X × M with divu + εp = 0F Lemma 6.1. nder the ssumptions @eIAD @ePAD c 0 ε ≤ 1 nd k 0 ∆t ≤ 1D the qlerkin projetion @R h , Q h A veri(es • ||u -R h (u, p)|| +h||A 1/2 (u -R h (u, p))|| + h||p -Q h (u, p)|| ≤ c ν h(||A 1/2 u|| + ||p||), ∀(u, p) ∈ X × M @TFRQA • ||u -R h (u, p)|| +h||A 1/2 (u -R h (u, p))|| + h||p -Q h (u, p)|| ≤ C ν h 2 (||Au|| + ||p|| 1 ), ∀(u, p) ∈ D(A) × (H 1 (Ω) ∩ M ) @TFRRA
with divu + εp = 0D where c ν is generl positive onstnt depending only on Ω nd νF roofF he qlerkin projetion hs its stility from @TFQSA nd @TFRPAD nd we hve

ν||A 1/2 R h (u, p)|| 2 ≤ ν 2 ||A 1/2 R h (u, p)|| 2 + c ν ||A 1/2 u|| 2 + ||p|| 2 @TFRSA ||Q h (u, p)|| ≤ c ν (||A 1/2 R h (u, p)|| + ||A 1/2 u|| + ||p||) @TFRTA
sing @TFRSAE@TFRTA nd the tringulr inequlityD yields to

||A 1/2 (u -R h (u, p))|| + ||p -Q h (u, p)|| ≤ c ν ||A 1/2 u|| + ||p|| @TFRUA
for ll (u, p) ∈ X × M with divu + εp = 0. vet onsider the dul prolemX (nd (Φ, Ψ) ∈ X × M suh tht for ll @vDqA∈ X × M a(v, Φ) + d(v, Ψ) -d(Φ, q) + ε(q, Ψ) = (v, u -R h (up)) @TFRVA fy using the vemm RFPD the prolem @TFRVA hs unique solution (Φ, Ψ) stisfying

||Au|| + ||Ψ|| 1 ≤ c ν ||u -R h (u, p)|| @TFRWA king (v, q) = (u -R h (u, p), p -Q h (u, p)
) in @TFRVA nd (v h , q h ) = (r h Φ, π h Ψ) in @TFRPA nd using @TFQQA nd @TFRWA we infer tht

|u -R h (u, p)| 2 = a(u -R h (u, p), Φ -r h Φ) + d(u -R h (u, p), Ψ -π h Ψ) -d(Φ -r h Φ, p -Q h (u, p)) + ε(p -Q h (u, p, Ψ -π h Ψ) ≤ c ν h(||A 1/2 (u -R h (u, p))|| + ||p -Q h (u, p)||)(||AΦ|| + ||ψ|| 1 ) ≤ c ν h(||A 1/2 (u -R h (u, p))|| + ||p -Q h (u, p)||)||u -R h (u, p)|| @TFSHA
he estimtion @TFRQA is onsequene of @TFSHA nd @TFRUA F e onsider (u, p) ∈ D(A) × H 1 (Ω) ∩ M with divu + εp = 0F sed on @TFQPAE@TFQQA nd @TFRPAD we dedue tht

||A 1/2 (r h u -R h (u, p))|| +||π h p -Q h (u, p)|| ≤ c ν ||A 1/2 (u -r h u)|| + ||π h p -p|| ≤ c ν h (||Au|| + ||p|| 1 )
hereforeD using the tringles inequlity nd @TFQQA yields to

||A 1/2 (u -R h (u, p))|| + ||p -Q h (u, p)|| ≤ c ν h (||Au|| + ||p|| 1 ) @TFSIA
gomining @TFSIA nd @TFSHA to otin

||u -R h (u, p)|| ≤ c ν h 2 (||Au|| + ||p|| 1 ) @TFSPA
gomining @TFSIAE@TFSPAD we otin the estimtion @TFRRA TFQF Error estimates.

