Very weak solutions for the stationary Oseen and Navier–Stokes equations

Ve y weak solu ions o he s a iona y Oseen and Na ie -S okes equa ions Ch´e i Am ouche a, Ma ´ıa ´ Angeles Rod ´ıguez-Bellido b,1 aLabo a oi e de Ma h´ema iques Appliqu´ees, CNRS UMR 5142, Uni e si ´e de Pau e des Pays de l’Adou , IPRA, A enue de l’Uni e si ´e- 64000 Pau (F ance) bDp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Ap do. de Co eos 1160 - 41080 Se illa (Spain) Recei ed *****; accep ed a e e ision +++++ P esen ed by Abs ac We conside he s a iona y Oseen and Na ie -S okes equa ions in a bounded domain o class C1,1o R3. He e we gi e a new and simple p oo o he exis ence o e y weak solu ions (u,q)∈Lp(Ω)×W−1,p(Ω) co esponding o bounda y da a in W−1/p,p(Γ). These solu ions a e ob ained wi hou imposing smallness assump ions on he ex e io o ces. We also ob ain egula i y esul s in ac ional Sobole spaces. To ci e his a icle: C. Am ouche, M. A. Rod ´ıguez-Bellido, C. R. Acad. Sci. Pa is, Se . I 340 (2005). R´esum´e Solu ions `es aibles pou les ´equa ions s a ionnai es d’Oseen e de Na ie -S okes. Nous consid´e ons les ´equa ions s a ionnai es d’Oseen e de Na ie -S okes dans un ou e bo n´e connexe e de classe C1,1de R3. Nous donnons ici une nou elle p eu e plus simple de l’exis ence de solu ions `es aibles (u,q)∈Lp(Ω)×W−1,p(Ω) co espondan `a des donn´ees au bo d dans W−1/p,p(Γ). Ces solu ions son ob enues sans hypo h`ese de pe i esse des o ces ex ´e ieu es. On ob ien aussi des ´esul a s de ´egula i ´e dans des espaces de Sobole ac ionnai es. Pou ci e ce a icle : C. Am ouche, M. A. Rod ´ıguez-Bellido, C. R. Acad. Sci. Pa is, Se . I 340 (2005). Ve sion an¸caise ab ´eg´ee L’obje de ce e no e consis e essen iellemen `a ´e udie l’exis ence de solu ions `es aibles (u,q)∈ Lp(Ω)×W−1,p(Ω) pou les ´equa ions d’Oseen (O) e de Na ie -S okes (NS). L’une des difficul ´es consis e Email add esses: [email p o ec ed] (Ch´e i Am ouche), ang[email p o ec ed] (Ma ´ıa ´ Angeles Rod ´ıguez-Bellido). 1Pa ially suppo ed by M.E.C. (Spain), P ojec MTM2006-07932, and by Jun a de Andaluc´ıa, P ojec P06-FQM-02373. P ep in submi ed o he Acad´emie des sciences 28 oc ob e 2009 `a donne un sens aux condi ions aux limi es de Di ichle . Le e me de con ec ion end les choses plus difficiles pou le p obl`eme (O) e complique s´e ieusemen la si ua ion pou l’´e ude du p obl`eme non lin´eai e (NS). Les ´esul a s conce nan