Note on Lagrangian-Eulerian Methods for Uniqueness in Hydrodynamic Systems

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1 Noe on Lagrangian-Eulerian Mehods for Uniqueness in Hydrodynamic Sysems Peer Consanin and Joonhyun La Deparmen of mahemaics, Princeon Universiy November 3, 8 Absrac We discuss he Lagrangian-Eulerian framework for hydrodynamic models and provide a proof of Lipschiz dependence of soluions on iniial daa in pah space. The paper presens a correced version of he resul in []. MSC Classificaion: 35Q3, 35Q35. Inroducion Many hydrodynamical sysems consis of evoluion equaions for fluid velociies forced by exernal sresses, coupled o evoluion equaions for he exernal sresses. In he simples cases, he Eulerian velociy u can be recovered from he sresses σ via a linear operaor u = Uσ and he sress marix σ obeys a ranspor and sreching equaion of he form σ + u σ = F u, σ, where F is a nonlinear coupling depending on he model. The Eulerian velociy gradien is obained in erms of he operaor x u = Gσ, and, in many cases, G is bounded in Hölder spaces of low regulariy. Then, passing o Lagrangian variables, τ = σ where is he paricle pah ransformaion, : R d R d, a volume preserving diffeomorphism, he sysem becomes { = U, τ, 3 τ = T, τ. wih U, τ = Uτ, T, τ = F Gτ, τ. In paricular, τ solves an ODE d τ = F g, τ 5 d where g = x u is of he same order of magniude as τ in appropriae spaces, and so he size of τ is readily esimaed from he informaion provided by he ODE model, analysis of G and of he cons@mah.princeon.edu joonhyun@mah.princeon.edu 4

2 operaion of composiion wih. The main addiional observaion ha leads o Lipschiz dependence in pah space is ha derivaives wih respec o parameers of expressions of he ype encounered in he Lagrangian evoluion 4, Uτ, Gτ, inroduce commuaors, and hese are well behaved in spaces of relaively low regulariy. The Lagrangian- Eulerian mehod of [] formalized hese consideraions leading o uniqueness and Lipschiz dependence on iniial daa in pah space, wih applicaion o several examples including incompressible D and 3D Euler equaions, he surface quasi-geosrophic equaion SQG, he incompressible porous medium equaion, he incompressible Boussinesq sysem, and he Oldroyd-B sysem coupled wih he seady Sokes sysem. In all hese examples he operaors U and G are ime-independen. The paper [] considered ime-dependen cases. When he operaors U and G are ime-dependen, in conras o he ime-independen cases sudied in [], G is no necessarily bounded in L, T ; C α. This was addressed in [] by using a Hölder coninuiy σ C β, T ; C α. While his reaed he Eulerian issue, i was acily used bu never explicily saed in [] ha his kind of Hölder coninuiy is ransferred o σ from τ by composiion wih a smooh ime-depending diffeomorphism close o he ideniy. This is false. In fac, we can easily give examples of C α funcions τ which are ime-independen hence analyic in ime wih values in C α and diffeomorphisms a = a + v wih consan v, such ha σ = τ is no coninuous in C α as a funcion of ime. In his paper we presen a correc version of he resuls in []. Insead of relying on he ime regulariy of τ alone, we also use he fac ha G is composed from a ime-independen bounded operaor and an operaor whose kernel is smooh and rapidly decaying in space. Then he ime singulariy is resolved by using he Lipschiz dependence in L of Schwarz funcions composed wih smoohly varying diffeomorphisms near he ideniy. A ypical example of he sysems we can rea is he Oldroyd-B sysem coupled wih Navier-Sokes equaions: u u = H div σ u u, u =, 6 σ + u σ = uσ + σ u T kσ + ρk u + u T, ux, = u x, σx, = σ x. Here x, R d [, T. The Leray-Hodge projecor H = I + R R is given in erms of he Riesz ransforms R = R,..., R d, and, ρk, k are fixed posiive consans. This sysem is viscoelasic, and he behavior of he soluion depends on he hisory of is deformaion. The non-resisive MHD sysem u u = H div b b u u, u =, b =, b + u b = ub, ux, = u x, bx, = b x. can also be reaed by his mehod. The sysems 6 and 7 have been sudied exensively, and a review of he lieraure is beyond he scope of his paper. The Lagrangian-Eulerian formulaion We show calculaions for 6 in order o be explici, and because he calculaions for 7 are enirely similar. The soluion map for ux, of 6 is 7 where ux, = L u x, + L u x, = g u x = g s H div σ u u x, sds. 8 R d 4π d e x y 4 u ydy. 9