sn this susetion we re foused on the error estimtes of (u n ε -u n εh , p n ε -p n εh ) nd then derive the optiml ound of the error (u(t n ) -u n εh , p(t n ) -p n εh )F Lemma 6.2. nder the ssumption of theorem TFQD we hve the following estimte

||u m ε -u m εh || 2 + ∆t m n=1 ||A 1/2 (u n ε -u n εh )|| 2 ≤ Ch 2 @TFSQA
for ll 1 ≤ m ≤ N roofF utrting @TFQTA from @TFQA with @vDqAa (v h , q h ) nd pplying @TFQPAD we get

((I -r h )d t u n ε + d t e n , v h ) + a((I -r h )u n ε + e n , v h ) -d(v h , (I -π h )p n ε + η n ) + d(e n , q h ) + b((I -r h )u n ε + e n , u n ε , v h ) + b(u n εh , (I -r h )u n ε + e n , v h ) +c α (u n ε , v h ) -c α (u n εh , v h ) +ε((I -π h )p n ε + η n , q h ) = 0, ∀(v h , q h ) ∈ X h × M h @TFSRA
where (e n , η n ) = (r h u n ε -u n εh , π h p n ε -p n εh )F etting (v h , q h ) = 2(e n , η n )∆t in @TFSRA nd using @PFRA nd @TFUAD we otin

||e n || 2 -||e n-1 || 2 + 2ν||A 1/2 e n || 2 ∆t + ε||η n || 2 ∆t + 2 b(e n , u n ε , e n )∆t +2 b((I -r h )u n ε , u n ε , e n )∆t + 2 b(u n εh , (I -r h )u n ε , e n )∆t +2c α (u n ε , e n )∆t -2c α (r h u n ε , e n )∆t + 2c α (u h u n ε , e n )∆t -2c α (u n εh , e n )∆t ≤ ||(I -π h )p n ε || 2 ∆t + 2d(e n , (I -π h )p n ε )∆t -2a((I -r h )u n ε , e n )∆t -2((I -r h )d t u n ε , e n )∆t
@TFSSA epplying @PFWA nd @TFQQAD we infer tht

2| b(e n , u n ε , e n )| ≤ c||e n || 2 L 4 ||A 1/2 u n ε || + c||A 1/2 e n |||u n ε || L 4 ||e n || L 4 ≤ ν 4 ||A 1/2 e n || 2 + d n ||e n || 2 2| b((I -r h )u n ε , u n ε , e n )| +2| b(u n εh , (I -r h )u n ε , e n )| ≤ c||A 1/2 (I -r h )u n ε || ||A 1/2 u n ε || + ||A 1/2 u n εh || ||A 1/2 e n || ≤ ν 4 ||A 1/2 e n || 2 + c ν h 2 ||Au n ε || 2 ||A 1/2 u n ε || 2 + ||A 1/2 u n εh || 2 ε||(I -π h )p n ε || 2 + 2|d(e n , (I -π h )p n ε )| ≤ ν 3 ||A 1/2 e n || 2 + c ν h 2 ||p n ε || 2 1 2|a((I -r h )u n ε , e n )| ≤ ν 12 ||A 1/2 e n || + c ν h 2 ||Au n ε || 2 2|((I -r h )d t u n ε , e n )| ≤ ν 12 ||A 1/2 e n || 2 + c ν h 2 ||A 1/2 d t u n ε || 2 2|c α (u n ε , e n ) -c α (r h u n ε , e n )| ≤ ν 12 ||A 1/2 e n || 2 + c ν h 2 ||Au n ε || 2 epling these inequlities in @TFSSA yields to ||e n || 2 -||e n-1 || 2 + ν||A 1/2 e n || 2 ∆t ≤ d n ||e n || 2 ∆t +c ν h 2 ||A 1/2 u n ε || 2 + ||A 1/2 u n εh || 2 ||Au n ε || 2 ∆t +c ν h 2 2||Au n ε || 2 + ||A 1/2 d t u n ε || 2 + |p n ε || 2 1 ∆t, ∀1 ≤ n ≤ N @TFSTA
umming this inequlity from I to m nd using theorem TFID theorem TFQ nd @PFIPA we dedue

||e m || 2 + ν∆t m n=1 ||A 1/2 e n || 2 ≤ ∆t m n=1
dn||e n || 2 + kh 2 @TFSUA sing @TFIRA nd disrete qronwll9s vemm to otin