l’exis ence de solu ions `es aibles son donn´es dans le h´eo `eme 2.4 pou (O) e les h´eo `ems 3.1 e 3.2 pou (NS). Les au es ´esul a s conce nen la ´egula i ´e de elles solu ions sous des hypo h`eses ad´equa es. Nons consid´e ons en pa iculie le cas o`u les donn´ees e donc les solu ions appa iennen `a des espaces de Sobole ac ionnai es. 1. In oduc ion Le Ωbe a bounded connec ed open se o R3o class C1,1wi h bounda y Γ. We a e in e es ed in some ques ions conce ning he s a iona y Oseen and Na ie -S okes equa ions, ha gene ally can be w i en as: (O)−∆u+ ·∇u+∇q= and ∇·u=hin Ω,u=gon Γ, (NS)−∆u+u·∇u+∇q= and ∇·u=hin Ω,u=gon Γ, whe e udeno es he eloci y ield and q he p essu e, bo h being unknown, and ,h,gand a e gi en. In he case o incomp essible luids, h= 0, i has been well-known since Le ay [6] ha i ∈W−1,p(Ω) and g∈W1−1/p,p(Γ) wi h p≥2, Γia e he connec ed componen s o he bounda y Γ,i=0,...,I, and �Γi g·ndσ=0,∀i=0, ..., I, (1) hen he e exis s a solu ion (u,q)∈W1,p(Ω)×Lp(Ω) sa is ying (NS). Se e p o ed [8] he exis ence o weak solu ion (u,q)∈W1,p(Ω)×Lp(Ω) o any 3 2<p<2whenh= 0 and gsa is ies he abo e condi ions. Recen ly, Kim [5] imp o es Se e’s exis ence and egula i y esul s on weak solu ions o (NS) o any 3 2≤p<2, when Γis connec ed (I= 0) p o ided his small in an app op ia e no m (due o (2), see below, gis also small in he co esponding app op ia e no m). Exis ence o e y weak solu ions (u,q)∈L3(Ω)×W−1,3(Ω), o h= 0, a bi a y la ge ∈H−1(Ω) and la ge g∈L2(Γ), wi hou assuming condi ion (1), was p o ed i s by Ma usic-Paloka in [7] (see Theo em 5) wi h Ωa bounded simply-connec ed open se o class C1,1. Bu he p oo o Theo em 5 becomes co ec only i ei he condi ion (1) o condi ion (12) hold. The same esul was p o ed by Kim [5] o a bi a y la ge ex e nal o ces ∈[W1,3/2 0(Ω)∩W2,3(Ω)]�, o small h∈[W1,3/2(Ω)]�and g∈W−1/3,3(Γ), wi h Γ supposed connec ed (I= 0). Obse e ha he space chosen o hand o a e no co ec ei he and led us o some e os (in pa icula , he equi alence gi en in Theo em 5 he e does no wo k). The pu pose o ou wo k is o gene alize he heo y o e y weak solu ions o he Di ichle p oblem om he S okes equa ions o he Oseen and Na ie -S okes equa ions, de ining igo ously he aces o he ec o unc ions which a e li ing in subspaces o Lp(Ω) (see [1], [2]), and he spaces o he da a. We p o e exis ence and egula i y o e y weak solu ions (u,q)∈Lp(Ω)×W−1,p(Ω) o Oseen equa ions o any p∈(1,+∞) wi h a bi a y la ge da a in Sobole spaces o nega i e o de . In he Na ie -S okes case, he exis ence o e y weak solu ion is p o ed o a bi a y la ge ex e nal o ces, bu wi h a smallness condi ion o bo h hand g. Uniqueness o e y weak solu ions is also p o ed o small enough da a. The de ailed p oo s o he esul s announced in his No e a e gi en in [2]. 