Existence and uniqueness results for a coupled problem related to the stationary Navier-Stokes System

Exis ence and Uniqueness Resul s o a Coupled P oblem Rela ed o he S a iona y Na ie -S okes Sys em B. Climen and E. Fe n´ andez-Ca a ∗ June 13, 2007 Abs ac In his pape , we conside some sys ems which a e close o he s a ion- a y Na ie -S okes equa ions. The s uc u e o hese sys ems is he ollow- ing: An N-dimensional equa ion o mo ion, he incomp essibili y condi- ion and a scala equa ion in ol ing an addi ional unknown, k=k(x). Among o he hings, hey se e o model he beha io o ce ain u bu- len lows. Ou main in e es conce ns exis ence and uniqueness. The main di icul ies a e due o he s uc u e o he scala equa ion; in pa ic- ula , he igh side is ypically in L1and, u he mo e, he e a e nonlinea e ms o he kind ∇ · (µ(k)∇k) and ∇ · (B(k)), whe e µand Ba e gene al con inuous unc ions. ∗Depa men o Di e en ial Equa ions and Nume ical Analysis, Uni e si y o Se illa, Ta ia s/n, E-41012 Se illa, Spain. Pa ially suppo ed by D.G.I.C.Y.T. (Spain), P oyec o PB92– 0696. 0 No a ion: •L1=L1(Ω), H1 0=H1 0(Ω), e c. • | · | ( esp. k·k) deno es he usual no m in L2( esp. H1 0). •H−1=H−1(Ω) is he dual space o H1 0;k · k∗deno es he usual no m in H−1. •z+= max(z, 0) o any eal z. •TM(s) = si s∈[−M, M] ; TM(s) = Msign so he wise. •Lnis he piecewise linea e en unc ion sa is ying Ln(s) = 1 i s∈[0, n] , Ln(s) = s n+ 2 i s∈[n, 2n] and Ln(s) = 0 i s > 2n. •S:D=PN i,j=1 SijDij o any S={Sij}and D={Dij}. •N0is he conjuga e exponen o N, i.e. N0=N N−1. •Fo each p∈[1,∞] , p∗is he associa ed Sobole embedding exponen : p∗=Np N−pi p<N; 1 < p∗<∞is a bi a y i p=Nand p∗=∞ o he wise. In pa icula , (N0)∗=N N−2i N≥3 . 1 1 In oduc ion. Desc ip ion o he p oblem This pape is conce ned wi h some nonlinea pa ial di e en ial sys ems s em- ming om luid mechanics. These a e a ian s o he s a iona y Na ie -S okes equa ions and ead as ollows:          −∇ · (νDu +kΦ0(Du)) + (u· ∇)u+∇p= , ∇ · u= 0 , −∇ · (µ(k)∇k+B(k)) + u· ∇k=ν0|Du|2+kΦ0(Du) : Du − |k|1/2kψ0(Du). (1) In (1), i is assumed ha Du =∇u+ ∇u. The unc ions D7→ Φ(D) , D7→ ψ0(D) , k7→ µ(k) and k7→ B(k) a e gi en. Once an open se Ω ⊂IRN and he da a ν > 0 , ν0∈[0, ν] and a e ixed, we sea ch o a solu ion {u, p, k} o (1), oge he wi h app op ia e bounda y alue condi ions. Sys ems like (1) a e mo i a ed by u bulence modelling. Mo e p ecisely, le U=U(x, ) and P=P(x, ) be espec i ely he eloci y ield and p essu e dis ibu ion o a iscous incomp essible luid in u bulen egime. Then, he couple (U, P) mus sa is y he ins a iona y Na ie -S okes equa ions. Deno ing by uand p he co esponding ime-a e aged a iables ( ha is o say, u=Uand p=P) and se ing U=u+u0, P =p+p0, i is cus oma y o eplace he sea ch o a solu ion o (1) by he analysis o a sys em ha should be sa is ied by uand p. A e some compu a ions, one inds: −∇ · (νDu +R)+(u· ∇)u+∇p= , ∇ · u= 0 ,(2) whe e is he ime-a e aged ex e nal o ces ield ac ing on he luid pa icles and Ris he so called Reynolds enso : R={Rij},wi h Rij =−u0 iu0 j. Since in (2) we