3 Thoroughou he paper we use The velociy gradien saisfies g x = 4π d e x 4. ux, = L u x, + We denoe he Eulerian velociy and gradien operaors Ufx, = Gfx, = g s H div σ u u x, sds. g s Hdiv fx, sds, g s H div fx, sds. Noe ha for a second order ensor f, Gf = x Uf = R R U x f. Le be he Lagrangian pah diffeomorphism, v he Lagrangian velociy, and τ he Lagrangian added sress, We also se v = = u, τ = σ. ga, = ua,, = L u a, +G τ a, U x v v a,. In Lagrangian variables he sysem is a, = a + V, τ, a, sds, τa, = σ a + T, τ, a, sds, va, = V, τ, where he Lagrangian nonlineariies V, T are { V, τ, a, s = L u a, s + U τ v v a, s, T, τ, a, s = gτ + τg T kτ + ρkg + g T a, s, and g is defined above in 3. The main resul of he paper is Theorem. Le < α < and < p <, be given. Le also v = u C +α,p and v = u C +α,p be given divergence-free iniial velociies, and σ, σ C α,p be given iniial sresses. Then here exiss T > and C > depending on he norms of he iniial daa such ha, τ, v,, τ, v, wih iniial daa Id, σ, u, Id, σ, u, are bounded in Id + Lip, T ; C +α,p Lip, T ; C α,p L, T ; C +α,p and solve he Lagrangian form 4 of 6. Moreover, Lip,T;C +α,p + τ τ Lip,T;C α,p + v v L,T,C +α,p C u u +α,p + τ τ α,p 6 Remark. The soluions Lagrangian sresses τ are Lipschiz in ime wih values in C α. Their Lagrangian counerpars σ = τ are bounded in ime wih values in C α and space-ime Hölder coninuous wih exponen α. The Eulerian version of he equaions 6 is saisfied in he sense of disribuions, and soluions are unique in his class. 3

4 The spaces C α,p are defined in he nex secion. The proof of he heorem occupies he res of he paper. We sar by considering variaions of Lagrangian variables. We ake a family ɛ, τ ɛ of flow maps depending smoohly on a parameer ɛ [, ], wih iniial daa u ɛ, and σ ɛ,. Noe ha v ɛ = ɛ. We use he following noaions u ɛ = ɛ ɛ, g ɛ = d dɛ g ɛ, ɛ = d dɛ ɛ, η ɛ = ɛ v ɛ = d dɛ v ɛ, σ ɛ = τ ɛ ɛ, ɛ, τ ɛ = d dɛ τ ɛ, δ ɛ = τ ɛ ɛ, 7 and u ɛ, = d dɛ u ɛ, σ ɛ, = d dɛ σ ɛ. 8 We represen where ɛ = a, a, = τ a, τ a, = v a, v a, = ɛdɛ, π ɛ dɛ, d dɛ V ɛdɛ, d dɛ V d ɛds, π ɛ = dɛ T ɛds + σ ɛ,, V ɛ = V ɛ, τ ɛ, T ɛ = T ɛ, τ ɛ. We have he following commuaor expressions arising by differeniaing in ɛ [], []: 9 where and d dɛ We noe, by he chain rule, Uτɛ ɛ ɛ ɛ = [η ɛ x, U]σ ɛ + Uδ ɛ, [η ɛ x, U]σ ɛ = η ɛ x Uσ ɛ U η ɛ x σ ɛ d dɛ Uv ɛ v ɛ ɛ ɛ ɛ = [η ɛ x, U]u ɛ u ɛ + Uv ɛ v ɛ + v ɛ v ɛ ɛ. 3 a V = a g. 4 Consequenly, differeniaing V ɛ, g ɛ and he relaion 4 we have d dɛ V ɛ ɛ = η ɛ L x u ɛ, + L u ɛ, +[η ɛ x, U]σ ɛ u ɛ u ɛ + Uδ ɛ v ɛ v ɛ + v ɛ v ɛ ɛ, g ɛ = L x u ɛ, ɛ + Gσ ɛ ɛ U x u ɛ u ɛ ɛ, g ɛ ɛ = η ɛ L x x u ɛ, + L x u ɛ, + [η ɛ x, G]σ ɛ + Gδ ɛ [η ɛ x, U] x u ɛ u ɛ U x v ɛ v ɛ + v ɛ v ɛ ɛ, d dɛ av ɛ = a ɛg ɛ + a ɛ g ɛ, d dɛ T ɛ = g ɛτ ɛ + g ɛ τ ɛ + τ ɛg ɛ T + τ ɛ g ɛ T kτ ɛ + ρkg ɛ + g ɛ T. 4 5