2. Oseen Equa ions Fo any 1 < ,p<∞, we de ine he spaces: Hp(Ω) = { ∈Lp(Ω); ∇· =0},X ,p(Ω) = {ϕ∈ W1, 0(Ω); ∇·ϕ∈W1,p 0(Ω)}, and Tp, (Ω) = { ∈Lp(Ω); ∆ ∈(X �,p�(Ω))�}, endowed wi h he opology 2 gi en by he no m � �Tp, (Ω)=� �Lp(Ω)+�∆ �[X �,p�(Ω)]�. As o he Na ie -S okes sys em, we can p o e ha i ∈H−1(Ω), ∈H3(Ω), h∈L2(Ω) and g∈H1/2(Γ) wi h hand g e i ying he compa ibili y condi ion �Ω h(x)dx=�Γ g·ndσ,(2) hen he p oblem (O) has a unique solu ion (u,q)∈H1(Ω)×L2(Ω)/R e i ying he ollowing es ima e: �u�H1(Ω)≤C�� �H−1(Ω)+�1+� �L3(Ω)�(�h�L2(Ω)+�g�H1/2(Γ))�. Theo em 2.1 (S ong solu ions) Conside p≥6 5, ∈Lp(Ω),h∈W1,p(Ω), ∈Hs(Ω)and g∈ W2−1/p,p(Γ),wi hs=3i p<3,s=pi p>3, o s=3+εi p=3, o some a bi a y ε>0, and sa is ying he compa ibili y condi ion (2). Then, he unique solu ion o (O) e i ies (u,q)∈W2,p(Ω)× W1,p(Ω). Mo eo e , he e exis s a cons an C>0such ha �u�W2,p(Ω)+�q�W1,p(Ω)/R≤C�1+� �Ls(Ω)��� �Lp(Ω)+�1+� �Ls(Ω)���h�W1,p(Ω)+�g�W2−1/p,p(Γ)��. P oo : Fi s , le (u,q)∈H1(Ω)×L2(Ω)/Rbe he unique solu ion o P oblem (O). Fo a gi en λ∈D(Ω) (λ>0) such ha ∇· λ= 0 and � λ− �Ls(Ω)≤λ,le (uλ,q λ)∈W2,p(Ω)×W1,p(Ω) be he unique solu ion o he p oblem (Oλ): −∆uλ− λ·∇uλ+∇qλ= and ∇·uλ=hin Ω,uλ=gon Γ(use he S okes egula i y and a boo s ap a gumen ). Secondly, we ocus on he ob en ion o a s ong es ima e o (uλ,q λ). I � is he ex ension by ze o o o R3and ρε he classical molli ie , we conside λ= ε 1+ ε λ,2whe e ε 1=� �ρ ε/2, ε λ,2= λ−� �ρ ε/2 o ε>0,and 0 <λ<ε/2.(3) By egula i y es ima es o he S okes p oblem, we ha e �uλ�W2,p(Ω)+�qλ�W1,p(Ω)/R≤C(� �Lp(Ω)+�h�W1,p(Ω)+�g�W2−1/p,p(Γ)+� λ·∇uλ�Lp(Ω)).(4) In o de o es ima e he e m � λ·∇uλ�Lp(Ω), we use (3) and Sobole embeddings. Fi s : � ε λ,2·∇uλ�Lp(Ω)≤� ε λ,2�Ls(Ω)�∇uλ�Lk(Ω)≤Cε�uλ�W2,p(Ω),wi h 1 k=1 p−1 s.