s ill ind he unknown a iables u0 i, i is easonable o in oduce closing hypo heses ela ing R o u. In he case o usual one-equa ion models, one imposes he ollowing hypo hesis o he Boussinesq kind: R=νTDu , whe e νT=F(k) (an algeb aic ela ion). (3) He e, k=1 2|u0|2is he mean u bulen kine ic ene gy. The p oblem is hus closed using (2), (3) and an addi ional PDE o k. Un o una ely, when one ies o deduce an equa ion o k, one inds again e ms in which he u bulen pe u ba ions u0 i(and k0) appea . Mo e p ecisely, one has: −∇ · ν∇k+ (−(p0+k0)u0)+u· ∇k=R:Du −ν 2|Du0|2.(4) 2 Consequen ly, one has o eplace (4) by an app oxima ion. This is made by in oducing new closing hypo heses: •The e is gene al ag eemen in he app oxima ion o he dissipa ion e m ν 2|Du0|2. I is usually eplaced by a cons an imes k3/2. •O cou se, (3) is used again in o de o app oxima e he p oduc ion e m R:Du. •The app oxima ion o −(p0+k0)u0has been achie ed by se e al au ho s in di e en ways. In mos pape s, his e m is eplaced by cνT∇k, whe e c is an expe imen al cons an ( o ins ance, see [13], [12] and he e e ences he ein). In o he s, i is eplaced by a ec o B(k) (see [7]). Hence, i is clea ha equa ions like (1) can be used o desc ibe he beha io o ce ain u bulen lows. Ano he mo i a ion o (1) can be ound in non New onian mechanics. In his se ing, {u, p}a e he ue eloci y ield and p essu e, kis he empe a u e and i is assumed ha he s ess enso τdepends on Du and kas ollows: τ=νDu +kΦ0(Du).(5) 2 The main esul s In he sequel, we will conside a simpli ied e sion o (1):      −ν∆u−∇·(kΦ0(∇u)) + (u· ∇)u+∇p= , ∇ · u= 0 , −∇ · (µ(k)∇k+B(k)) + u· ∇k=ν|∇u|2+kΦ0(∇u) : ∇u . (6) This is made o con enience only; he esul s in his sec ion also hold o (1) wi h app op ia e changes. In (6), he i s , second and hi d equa ions will be espec i ely known as he mo ion equa ion, he incomp essibili y condi ion and he ene gy equa ion. Ou assump ions a e he ollowing: •Ω⊂IRNis a bounded, connec ed, open and egula se ; ν > 0 and ∈H−1. •D7→ Φ(D) is C1, Φ0(0) = 0, |Φ0(D)| ≤ Cons . and D7→ Φ0(D) : Dis con- ex (consequen ly, i is also locally Lipschi z-con inuous). In pa icula , D7→ Φ(D) is con ex and one has (Φ0(D1)−Φ0(D2)) : (D1−D2)≥0 o all D1and D2. •k7→ µ(k) and k7→ B(k) a e con inuous unc ions; u he mo e, µ(k)≥ µ0>0 o all k. 3 We wan o sol e (6) oge he wi h Di ichle condi ions o uand k: u= 0 and k= 0 on ∂Ω.(7) Ou main in e es conce ns gene al con inuous unc ions µand B. This is o cou se mo i a ed by he ac ha , in u bulence modelling, an equa ion exac ly sa is ied by kis unknown. Besides he usual spaces L2,H1 0,V, e c., we will use he ollowing: L={ψ∈L1;TM(ψ)∈H1 0∀M > 0,lim n→∞ 1 nZn≤|ψ|≤2n |∇ψ|2dx = 0 } (see he No a ion). Theo em 1–Unde he p e ious assump ions, he e exis s {u, p, k}, wi h u∈V,p∈L2and k∈ L such ha : 1. The couple {u, p}sol es he i s wo equa ions in (6) in he usual weak o dis ibu ional sense. 