5 3 Funcions, operaors, commuaors We consider funcion spaces wih norm C α,p = C α R d L p R d 6 f α,p = f C α R d + f L p R d 7 for α,, p,, C +α R d wih norm and wih norm f C +α R d = f L R d + f C α R d, 8 We also use spaces of pahs, L, T ; Y wih he usual norm, spaces Lip, T ; Y wih norm f Lip,T ;Y = C +α,p = C +α R d W,p R d 9 f +α,p = f C +α R d + f W,p R d. 3 f L,T ;Y = f fs sup Y s,,s [,T ] s where Y is C α,p or C +α,p in he following. We use he following lemmas. sup f Y, 3 [,T ] + f L,T ;Y 3 Lemma []. Le < α <, < p <. Le η C +α R d and le Kσx = P.V. kx yσydy 33 R d be a classical Calderon-Zygmund operaor wih kernel k which is smooh away from he origin, homogeneous of degree d and wih mean zero on spheres abou he origin. Then he commuaor [η, K] can be defined as a bounded linear operaor in C α,p and [η, K]σ C α,p C η C +α R d σ Cα,p. 34 Lemma Generalized Young s inequaliy. Le q and C >. Suppose K is a measurable funcion on R d R d such ha sup Kx, y dy C, sup x R d R d y R d Kx, y dx C. R d 35 If f L q R d, he funcion T f defined by T fx = Kx, yfydy R d 36 is well defined almos everywhere and is in L q, and T f L q C f L q. The proof of his lemma for < q < is done using dualiy, a sraighforward applicaion of Young s inequaliy and changing order of inegraion. The exreme cases q = and q = are proved direcly by inspecion. For simpliciy of noaion, le us denoe M = + Id L,T ;C +α. 37 5

6 Theorem. Le < α <, < p < and le T >. Also le be a volume preserving diffeomorphism such ha Id Lip, T ; C +α. Then τ L,T ;C α,p τ L,T ;C α,p M α. 38 If Lip, T ; C +α, hen L,T ;C +α L,T ;C +α M +α. 39 If v Lip, T ; W,p, hen v L,T ;W,p v L,T ;W,p M. 4 If in addiion, exis in L, T ; C +α, hen Lip,T ;C α Lip,T ;C +α Id Lip,T ;C +α M +3α. 4 Proof. τ = τ Lp L L p L, 4 and, denoing he seminorm we have [τ] α = τa τb sup a b,a,b R a b α [ τ ] α [τ] α x α L [τ] α + Id L,T ;C +α α. 43 Noe ha his shows ha he same bound holds when we replace by. For he second and hird par, i suffices o remark ha x = a a 44 and he previous par gives he bound in erms of Lagrangian variables. For he las par, we noe ha where = Now noing ha we have so ha x,, x, s, s s x, β τ, β τ + x, β τ a 45 x, β τ, β τ dτ, β τ = τ + τs. 46 = a 47 s s C α L,T ;C α + L,T ;C α L,T ;C +α Lip,T ;C α Lip,T ;C +α Id Lip,T ;C +α +3α 48 + Id L,T ;C +α + Id L,T ;C +α +3α. 49 6