(5) Fo he es ima e on ε 1, we conside wo cases: I p≤2, le ∈]3,∞] be such ha 1 p=1 +1 2, and ≥1 such ha 1+1 =1 3+1 sa is ying: � ε 1·∇uλ�Lp(Ω)≤� ε 1�L (Ω)�∇uλ�L2(Ω)≤� �L3(Ω)�ρε/2�L (R3)�∇uλ�L2(Ω). Using he es ima e (5), we deduce om (4) ha �uλ�W2,p(Ω)+�qλ�W1,p(Ω)/R≤C�1+� �L3(Ω)�(� �Lp(Ω)+�1+� �L3(Ω)�(�h�W1,p(Ω)+�g�W2−1/p,p(Γ))). I p>2, using he compac embedding W2,p(Ω)�→W1,q(Ω), wi h q<p ∗, o any ε�>0, we known ha he e exis s Cε�>0 such ha �∇uλ�Lq(Ω)≤ε��uλ�W2,p(Ω)+Cε��uλ�H1(Ω).Conside ing he case p<3 and hen he case p≥3, we can choose he exponen qand ix ε>0 and ε�>0 small enough o ob ain �uλ�W2,p(Ω)+�qλ�W1,p(Ω)/R≤C�� �Lp(Ω)+�h�W1,p(Ω)+�g�W2−1/p,p(Γ) +Cε�� �Ls(Ω)�ρε/2�L (Ω)�� �Lp(Ω)+�1+� �Ls(Ω)�(�h�W1,p(Ω)+�g�W2−1/p,p(Γ))��. Thus, we deduce ha he e exis s a sequence o eal numbe s kλsuch ha (uλ,q λ+kλ) con e ges weakly in W2,p(Ω)×W1,p(Ω) o (u,q), solu ion o P oblem (O) wi h he co esponding es ima e. Theo em 2.2 Le ∈W−1,p(Ω), ∈H3(Ω),h∈Lp(Ω)and g∈W1−1/p,p(Γ) e i y he compa ibili y condi ion: �Ω h(x)dx=�g·n,1�W−1/p,p(Γ)×W1/p,p�(Γ).(6) 3 Then, he p oblem (O)has a unique solu ion (u,q)∈W1,p(Ω)×Lp(Ω)/R. Mo eo e , he e exis s some cons an C>0such ha , o α=1i p≥2and α=1+� �L3(Ω)i p<2, we ha e �u�W1,p(Ω)+�q�Lp(Ω)/R≤C�1+� �L3(Ω)�2�� �W−1,p(Ω)+α�h�Lp(Ω)+�g�W1−1/p,p(Γ)�.(7) Ske ch o he p oo : We spli i in wo cases. I p≥2, we decompose he solu ion (u,q) as (z,θ)+(u0,q 0), being (u0,q 0)∈W1,p(Ω)×Lp(Ω) sa is ying −∆u0+∇q0= and ∇·u0=hin Ω,u0=gon Γ, and (z,θ)∈W2, (Ω)×W1, (Ω) sa is ying −∆z+ ·∇z+∇θ=− ·∇u0and ∇·z= 0 in Ω,z=0on Γ, whe e 1 =1 3+1 p. The co esponding es ima es (see Theo em 2.1) and he embedding W2, (Ω)�→W1,p(Ω) conclude he p oo in his case. Secondly, i p<2, we a e able o conclude by a duali y a gumen . Rema k 1 Es ima e (7) can be imp o ed o p∈[6 5,6], and o any p>1 i ·n= 0 on Γas: �u�W1,p(Ω)+�q�Lp(Ω)/R≤C�1+� �L3(Ω)��� �W−1,p(Ω)+�1+� �L3(Ω)���h�Lp(Ω)+�g�W1−1/p,p(Γ)��. Co olla y 2.3 Conside 1<p<6/5and ∈Lp(Ω), ∈H3(Ω),h∈W1,p(Ω)and g∈W2−1/p,p(Γ) e i ying he compa ibili y condi ion (6). Then, he solu ion gi en by Theo em 2.2 sa is ies (u,q)∈ W2,p(Ω)×W1,p(Ω)and he ollowing es ima e holds: �u�W2,p(Ω)+�q�W1,p(Ω)/R≤C�1+� �L3(Ω)��� �Lp(Ω)+�1+� �L3(Ω)���h�W1,p(Ω)+�g�W2−1/p,p(Γ)��. Using he p e ious esul s and ollowing a gumen s in [2], we ob ain: Theo em 2.4 (Ve y weak solu ion o Oseen equa ions) Le ∈(X �,p�(Ω))�,h∈L (Ω),g∈ W−1/p,p(Γ),wi h1 =1 p+1 s, be gi en, sa is ying he compa ibili y condi ion (6), and ∈Hs(Ω)wi h s=3i p>3/2,s=p�i p<3/2, o s=3+εi p=3/2.Then, he Oseen p oblem (O)has a unique solu ion (u,q)∈Tp, (Ω)×W−1,p(Ω)/R e i ying he es ima es �u�Tp, (Ω)≤C�1+� �Ls(Ω)��� �[X �,p�(Ω)]�+�h�L (Ω)+�g�W−1/p,p(Γ)�,(8) �q�W−1,p(Ω)/R≤C�1+� �Ls(Ω)�2�� �[X �,p�(Ω)]�+�h�L (Ω)+�g�W−1/p,p(Γ)�. Conce ning he