2. k≥0and sol es he hi d equa ion in (6) in he ollowing sense: (−∇ · (β(k)(µ(k)∇k+B(k))) + β0(k)∇k·(µ(k)∇k+B(k)) +β(k)(u· ∇k) = β(k)ν|∇u|2+kΦ0(∇u) : ∇u(8) in D0(Ω) o e e y β∈W1,∞(IR) wi h compac suppo . A iple {u, p, k}as abo e will be called a weak- eno malized solu ion o (6). Reno malized solu ions o PDE’s seem o ha e been in oduced by R. DiPe na and P.L. Lions in [8], in he amewo k o he Bol zmann equa ion. They ha e been used in connec ion wi h a ious nonlinea ellip ic equa ions by P. Benilan e al. [3], L. Bocca do e al. [6] and P.L. Lions and F. Mu a [10] (see also [11]). In he analysis o exis ence esul s o p oblems simila o (1) and (6), weak- eno malized solu ions we e conside ed by R. Lewandowski [9] (see also [2]). Tha we look o a eno malized solu ion kis mo i a ed by he s uc u e o he igh side o he ene gy equa ion in (6) ( ypically in L1) and also by ou in e es in keeping µand Bas gene al as possible. Le us deno e by ˜µ he ollowing unc ion: ˜µ(s) = Zs 0 µ(σ)dσ ∀s∈IR . Assume ha , in heo em 1, one has B≡0 . Then i is no di icul o see ha he solu ion {u, p, k} u nished by heo em 1 sa is ies ˜µ(k)∈ q<N0 W1,q 0,∇˜µ(k) = µ(k)∇k(9) 4 and also he ollowing:   ZΩ µ(k)∇k· ∇φ+ZΩ (u· ∇k)φ=ZΩν|∇u|2+kΦ0(∇u) : ∇uφ ∀φ∈ D(Ω) . (10) In his case, i will be said ha {u, p, k}is a weak solu ion o (6). Theo em 2–Assume ha , in heo em 1,B≡0and D→Φ0(D) : Dis globally Lipschi z-con inuous. Then, he e exis s ν0>0such ha , when ν≥ν0, he e exis s a mos one weak solu ion {u, p, k} o (6) wi h k≥0. Be o e gi ing he p oo s o hese esul s, le us make some ema ks: 1. A e y in e es ing ques ion emains: When νis la ge and Bis no ze o, is i s ill possible o p o e he uniqueness o a eno malized solu ion ? 2. The e a e se e al o he possible condi ions o uniqueness, di e en om he assump ion ν≥ν0. Fo ins ance, o ixed ν, one can also ob ain a mos one weak solu ion {u, p, k} o (6) wi h k≥0 i k k∗is su icien ly small. 3. Resul s simila o hose abo e can be p o ed o he ins a iona y a ian o (6). This will be analyzed in a o hcoming pape . 4. Se e al mo e o less ob ious gene aliza ions a e possible. In pa icula , we ind an in e es ing si ua ion when we simply assume µ(k)≥0 in (6). This case is a om i ial and will also be he subjec o u u e wo k. 3 The p oo o heo em 1 In his sec ion, Cdeno es a cons an which may depend on N, Ω and he da a in (6). The p oo o heo em 1 consis s o six s eps: Fi s s ep: The in oduc ion o a amily o app oxima ions. Fo each ε > 0, we conside he ollowing app oxima ion o (6):        −ν∆uε− ∇ · (T1 ε(kε)+Φ0(∇uε)) + (uε· ∇)uε+∇pε= , ∇ · uε= 0 , −∇ · T1 ε(µ(kε))∇kε+B(T1 ε(kε)+uε· ∇kε=T1 ε(τε:∇uε). (11) He e, we ha e used he ollowing no a ion: τε=ν∇uε+T1 ε(kε)+Φ0(∇uε). 5 O cou se, hese equa ions a e equi ed o be sa is ied in Ω , oge he wi h ho- mogeneous Di ichle condi ions o uεand kεon ∂Ω . The exis ence o a iple {uε, pε, kε}can be es ablished using ( o ins ance) a Gale kin me hod. In ac , some non i ial di icul ies a e ound wi h his echnique ha can be sol ed a - guing as in he ollowing s eps. One inds ha he solu ion belongs o he space V×L2×H1 0(Ω) and, also, ha kε≥0 . Second s ep: A p io i es ima es and weak con e gence. Using uεas a es unc ion in he