7 Theorem 3. Le < α <, < p < and le T >. There exiss a consan C independen of T and such ha for any < < T, L u L,T ;C α,p C u α,p, L u L,T ;C +α,p C u +α,p, L u α,p C u α,p, 5 L u L,T ;C α,p C u +α,p hold. Proof. L u α,p g L u α,p = u α,p, L u +α,p g L u +α,p = u +α,p, L u α,p g L u +α,p = C u α,p, L u α,p g L u α,p u +α,p. 5 Theorem 4. Le < α <, < p < and le T >. There exiss a consan C such ha Uσ L,T ;C α,p C T σ L,T ;C α,p. 5 Proof. Uσ C α,p C g s L σs α,p ds ds σ L,T ;C α,p C T σ L,T ;C α,p. 53 C s Theorem 5. Le < α <, < p < and le T >. There exis consans C, C depending only on α and, and C 3 T,, C 4 T, such ha Gτ C L,T ;C α,p Id α Lip,T ;C +α τ α,p + C 3T, +C τ Lip,T ;C α,p C 54 4T, where C 3 T, and C 4 T, are of he form CT Proof. Since G = R RHΓ where Γτ = we can replace G by Γ. Then Γτ can be wrien as Id α Lip,T ;C +α + Id 4 Lip,T ;C +α. g s τ sds, 55 Bu Γτ = + g s τ s τ ds g s τ ds. 56 g s τ ds = τ g τ 57 7

8 so he second erm is bounded by τ L,T ;C α,p M α by Theorem. Now we le where Bu since τ x, s τ x, = τx, s, + τx, s,, 58 τx, s, = τ x, s, s τ x, s,, τx, s, = τ x, s, τ x,,. 59 τs, C α,p s M α τ Lip,T ;C α,p, 6 by he proof of Theorem we ge g s τs, ds C α,p τ Lip,T ;C α,p M, α 6 On he oher hand, where We use he following lemma. g s τs, ds = R d Kx, z,, sτz, dzds, 6 Kx, z,, s = g s x z, s g s x z,. 63 Lemma 3. Kx, z,, s is L in boh he x variable and he z variable, and sup z K, z,, s L, sup x Kx,,, s L C Id Lip,T ;L. 64 s 3 Proof. We define Sx = 4πe x x d 65 so ha Then g s = 4π s d + x S 4 s. 66 Kx, z,, s dz = 4π s d + x z, s x z, S S 4 s 4 s dz = 4π s d + x y x y, s S S 4 s 4 s dy = 4π s π d + Su S u Idx 4 s u, s 4 s du. 67 However, for each u Su S u Idx 4 s u, s 4 s Idx 4 s u, s 4 s { } 68 Idx 4 s u, s sup Su z : z 4 s and we have Idx 4 s u, s 4 s Id s Lip,T ;L CT 69 8

9 and obviously Su = is inegrable in R d ; because S is Schwarz, Sx sup z CT for some consan C d, bu if z CT, hen u z u C T and Su z Su z 7 C d + C T + x d 7 C d + C T + u d 7 and he righ side of above is clearly inegrable wih bound depending only on d and T. Therefore, we have Kx, z,, s dz s 3 Id Lip,T ;L Cd, T. 73 Similarly, Kx, z,, s dx = 4π s d + x z, s x z, S S 4 s 4 s dx = 4π s π d + 74 z, s z, Sy Sy + 4 s dy and again we have z, s z, 4 s Id Lip,T ;L s CT. 75 Therefore, we have he bound Kx, z dx s 3 Id Lip,T ;L Cd, T. 76 From Lemma 3 and generalized Young s inequaliy, we have g s τs, ds C Id Lip,T ;C +α τ L,T ;L p L. 77 Lp L For he Hölder seminorm, we measure he finie difference. Le us denoe δ h fx, = fx+h, fx,. If h <, hen δ h g s τs, ds = δ h g s τs, ds. 78 If < s < h, hen δ h g s L g s L C s we have h and since τs, L s α Id α Lip,T ;C +α τ L,T ;C α,p 79 δ h g s τs, ds L C α h α Id α Lip,T ;C +α τ L,T ;C α,p. 8 If h < s <, hen following lines of Lemma 3 δ h g s is a L funcion wih δh g s L C h s 3 8 9