egula i y o solu ions o he Oseen equa ions in ac ional Sobole spaces, we ob ain: Theo em 2.5 (Regula i y o Oseen equa ions) Conside σ∈(1/p, 2]. Le ∈Wσ−2,p(Ω),h∈ Wσ−1,p(Ω),g∈Wσ−1/p,p(Γ)be gi en sa is ying he compa ibili y condi ion (6), and ∈Hs(Ω)wi h sas in Theo em 2.4. Then, he Oseen p oblem (O) has a unique solu ion (u,q)∈Wσ,p(Ω)×Wσ−1,p(Ω)/R sa is ying �u�Wσ,p(Ω)+�q�Wσ−1,p(Ω)/R≤C(� �Wσ−2,p(Ω))+�h�Wσ−1,p(Ω)+�g�Wσ−1/p,p(Ω)). 3. Na ie -S okes Equa ions Now, we p esen wo heo ems gi ing exis ence o e y weak solu ions o he Na ie -S okes equa ions in L3(Ω)×W−1,3(Ω), i s one o he small da a case, and second one o a bi a y la ge bu hand g small enough in a domain possibly mul iply-connec ed. Theo em 3.1 (Ve y weak solu ion o Na ie -S okes, small da a case) Le ∈(X3,3/2(Ω))�,h∈ L3/2(Ω)and g∈W−1/3,3(Γ) e i y (6). Then, i) he e exis s a cons an α1>0such ha , i � �[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)≤α1, hen, he e exis s a e y weak solu ion (u,q)∈L3(Ω)×W−1,3(Ω) o p oblem (NS) e i ying he es ima es �u�L3(Ω)≤C�� �[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)�,(9) �q�W−1,3/R≤C1� �[X3,3/2)]�+ 2(1 + C2)C�� �[X3,3/2]�+�h�L3/2+�g�W−1/3,3�,(10) 4 whe e C>0is he cons an gi en in (8), α1= min �(2C)−1,(2C2)−1�, and C1and C2cons an s o Sobole embeddings. ii) Mo eo e , he e exis s a cons an α2∈]0,α 1]such ha i � �[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)≤ α2, hen his solu ion is unique, up o an addi i e cons an o q. P oo : We p o e exis ence o a e y weak solu ion by applying Banach’s ixed poin heo em o e he Oseen equa ions. Indeed, le T:H3(Ω)→H3(Ω) be he applica ion de ined as �→ T =u, whe e uis he unique solu ion o (O) p o ided by Theo em 2.4. We se B ={ ∈H3(Ω); � �L3(Ω)≤ }. We will p o e ha he e exis s θ∈]0,1[ such ha �T 1−T 2�L3(Ω)=�u1−u2�L3(Ω)≤θ� 1− 2�L3(Ω).(11) In o de o es ima e �u1−u2�L3(Ω), we obse e ha o each i=1,2, (ui,q i) is he solu ion o −∆ui+ i·∇ui+∇qi= and ∇·ui=hin Ω,ui=gon Γ, wi h he es ima es �ui�L3(Ω)≤C�1+� i�L3(Ω)��� �[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ)�, being C>0 he cons an gi en in (8). Howe e , o es ima e he diffe ence u1−u2, we ha e o a gue diffe en ly. Conside he p oblem ul illed by (u,q)=(u1−u2,q 1−q2), which is −∆u+ 1·∇u+∇q= − ·∇u2and ∇·u= 0 in Ω,u=0on Γ, whe e u1=T 1,u2=T 2and = 1− 2. Using he e y weak es ima es (8) o he Oseen p oblem successi ely o uand o u2, we ob ain ha �u�L3(Ω)≤C�1+� 