||e m || 2 + ν∆t m n=1
||A 1/2 e n || 2 ≤ Ch 2 @TFSVA fy using @PFIPAD @TFQQA nd theorem TFI we get

||(I -r h )u m ε || 2 + ν∆t m n=1 ||A 1/2 (I -r h )u n ε || 2 ≤ ch 2 ||A 1/2 u n ε || 2 + ν∆t m n=1 ||Au n ε || 2 ≤ Ch 2
@TFSWA e end the proof of lemm TFI y omining @TFSVA nd @TFSWAF Lemma 6.3. nder the ssumptions of heorem TFQD the following error estimte

||A 1/2 (u m ε -u m εh )|| 2 + ∆t m n=1 ||p n ε -p n εh || 2 ≤ Ch 2 , ∀1 ≤ m ≤ N @TFTHA
holds roofF utrting @TFQTA from @TFQA with (v, q) = (v h , q h ) nd tking

e n = R n h -u n εh D η n = Q n h -p n εh D R n h = R h (u n ε , p n ε )D Q n h = Q h (u n ε , p n ε )D we get (d t e n , v h ) +a(e n , v h ) -d(v h , η n ) + d(d t e n , q h ) + ε(d t η n , q h ) + b(u n ε -R n h + e n , u n ε , v h ) + b(u n ε , u n ε -R n h + e n , v h ) -b(u n ε -u n εh , u n ε -u n εh , v h ) + a α (u ε , v h ) -c α (u n εh , v h ) = -(d t u n ε -R h (d t u n ε , d t p n ε ), v h ) @TFTIA for ll (v h , q h ) ∈ X h × M h D with 1 ≤ n ≤ N F etting (v h , q h ) = 2(d t e n , η n )∆t in @TFTIAD we otin 2||d t e n || 2 ∆t +ν ||A 1/2 e n || 2 -||A 1/2 e n-1 || 2 + ε(||η n || 2 -||η n-1 || 2 ) +2 b(u n ε -R n h + e n , u n ε , d t e n ) + 2 b(u n εh , u n ε -R n h + e n , d t e n )∆t -2 b(u n ε -u n εh , u n ε -u n εh , d t e n )∆t + c α (u n ε , d t e n )∆t -c α (R n h , d t e n )∆t +c α (R n h , d t e n )∆t -c α (u n εh , d t e n )∆t = -2(d t u n ε -R h (d t u n ε , d t p n ε ), d t e n )∆t
@TFTPA sing the ft tht (u n ε , p 0 ε ) = (u 0 , 0)D we estlish from @TFQQA nd @TFRHA tht

||A 1/2 (u 0 ε -R h (u 0 ε , p n ε ))|| + ||p 0 ε -Q h (u 0 ε , p 0 ε )|| ≤ c ν h||Au 0 || @TFTQA ||A 1/2 e n || + ||η n || ≤ ||A 1/2 (u 0 -r h u 0 ε )|| + ||A 1/2 (u 0 ε -R h (u 0 ε , p 0 ε ))|| ||Q h (u 0 ε , p 0 ε )|| ≤ c ν h||Au 0 || @TFTRA
epplying ones more @PFWAE@PFIIA nd @TFQRAD to hve

2| b(e n , u n ε , d t e n )| + 2| b(u n ε , e n , d t e n )| ≤ c||Au n ε ||||A 1/2 e n ||||d t e n || ≤ 1 4 ||d t e n || 2 + c||Au n ε || 2 ||A 1/2 e n || 2 2| b(u n ε -R n h , u n ε , d t e n )| +2| b(u n ε , u n ε -R n h , d t e n )| ≤ c||Au n ε ||||A 1/2 (u n ε -R n h )||||d t u n ε || ≤ 1 4 ||d t e n || 2 + c||Au n ε || 2 ||A 1/2 (u n ε -R n h )|| 2 2| b(u ε -u n εh u n ε -u n εh , d t e n )| ≤ c(h -1 ||A 1/2 (u n ε -u n εh )||||u n ε -u n εh || +||A 1/2 (u n ε -u n εh )|| 2 )||d t e n || ≤ 1 8 ||d t e n || 2 + c(h -2 ||A 1/2 (u n ε -u n εh )|| 2 ||u n ε -u n εh || 2 +||A 1/2 (u n ε -u n εh )|| 4 ) 2|c α (u n ε , d t e n ) -c α (R n h , d t e n )| ≤ 1 8 ||d t e n || 2 + c||A 1/2 (u n ε -R n h )|| 2 2|c α (u n εh , d t e n ) -c α (R n h , d t e n )| ≤ 1 8 ||d t e n || 2 + c||A 1/2 (u n εh -R n h )|| 2 2|(d t u n ε -R h (d t u n ε , d t p n ε ), d t e n )| ≤ 1 8 ||d t e n || 2 + ||d t u n ε -R h (d t u n ε , d t p n ε )|| 2
ustituting these inequlities in @TFTPA nd using vemm TFI results in