10 and we have C 3 C 3 h δ h g s τs, ds L Id α Lip,T ;C +α τ L,T ;C α,p h α α α, Id α Lip,T ;C +α τ L,T ;C α,p h α α α >. If h, hen we only have he firs erm. Therefore, we have h α δ h g s τs, ds Cα Id α Lip,T ;C +α τ L,T ;C α,p. 83 We noe ha To summarize, we have Cα + + Cα L 8 τ α,p τ α,p + τ Lip,T ;C α,p. 84 Γτ L,T ;C α,p Id αlip,t ;C+α τ α,p + Cα + Id α Lip,T ;C +α T τ Lip,T ;C α,p T α max{ Id Lip,T ;C +α, Id 4 Lip,T ;C +α } τ α,p + T τ Lip,T ;C α,p, 85 and his complees he proof. Theorem 6. Le < α <, < p < and le T >. Le Lip, T ; C +α wih L, T ; C +α. There exiss a consan C such ha [, U ] σ L,T ;C α,p T T C + Id Lip,T ;C +α M +3α Lip,T ;C +α σ 86 L,T ;C α,p Proof. Firs, we denoe Then we have η =. 87 = η = [η, H] +H [η, U] σ g s Hdiv σsds g s div σsds + H H g s Hdiv ηs σsds g s ηs η σsds g s ηsσs ds η g s σs g s ησs ds, 88 where g s ηs ησs, η g s σs, and g s ηsσs represen i j g s η i s η i σ jk s, i,j 89 i j g s ηi sσ jk s. i,j η i i j g s σjk s, and respecively i,j

11 The firs erm is bounded by Lemma and he second erm is esimaed direcly [η, H] g s divσsds C η C +α α,p H g s ηsσsds C α,p The hird erm is bounded by σ L,T ;C α,p, 9 η L,T ;C +α σ L,T ;C α,p. C η Lip,T ;C α σ L,T ;C α,p 9 by he virue of Theorem. For he las erm, noe ha η g s σs g s ησs x = g s zz ηx λz, dλ σx z, sdz R d and noe ha g s zz is a L funcion wih Therefore, g s zz L 9 C. 93 s η g s σs g s ησs α,p C s so ha he las erm is bounded by η C +α σs α,p 94 We finish he proof by replacing η by using Theorem. C η C +α σ L,T ;C α,p. 95 Theorem 7. Le < α <, < p < and le T >. Le Lip, T ; C +α wih L, T ; C +α. There exiss a consan Cα depending only on α such ha [, G ] τ L,T ;C α,p L,T ;C +α + Lip,T ;C +α T R 96 where R is a polynomial funcion on τ Lip,T ;C α,p, Id Lip,T ;C +α, whose coefficiens depend on α,, and T, and in paricular i grows polynomially in T and bounded below. Proof. Again we denoe η =. Also i suffices o bound where Γ is as defined in 55, since [η, Γ] τ = η Γ τ Γ η τ 97 [η, G] = R RH [η, Γ] + [η, R RH] Γ 98 and he second erm is bounded by Lemma. For he firs erm, we have [η, Γ] τ = I + I + I 3 + I 4 + I 5 + I 6, 99

12 where I = η g s τ g s ητ ds, I = η g s τ s τ g s η τ s τ ds, I 3 = I 5 = I 4 = I 6 = g s ηs η τ s ds, g s ηs η τ s ds, g s η τ s τ ds, ητ g ητ. Firs, I + I 6 can be bounded: I + I 6 = η g τ g ητ g η τ and he firs erm is reaed in he same way as 9. Since he firs erm is and g yy R d he C α,p -norm of he firs erm is bounded by ηx λy, dλ τ x y, dy g yy L C, 3 C η C +α τ α,p. 4 The C α,p -norm of he second erm is also bounded by he same bound. Therefore, I + I 6 L,T ;C α,p C M +3α L,T ;C +α τ L,T ;C α,p. 5 The erm I 3 is bounded due o Theorem. Since η Lip, T ; C α we have I 3 L,T ;C α,p C T M +4α Id Lip,T ;C +α 6 Lip,T ;C +α τ L,T ;C α,p. The erms I 4, and I 5 are reaed in he spiri of Theorem 5. We rea L p L norm and Hölder seminorm separaely. For he erm I 5, we have I 5 = g s η τs, + τs, ds 7 where τ and τ are he same as 59. From he same argumens from he above, g s η τs, ds α,p C η L,T ;C +α τ Lip,T ;C α,p M α. 8