1�L3(Ω)��( ·∇)u2�[X3,3/2(Ω)]�≤C2β�1+� 1�L3(Ω)��1+� 2�L3(Ω)�� �L3(Ω), whe e β=� �[X3,3/2(Ω)]�+�h�L3/2(Ω)+�g�W−1/3,3(Γ). Thus, we ob ain es ima e (11) i we con- side C2β(1 + )2<1, and (9)-(10) hold o C1 he con inui y cons an o he Sobole embedding [X3,3/2(Ω)]��→W−2,3(Ω) and C2 he con inui y cons an o he Sobole embedding W1,3/2 0(Ω)�→L3(Ω). The uniqueness esul is a simple consequence o Sobole embeddings and he S okes es ima es. Theo em 3.2 (Ve y weak solu ion o Na ie -S okes, a bi a y o ces) Le ∈(X3,3/2(Ω))�, h∈L3/2(Ω)and g∈W−1/3,3(Γ)be gi en, and sa is ying he compa ibili y condi ion (6). The e ex- is s a cons an δ>0(depending only on Ω) such ha he p oblem (NS)has a e y weak solu ion (u,q)∈L3(Ω)×W−1,3(Ω)i �h�L3/2(Ω)+ i=I � i=0 |�g·n,1�Γi|≤δ.(12) Ske ch o he p oo : We decompose (NS) in o wo p oblems. One sys em, deno ed (NS1), o small da a: −∆ ε+ ε·∇ ε+∇q1 ε= − ε,∇· ε=h−hεin Ω,and ε=g−gεon Γ. wi h ε>0 and he (NS2) sys em: −∆zε+zε·∇zε+zε·∇ ε+ ε·∇zε+∇q2 ε= ε,∇·zε=hεin Ω,zε=gεon Γ. whe e ε∈H−1(Ω),h ε∈L2(Ω) and gε∈H1/2(Γ) sa is y � − ε�[X3,3/2(Ω)]�+�h−hε�L3/2(Ω)+�g−gε�W−1/3,3(Γ)≤εand �hε�L3/2(Ω)+ i=I � i=0 |�gε·n,1�Γi|≤2δ, (he e, we ha e used densi y a gumen s). Finally, we use an ex ension o Hop ’s lemma: (see [3], Rema k VIII.4.4 o ins ance) o any α>0, he e exis s yε∈H1(Ω), depending on α, such ha o C1>0 depending only on Ω,∇·yε=hεin Ω,yε=gεon Γand o any w∈H1 0(Ω), �����Ω (w·∇)yε·wdx����≤�α+�hε�L3/2(Ω)+C i=I � i=0 |�gε·n,1�Γi|��w�2 H1(Ω)≤(α+2C1δ)�w�2 H1(Ω). 5 To inish, we p o e some egula i y esul s on e y weak solu ions o he Na ie -S okes equa ions by using he egula i y esul s o he S okes and Oseen p oblems. Theo em 3.3 (Regula i y o Na ie -S okes equa ions) Le (u,q)∈L3(Ω)×W−1,3(Ω)be he so- lu ion gi en by Theo em 3.2. Then, he ollowing egula i y esul s hold: i) I ∈(X �,p�(Ω))�,h∈L (Ω)and g∈W−1/p,p(Γ), wi h 1 ≤1 p+1 3and max{ , 3}≤p, hen (u,q)∈ Lp(Ω)×W−1,p(Ω). ii) Conside ≥3/2, ∈W−1, (Ω),h∈L (Ω)and g∈W1−1/ , (Γ). Then (u,q)∈W1, (Ω)×L (Ω). iii) Fo ∈(1,+∞), i ∈L (Ω),h∈W1, (Ω)and g∈W2−1/ , (Γ), hen (u,q)∈W2, (Ω)×W1, (Ω). i ) Suppose ha 3/2≤p≤3, =∇·F0+∇ 1 o F0∈Wσ, (Ω)and 1∈Wσ−1,p(Ω),h∈Wσ, (Ω), and g∈Wσ−1/p,p(Γ), wi h σ=3 p−1,1 ≤1 p+1 3and ≤p. Then (u,q)∈Wσ,p(Ω)×Wσ−1,p(Ω). ) Le σbe such ha 1/p<σ≤1and σ≥3/p −1. Suppose ha ∈Wσ−2,p(Ω),h∈Wσ−1,p(Ω), and g∈Wσ−1/p,p(Γ). Then (u,q)∈Wσ,p(Ω)×Wσ−1,p(Ω). Rema k 2 i) Poin i) shows in pa icula ha o any p≥3, i ∈W−1, (Ω) and g∈W1−1/ , (Γ), wi