ν||A 1/2 e n || 2 -ν||A 1/2 e n-1 || 2 + ||d t e n || 2 ∆t + ε(||η n || 2 -||η n-1 || 2 ) ≤ c||Au n ε || 2 (||A 1/2 e n || 2 + h 2 ||Au n ε || 2 + h 2 ||p n ε || 2 1 )∆t +ch 4 (||Ad t u n ε || 2 + ||Ad t u n εh || 2 + ||d t p n ε || 2 )∆t +c(h -2 ||A 1/2 (u n ε -u n εh )|| 2 ||u n ε -u n εh || 2 + ||A 1/2 (u n ε -u n εh )|| 4 )∆t @TFTSA
sing theorem TFI nd TFQD vemm TFP nd the propriety

h 2 ≤ c 1 ∆t ≤ c 1 τ (t n )D we get ν||A 1/2 e n || 2 -ν||A 1/2 e n-1 || 2 + ||d t e n || 2 ∆t + ε(||η n || 2 -||η n-1 || 2 ) ≤ k(||A 1/2 e n || 2 + h 2 ||Au n ε || 2 + h 2 ||p n ε || 2 1 )∆t +ch 2 τ (t n )(||Ad t u n ε || 2 + ||Ad t u n εh || 2 + ||d t p n ε || 2 )∆t +k(||A 1/2 (u n ε -u n εh )|| 2 + ||A 1/2 (u n ε -u n εh )|| 4 )∆t @TFTTA
umming this inequlity from I to m nd using heorem TFI nd TFPD lemm TFQ nd @TFTRA yields to

ν||A 1/2 e m || 2 + ∆t m n=1 ||d t e n || 2 ≤ kh 2 , ∀1 ≤ m ≤ N @TFTUA fy pplying vemm TFID heorem TFID heorem TFP nd h 2 ≤ c 1 τ (t n )D we otin ν||A 1/2 (u n ε -R h (u n ε , p n ε ))|| 2 + ∆t m n=1 ||d t u n ε -R h (d t u n ε , d t p n ε )|| 2 ≤ c ν h 2 ||Au n ε || 2 + ||p n ε || 2 1 + c ν h 2 ∆t m n=1 τ (t n ) ||Ad t u n ε || 2 + ||d t p n ε || 2 1 ≤ kh 2 , ∀1 ≤ m ≤ N @TFTVA
gomining @TFTUA with @TFTVA leds to

||A 1/2 (u n ε -u n εh )|| 2 + ∆t m n=1 ||d t u n ε -d t u n εh || 2 ≤ kh 2 , ∀1 ≤ m ≤ N @TFTWA
xextD y @TFSRA @TFQSA nd @PFWA we get

||π h p n ε -p n εh || 2 ≤ c||d t u n ε -d t u n εh || + cν||A 1/2 (u n ε -u n εh )|| +c||A 1/2 (u n ε -u n εh )|| ||A 1/2 u n ε || + ||A 1/2 u n εh || +c||(I-)p n ε || @TFUHA
epplying theorem TFID theorem TFQ nd lemm TFI results for estlish from @TFUHA tht

||p n ε -p n εh || 2 ∆t ≤ c||d t u n ε -d t u n εh || 2 ∆t + +k||A 1/2 (u n ε -u n εh )|| 2 ∆t +c||(I -π h )p n ε || 2 ∆t @TFUIA
umming this inequlity from I to m nd using @TFTWAD @TFQQA nd heorem TFI we infer tht