13 On he oher hand, g s η τs, x = Kx, z,, s η z,, R d + g s x z, η z, s, η z,, dz, 9 where K is as in 63. Then as in he proof of Lemma 3, by he generalized Young s inequaliy we have g s η τs, ds C τ L p L η L,T ;C +α L p L α Id Lip,T ;C +α α Id 3 Lip,T ;C +α. For he Hölder seminorm, we repea he same argumen in he proof of Theorem 5, using he bound 8. Then we obain h α δ h g s τs, ds L Cα + + Id αlip,t ;C+α τ L,T ;Cα,p η L,T ;C+α. Therefore, I 5 L,T ;C α,p Cα Id Lip,T ;C +α M +α L,T ;C +α τ Lip,T ;C α,p. The erm I 4 is reaed in he exacly same way, by noing ha ηs η = x s : a s, + x s : a s, + x s x : a 3, where as in 59 a x, s, = a x, s, s a x, s,, 4 a x, s, = a x, s, a x,,, and x x, s x, = a x, a Id x,, s 5 so ha x s x +α C α s Id Lip,T ;C+α M. 6 Also noe ha a s, L a C α Id α Lip,T ;L s α 7 so ha The final resul is g s x s : a s, τ s ds Cα + α + M +α Id α Lip,T ;C +α I 4 α,p Cα L,T ;C +α τ L,T ;C α,p. + + C α,p M +4α L,T ;C +α τ L,T ;C α,p +C M +3α Lip,T ;C +α τ L,T ;C α,p

14 Finally, I can be bounded using he combinaion of he echnique in Theorem 5 and Theorem 6. Firs, we have + I x, = g s y y ηx λy, dλ τx y, s, dyds R d g s x z x z ηλx + λz, dλ τz, s, dzds. R d Then applying he argumen of he proof of Theorem 6, he firs erm is bounded by C M α η L,T ;C +α τ Lip,T ;C α,p. The second erm is reaed using he mehod used in Theorem 5. By changing variables o form a kernel similar o 63, and applying generalized Young s inequaliy, he L p L norm of he second erm is bounded by Cα α Id Lip,T ;C+α η L,T ;C +α τ L,T ;Lp L. Finally, he Hölder seminorm of he second erm is bounded by he same mehod as Theorem 5. The only addiional poin is he finie difference of η erm, bu his erm is bounded by a sraighforward esimae. The bound for he Hölder seminorm of he second erm is Cα + α + To sum up, we have + Id α Lip,T ;C +α η L,T ;C +α τ L,T ;C α,p. 3 I α,p Cα Id Lip,T ;C +α M +3α 4 L,T ;C +α τ Lip,T ;C α,p. If we pu his ogeher, [, G ] τ L,T ;C α,p C L,T ;C +α M +α Γτ L,T ;C α,p 5 + L,T ;C +α + Lip,T ;C +α T F, α,, τ Lip,T ;C α,p, T where F depends on he wrien variables and grows like polynomial in T, τ Lip,T ;C α,p, and Id Lip,T ;C +α. The bound on Γτ is given by Theorem 5. 4 Bounds on variaions and variables Using he resuls from he previous secion we find bounds for variaions and variables. For simpliciy, we adop he noaion M ɛ = + ɛ Id L,T ;C +α. 6 Firs, we bound d dɛ V ɛ. Noe ha ɛ = Id, so ɛ = and by Theorem and since ɛ Lip, T ; C +α,p we have ɛ L,T ;C +α T ɛ Lip,T ;C +α,p, η ɛ C α Lip,T ;C +α,p M α ɛ. 7 4