h 3p 3+p≤ ≤p, and �Γig·n= 0 o any i=1,...,I and h= 0, hen P oblem (NS) has a solu ion (u,q)∈Lp(Ω)×W−1,p(Ω). Se e [8] p o es ha o any 3/2< <2 (and hen o >3/2), i ∈W−1, (Ω), g∈W1−1/ , (Γ), h= 0 and (1) is e i ied o any i=0,...,I, hen (NS) has a solu ion (u,q)∈W1, (Ω)×L (Ω). Poin ii) p o es ha his esul holds i =3/2 wi hou assuming ho he lux g h ough Γi o be equal o 0. Ac ually, i suffices o assume he smallness condi ion (12). ii) F om ela ion (6), condi ion (12) is au oma ically ul illed when he no m �h�L3/2(Ω)is small enough and I= 0, ha means ha he bounda y Γis connec ed, which is he case conside ed by Kim [5]. iii) Ma usic-Paloka [7] p o es Theo em 3.2 wi h ∈H−1(Ω)⊆(X3,3/2(Ω))�,h= 0 and g∈L2(Γ)⊆ W−1/3,3(Γ) wi h �g�L2(Γ)small, in a domain Ωsimply-connec ed. In ac , he solu ion u∈L3(Ω) ob ained in [7] is mo e egula and belongs o H1/2(Ω) by poin i ) wi h p= 2. i ) Galdi e al. [4] p o e Theo em 3.2 and Theo em 3.3 poin i) wi h = di F0,F0∈L (Ω), h∈Lp(Ω) and g∈W−1/p,p(Γ) wi h 1 ≤1 p+1 3and max{2 , 3}≤p, in a domain Ωo class C2,1, assuming ha , hand ga e small enough in hei espec i e no ms. The smallness condi ion on is in ac unnecessa y. Re e ences [1] C. Am ouche and V. Gi aul ,Decomposi ion o ec o spaces and applica ion o he S okes p oblem in a bi a y dimension, Czechoslo ak Ma hema ical Jou nal, 44, 119 (1994), pp. 109–140. [2] C. Am ouche and M. A. Rod ´ ıguez-Bellido,S okes, Oseen and Na ie -S okes equa ions wi h singula da a, Submi ed. [3] G. P. Galdi,An In oduc ion o he Ma ema ical Theo y o he Na ie -S okes Equa ions, Vol 2: Nonlinea S eady P oblems. Sp inge T ac s in Na u al Philosophy, ol. 39. Sp inge , New Yo k (1994). [4] G. P. Galdi, C. G. Simade and H. Soh ,A class o solu ions o s a iona y S okes and Na ie -S okes equa ions wi h bounda y da a in W−1/q,q,Ma h. Ann. , 331 (2005), pp. 41–74. [5] H. Kim, Exis ence and egula i y o e y weak solu ions o he s a iona y Na ie -S okes equa ions, A ch. Ra ional Mech. Anal., 193 (2009), 117-152. [6] J. Le ay, E ude de di es es ´equa ions in ´eg ales non lin´eai es e de quelques p obl`emes que pose l’hyd odynamique, J. Ma h. Pu es Appl. 12 (1933), pp. 1–82. [7] E. Ma usiˇ c-Paloka,Sol abili y o he Na ie -S okes sys em wi h L2bounda y da a, Appl. Ma h. Op im. 41 (2000), pp. 365-375. [8] D. Se e,´ Equa ions de Na ie -S okes s a ionnai es a ec donn´ees peu ´eguli`e es, Ann. Sc. No m. Sup. Pisa 10-4, (1983), pp. 543-559. 6