∆t m n=1 ||p n ε -p n εh || 2 ≤ Ch 2 , ∀1 ≤ m ≤ N @TFUPA
edding the inequlities @TFUPA nd @TFTWA gives @TFTHAF gomining vemm TFQ with heorem TFQ gives the following optiml error estimte Theorem 6.4. nder the ssumption of heorem TFQD we hve the following error estimte

τ 2 (t m )||A 1/2 (u(t m ) -u m εh )|| 2 +∆ m n=1 τ 2 (t n )||p(t n ) -p n εh || 2 ≤ C(ε 2 + ∆t 2 + h 2 ), ∀1 ≤ m ≤ N @TFUQA
UF Numerical approximation and solution algorithms prom now on we will restrit ourselves to the pproximtion of optimiztion prolems involving (shE pssesF @UFRA he optiml position y ∈ Ω 0 is found through the following lgorithm using the omputtion of the ost funtion grdient nd the projetion funtionsF A fegin with ȳ ∈ Ω nd ε > 0 positive tolerneF A gompute the projetion d = P Ω0 (ȳ -η∇J(ȳ)) -ȳD where ∇J is the grdient of the ost funtion with η is positive onstnt hosen ording to prtiulr onditionsF A top if d a H @in prtieD d < εAF eA he(ne ỹ = ȳ + ξdD with ξ is the desent stepF qo to @A with ȳ = ỹF he vlue of J in this lgorithm is given y the ensuing disrete formul

Φ(y) = ∆t 2 N n=1 e∈Ti e u n,i -- → v 2 + σ e | curl(u n,i ) | 2 . @UFSA y = P Ω0 (z) is the projetion of z ∈ R 4 in Ω onto Ω 0 F st is omputed y minimizing qudrti funtion of the distne of z to Ω s follow min y∈Ω 1 2 y -z 2 = min y∈Ω 1 2 z T z -z T y + 1 2 y T y @UFTA whih is equivlent to              min (y1,y2,y3,y4) 4 i=1 1 2 z 2 i -z i y i + 1 2 y 2 i subject to a i ≤ y i ≤ b i , i = 1, ..., 4
y 3 -y 1 ≥ d 1 , y 2 -y 4 ≥ d 2 e seprte
the previous qudrti prolem into two qudrti prolems the (rst one for the sessesD the seond for the ordintes nd we write min (y1,y3)

1 2 (z 2 1 + z 2 3 ) -(z 1 y 1 + z 3 y 3 ) + 1 2 (y 2 1 + y 2 3 ) subject to a i ≤ y i ≤ b i , i = 1, 3 y 3 -y 1 ≥ d 1 , and
min (y2,y4) 1 2 (z 2 2 + z 2 4 ) -(z 2 y 2 + z 4 y 4 ) + 1 2 (y 2 2 + y 2 4 ) subject to a i ≤ y i ≤ b i , i = 2, 4 y 2 -y 4 ≥ d 2 ,
e urushEuuhnEuker @uuA tehnique is used to solve the two qudrti prolems oveF e egin y the (rst one nd we hoose a = a 1 = a 3 D nd b = b 1 = b 3 F e kept only three onstrints nd write the optimiztion prolem to e solved s min

(x1,x2) c -(l 1 x 1 + l 2 x 2 ) + 1 2 (x 2 1 + x 2 2 ) subject to x 2 ≤ b, x 1 ≥ a, x 2 -x 1 ≥ d 1 , where DD D d 1 D D nd l i D iaIDPD re rel numersF e hoose the following mtrix A =    0 -1 1 0 -1 1   D b =    -b a d 1    ,
hen the liner onstrints write