i s equa ion in (11), one inds: ZΩ τε:∇uε≤C . (12) In pa icula , kuεk ≤ C . (13) In he ene gy equa ion in (11), le us use TM(kε) as es unc ion. This gi es: kTM(kε)k2≤C·M(14) On he o he hand, i we choose ξn(kε) = T2n(kε)−Tn(kε) as es unc ion in he same equa ion, i is no di icul o check ha 1 nZn≤kε≤2n T1 ε(µ(kε))|∇kε|2≤Zkε≥n T1 ε(τε:∇uε),(15) whence 1 nZn≤kε≤2n |∇kε|2≤C . (16) ¿F om (14) and (16), a guing as in [5] and [11], one deduces he ollowing: kkεkW1,q 0≤Cq,∀q < N0.(17) Consequen ly, passing o a subsequence i necessa y, i can be assumed ha uε→uweakly in V, s ongly in L ∀ < 2∗and a.e., kε→kweakly in W1,q 0∀q < N0, s ongly in Lp∀p < (N0)∗and a.e., TM(kε)→TM(k) weakly in H1 0∀M > 0. Ob iously, one has k≥0 . Thi d s ep: uis, oge he wi h some p, a solu ion o he mo ion equa ion. Fo each ε > 0 , uεis a solu ion o he ollowing a ia ional inequali y:        νZΩ ∇uε:∇ +ZΩ (uε· ∇)uε· +ZΩ T1 ε(kε)Φ(∇ ) ≥νZΩ |∇uε|2+ZΩ T1 ε(kε)Φ(∇uε) + h , −uεi ∀ ∈V , uε∈V . 6 Taking limi s as ε→0 , one ob ains: νZΩ ∇u:∇ +ZΩ (u· ∇)u· +ZΩ kΦ(∇ ) ≥νlim in ε→0ZΩ |∇uε|2+ lim in ε→0ZΩ T1 ε(kε)Φ(∇uε) + h , −ui The i s e m in he igh is bounded om below by νZΩ |∇u|2. In wha conce ns he second e m, le us i s no ice ha ZΩ T1 ε(kε)Φ(∇uε) = ZΩ (T1 ε(kε)−k)Φ(∇uε) + ZΩ kΦ(∇uε). Thus, aking in o accoun ha he unc ion 7→ ZΩ kΦ(∇ ) is lowe semicon inous, we ind: lim in ε→0ZΩ T1 ε(kε)Φ(∇uε)≥lim ε→0ZΩ (T1 ε(kε)−k)Φ(∇uε) + lim in ε→0ZΩ kΦ(∇uε) ≥ZΩ kΦ(∇u). Consequen ly, uis a solu ion o he a ia ional inequali y        νZΩ ∇u: (∇ − ∇u) + ZΩ (u· ∇)u·( −u) + ZΩ kΦ(∇ ) −ZΩ kΦ(∇u)≥ h , −ui ∀ ∈V , u ∈V . (18) Now, aking in (18) he unc ion o he o m u+ w , whe e w∈Vand ∈IR and le ing →0 , i is a s anda d ma e o p o e ha usol es, oge he wi h some p∈L2, he i s wo equa ions in (6) in he usual weak sense. Fou h s ep: uεcon e ges s ongly in V. ¿F om he mo ion equa ion in (6), i is clea ha ZΩν|∇u|2+kΦ0(∇u) : ∇u=h , ui. On he o he hand, choosing uεas es unc ion in he i s equa ion in (11), one has: ZΩν|∇uε|2+T1 ε(kε)Φ0(∇uε) : ∇uε=h , uεi. 7 Consequen ly, lim ε→0ZΩν|∇uε|2+T1 ε(kε)Φ0(∇uε) : ∇uε=ZΩν|∇u|2+kΦ0(∇u) : ∇u, whence i is also clea ha lim ε→0ZΩν|∇uε|2+kΦ0(∇uε) : ∇uε=ZΩν|∇u|2+kΦ0(∇u) : ∇u ( ecall ha Φ0is uni o mly bounded). Hence, 0 = lim ε→0ZΩν|∇uε|2+kΦ0(∇uε) : ∇uε−ZΩν|∇u|2+kΦ0(∇u) : ∇u ≥lim sup ε→0νZΩ |∇(uε−u)|2 + lim in ε→0ZΩ kΦ0(∇uε) : ∇uε−ZΩ kΦ0(∇u) : ∇u). He e, he las e m is ≥0 , in iew o he lowe semicon inui y o he unc ion 7→ ZΩ kΦ0(∇ ) : ∇ . Thus, lim ε→0ZΩ |∇(uε−u)|2= 0 . Fi h s ep: Fo all M > 0 , TM(kε) con e ges s ongly in H1 0. We will use an a gumen due o P.L. Lions and F. Mu a (see [10], [11]). Le us see ha T1 ε(µ(kε))1 2∇TM(kε)→µ(k)1 2∇TM(k) s ongly in L2∀M > 0 (19) (obse e ha µ(k)1 2∇TM(k) has a sense). O cou se, (19) will su ice o ou pu poses. I has al eady been p o ed ha T1 ε(τε:∇uε)→τ:∇us ongly in L1and a.e. He e, we ha e in oduced τ=ν∇u+kΦ0(∇u) . Choosing TM(kε) as es unc ion in he ene gy equa ion in (11), one inds: ZΩ T1 ε(µ(kε))∇kε· ∇TM(kε) + ZΩ B(T1 ε(kε)) · ∇TM(kε) +ZΩ (uε· ∇kε)TM(kε) = ZΩ T1 ε(τε:∇uε)TM(kε). 8