Ax -v = b, v ≥ 0, where v = (v 1 , v 2 v 3 )
T is the slk vrilesF he optimiztion prolem hs n unique solution due to the onvexity of the ost funtion in R 2 F his solution stis(es the uu onditions

v = Ax -b -l + x = A T λ, v ≥ 0, λ ≥ 0, v T λ = 0,
where l = (l 1 , l 2 , l 3 )D nd λ = (λ 1 , λ 2 , λ 3 ) is the vetor of the vgrnge multipliers ssoited to the three onstrints oveF he unique optiml solution writes

x = l A T λ whih gives   x 1 = l 1 + λ 2 -λ 3
x 2 = l 2 -λ 1 + λ 3 por omputing the vgrnge multipliers λ i D we solve the following liner omplementrity prolem

v = (-b + Al) + AA T λ v ≥ 0, λ ≥ 0 v T λ = 0
UFPF Numerical results. sn this susetion we present some results out the optimiztion of (shwy using (nite element method investigted in the previous setions to solve the stte prolem omined with projeted grdient lgorithm for the resolution of the optimiztion prolemF he numeril imE plementtion is rried out in the (nite element pkge preepemCCVD using the sprse liner solver for the liner systemsFhe omputtionl grids re uniformD unstruturedF P 2 EP 1 (nite elements hve een used for the sptil disretiztionF he domin )uid is onsidered s retngulr hnnel onsisting of ten poolsF snside eh poolD two 1es in form of s or v re uilt nd they re perpendiulr to the lterl oundryF he length nd width of 1es vry ording to the nture of the studied (shwy struturesF sn (gure QD we give desription of the (shwy strutureF Figure 3. ground pln ΩX eh pool is designed y dshed lines e ondut two QEh tests of (shwy design mostly used for the modeling of (shwy pssgeF pirstlyD we egin y QEh hnnel with s shped 1e @pigure P @leftAAF xextD we onsider v shped 1e design @pigure P @rightAAF he numeril simultion for the two tests is done with the sme physil nd numeril onditionsF ell initil nd oundry onditions re tken onstntsF the in)ow veloity is (0.1, 0, 0)D the trget veloity is u d = (0.8, 0, 0) to e s lose s possile to typil horizontl veloity suitle for (sh leping nd swimming pilitiesF he vortiity prmeter is σ = 0.3F he visosity is ν = 0.01F he exterior funtion re)eting the grvity e'et is f = (0, 0, 9.81)F he nonEslip oundry ondition ws pplied to ll wlls @ottomD lterl wlls nd 1esAF UFPFIF lot pishwy @QEh 4s4 shped 1eEretngulr slotsAF he stte penlized prolem @QFIAE@QFPA is solved y the (nite element method @TFQTA omined with spetrl projeted grdient lgorithm for the resolution of the shpe optimiztion prolem @SFQAF his test onsists to onsider QEh geometry of pools with two retngulr 1es s shown in the pigure RF he oming results gives desription of the veloity for ten pools relted to the ntures of the studied shpe of (shwy struturesF 5. snitil nd rndom shpe nd orresponding veloity for ten pools @aPHA Figure 6. yptiml shpe nd orresponding veloity for ten pools @aPHA e oserve tht the reirultion res ner of slots re removed for the optiml shpe @pigure TA omE pred to the initil rndom struture @pigure SAF he veloity is lmost null in the res ner from the oundries ehind slots for the optiml shpe whih provides rest zones for (sh efore to egin the pssge from the urrent pool to the next oneF hese results ppers lerly in the following exmples for the entrl pool @pigure U @rightAA UFPFPF lot pishwy @QEh 4v4 shped 1eEolique slotsAF sn this susetion we give results of numeril simultions in QEh dimension with v shpe 1eF e onsider hnnel of ten poolsF ih pool hs two vertil slotsF he smll one is olique nd the longest is in the form of v nd they re vertil to the lterl oundryF he di'erent vlues orresponding to the geometry of pools re shown in the (gure PA @leftAF en exmple of the (rst pool geometry is given in the pigure VF he following results represent the veloity for ten pools in the initil rndom shpe @pigure WA nd the optiml shpe @pigure IHAF he nture of the studied struture tkes prt of the minimiztion of turulene y the use of the form v of 1es whih re onsidered s rek wves of wterF he veloity is lmost lose from the uniform trget veloity u d for the optiml shpe @pigure IHAF Figure 7. nd rndom shpe @leftA shpe @rightA nd orresponding veloity for the entrl pool Figure 8. QEh exmple of (shwy with porous medium @leftAD wesh of the entrl pool @rightA Figure 9. xon optiml shpe nd orresponding veloity for ten pools @aPHA Figure 10. yptiml shpe nd orresponding veloity for ten pools @aPHA e give numeril results for the entrl pool in the ensuing (guresF urulene ppers lerly in the res ner from slots in the (gure II @leftAF por the optiml shpe @pigure II@rightAAD the nture of veloity permits suitle pssge for (sh nd provides for them rest in the zones where the veloity is lmost nullF 11. snitil rndom shpe @leftA yptiml shpe @rightA nd orresponding veloity for the entrl pool pigure IP represents the fst onvergene of the ost funtionl for the vrious porhheimer oe0ients for the v shped 1e designF he prolem of the optiml shpe design of vertil slot (shwys hs een ddressedF por this purposeD omplete penlty method hs een introdued nd di'erent properties were derivedF e (nite element method hs een used to disretize the )uid domin nd otin n error estimte for the veloity nd the pressure in term of H 1 nd L 2 norms respetivelyF he suitility nd omputtionl e0ieny of the proposed optimiztion methodology hs een suessfully demonstrted for QEh (shwys struturesF Appendix eF control problem of mixed boundary conditions por some numeril simultions nd in the se where the oth hirihletEtype nd xeumnnEtype oundry onditions re onsideredD we give the desription of the shpe derivtiveF e express the shpe grdients of the ost funtionls y introduing the orresponding liner djoint stte systemsF we de(ne the following funtions depend on the prmeter

l 1 (t) = 1 2 T 0 Ωt |u ε • T -1 t -u d | 2 dx + σ 2 T 0 Ωt |curl(u ε • T -1 t )| 2 dtdx, l 2 (t) = T 0 Ωt [ν∇(u ε • T -1 t ) : ∇(v • T -1 t ) + ((u ε • T -1 t ).∇)(u ε • T -1 t ).(v • T -1 t ) + a|u ε • T -1 t | α (u ε • T -1 t ).(v • T -1 t ) -(p ε • T -1 t )∇.(v • T -1 t ) -(f • T -1 t ).(v • T -1 t )]dtdx - T 0 Ωt (q • T -1 t )∇.(u ε • T -1 t )dtdx -ε T 0 Ωt (q • T -1 t )(p ε • T -1 t )dtdx
he vgrngin funtionl writes

L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t ) = l 1 (t) -l 2 (t)
e onsider V(0, X) ∈ V ad D nd oserve tht V(0, X) = VF herefore we n derive the shpe grdient using the formul @SFVA [(-∇u ε .V) t v + (-∇v.V) t u ε + ν∇(-∇u ε .V) : ∇v + ν∇u ε : ∇(-∇v.V) + ((-∇u ε .V).∇u ε ).v + (u ε .∇(-∇u ε .V).v + (u ε .∇u ε ).(-∇v.V) [ν∇u ε : ∇v + (u ε .∇u ε ).v + a|u ε | α u ε .v -p ε ∇.v -q∇.u ε ]V.n dtds @eFWA epling @eFVA nd @eFWA in @eFUA nd using the ft tht (u ε , p ε ) is solution of @IFPA nd (v, q) is solution of @eFQA respetively yields dJ(Ω; V) = ∂v ∂n (V.n) = (∇u ε : ∇v)V.n @eFIQA ustituting @eFIIAD @eFIPA nd @eFIQA in @SFIQA the shpe derivtive tkes the form

d dt L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t )| t=0 = l 1 (0) -l 2 (
dJ(Ω; V) = T 0 Γ1

  w s , e) + ν(∆w, e) + (∇φ, e) = d ds (w, e) -(e s , w) -ν(∇e, ∇w) + (∇φ, e) = d ds (w, e) + (∇q, w) -(φ, ∇.e) = d ds (w, e) -ε(φ, p ε ) e integrte from 0 to tD nd the ft w(t) = e(0) = 0 gives t 0 ||e|| 2 ds = -

Figure 1 .Figure 2 . 1 y 2 - 1 = a 3 a 2 = a 4 = 0D b 2 = b 4

 1212132424 Figure 1. hemti desription of (shwy struture he geometry of the vertil slot sed on the use of guide elements to led smooth hydruli )ow into the next slotF he positioning of the guide elements ws rried out t two di'erent lotionsD a nd bD whih on(gure the shpe of the (sh ldder Ω @pigure PAF he design vriles a nd b re sujet to onstrints in order to ensure positive in)uene on the )ow in the individul poolsF hese onstrints re formulted s 1 4 1.213 ≤ y 1 , y 3 ≤ 3 4 1.213 0 ≤ y 2 , y 4 ≤ 1 4 0.97

Figure 4 .

 4 Figure 4. of entrl

Figure

  Figure 5. snitil nd rndom shpe nd orresponding veloity for ten pools @aPHA

Figure 12

 12 Figure 12. gonvergene history

(

  u d ).(-∇u ε .V)dx + σ T 0 Ω curl(u ε )curl(-∇u ε .V)dx |u ε -u d | 2 V.n)ds +

1 2 ∇

 12 .(-∇u ε .V)u ε .v + 1 2 ∇.u ε (-∇u ε .V).v + 1 2 ∇.u ε u ε .(-∇v.V) + aα|u ε | α-2 (-∇u ε .V.u ε )(u ε .v) + a|u ε | α (-∇u ε .V).v + a|u ε | α u ε .(-∇v.V) -(f.V).v -f.((-∇v).V) -(∇p ε .V)∇.v -p ε ∇.(-∇v.V) -q∇.(-∇u ε .V) -(∇.u ε )(-∇q.V)]dxsing qreen9s formul nd the ondition u = 0 Γ 1 D we otinl 2 (0) = -ε t -ν∆u ε + (u ε .∇)u ε + 1 2 ∇.u ε u ε + a|u ε | α u ε + ∇p ε -f ).(∇v.V)]dtdx + t -ν∆v + (∇u ε ) T .v -(u ε .∇)v -u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) -∇q].(∇u ε .V)dtdx u ε .n)v -nq].(∇u ε .V)dtds -

-

  εu d | 2 + σ|curl(u ε )| 2 )V.n)ds + np ε ].(∇v.V)dtds -(u ε ) ∧ n))V.n dtds @eFIHA xoting u ε = 0 nd v = 0 in Γ 1 to hve n.(∇u ε .V) = ∇u ε .(n ⊗ n).V.n = ∇u ε .n.n(V.n) = (∇.u ε )(V.n) = 0, ∀x ∈ Γ 1 @eFIIA ∂v ∂n .(∇u ε .V) = ∇u ε .(n ⊗ n).V. n) = (∇u ε : ∇v)V.n @eFIPAimilrly we otin n.(∇v.V) = 0, ∂u ε ∂n .(∇v.V) = ∂u ε ∂n .

eFIF Adjoint sn this setion we n djoint eqution relted to the eqution

@eFIA

where ∂Ω = Γ 1 ∪ Γ 2 F e keep the sme de(nition of the ost funtion @SFPAF he optimiztion prolem n e expressed s min

× Y e solution of the penlized prolem @eFIAF hen the djoint eqution ssoited to the eqution @eFIA tkes the form

roofF e de(ne lso the following funtion

we (rst derive L with respet to the stte vrile p in ny diretion p ∈ M (Ω), we otin

the vrition p is ritrryD we get ∇.v + εq = 0 on Ω. @eFSA xextD we derive L with respet of the stte vrile u ε in the ritrry diretion ũ ∈ V 0 (Ω) we hve

gonsidering n ritrry diretion ũ whih vnishes in neighorhood of the oundry Γ 2 D we otin

pinlly we otin the djoint eqution

@eFTA eFPF Shape gradient. sn this susetionD we will try to express the shpe grdient using the veloity method @see gé IAD vi the stte nd djoint prolemsF vet us (nd n expression of the derivtive of the sddle point prolem j@tA with respet to t where

with (u, p) nd (v, q) re solutions of @IFIA nd @eFQA in the pertured domin Ω t D respetivelyF e onsider the rilert spes whih depend on the prmeter t de(ned y

sine T t nd T -1 t re di'eomorphismsD the prmetristion do not in)uene j@tAF end we hve