15 Then by he Theorem 3, we have T η ɛ L x u ɛ, L,T ;C α,p C M α ɛ ɛ Lip,T ;C +α,p u ɛ, +α,p, L u ɛ, C L u,t ;C α,p ɛ, α,p. 8 By Theorem 6, we have and by Theorem 4, we have T [η ɛ x, U] σ ɛ u ɛ u ɛ L,T ;C α,p C T + Mɛ +4α 9 ɛ Lip,T ;C +α τ ɛ v ɛ v ɛ L,T ;C α,p, U δɛ v ɛ v ɛ + v ɛ v ɛ ɛ L T,T ;C α,p C M α ɛ 3 τ ɛ v ɛ v ɛ + v ɛ v ɛ L,T ;C α,p. Therefore, d dɛ V ɛ C u α,p ɛ, L,T ;C α,p +S T ɛ Lip,T ;C +α,p + v ɛ L,T ;C α,p + σ α,p ɛ, + τ ɛ Lip,T ;C α,p Q 3 where S T vanishes as T as T and Q is a polynomial in u ɛ, +α,p, ɛ Id Lip,T ;C +α,p, τ ɛ L,T ;C α,p, and v ɛ L,T ;C α,p, whose coefficiens depend on. Similarly, g ɛ L,T ;C α,p M α u +α,p + C Id α Lip,T ;C +α σ ɛ, α,p + S T Q, 3 where S T vanishes as T as T and Q is polynomial in τ Lip,T ;C α,p and Id Lip,T ;C +α, whose coefficiens depend on α and. Also g ɛ L,T ;C α,p C u +α,p ɛ, + Id α Lip,T ;C +α τ α,p ɛ, +S 3 T ɛ Lip,T ;C +α,p + σ ɛ, α,p + τ ɛ Lip,T ;C α,p + v ɛ L,T ;C +α,p Q 33 3, where S 3 T vanishes as T as T and Q 3 is polynomial in u ɛ, +α,p, Id Lip,T ;C +α,p, τ Lip,T ;C α,p, and v ɛ L,T ;C +α,p, whose coefficiens depend on and α. Then we have d a dɛ V ɛ T ɛ Lip,T ;C +α g ɛ L,T ;C α,p + M ɛ g ɛ L,T ;C α,p 34 L,T ;C α,p and d dɛ T ɛ g ɛ L,T ;C α,p τ ɛ L,T ;C α,p + ρk L,T ;C α,p + τ ɛ L,T ;C α,p g ɛ L,T ;C α,p + k Local exisence We define he funcion space P and he se I, P = Lip, T ; C +α,p Lip, T ; C α,p L, T ; C +α,p I = {, τ, v : Id, τ, v P Γ, v = d d }, 36 5

16 where Γ > and T > are o be deermined. Now, for given u C +α,p divergence free and σ C α,p we define he map, τ, v S, τ, v = new, τ new, v new 37 where new = Id + Vs, τs, vsds, τ new = σ + T s, τs, vsds, v new = V, τ, v. 38 If Id, τ, v P, hen new Id, τ new, v new P for any choice of T >. Moreover, we have he following: Theorem 8. For given u C +α,p divergence free and σ C α,p, here is a Γ > and T > such ha he map S of 38 maps I o iself. Proof. I is obvious ha d d new = v new. For he size of S, τ, v, firs noe ha if Id, τ, v P Γ, hen M = + Id L,T ;C +α + T Γ. 39 Applying Theorem 3 and Theorem 4, we know ha V L,T ;C α,p u α,p + A T B Γ, u α,p, σ α,p, 4 where A T vanishes like T for small T > and B is a polynomial in is argumens, and some coefficiens depend on. We esimae g L,T ;C α,p u +α,p + C Γ α σ α,p + A T B Γ, u +α,p, σ α,p, 4 where C is as in Theorem 5, depending only on α and, A T vanishes in he same order as A T as T, and B is a polynomial in is argumens, and some coefficiens depend on and α. From 4 we conclude V L,T ;C +α,p K u +α,p + Γ α σ α,p + A 3 T B 3 Γ, u +α,p, σ α,p, 4 where K is a consan depending only on and α, and A 3 and B 3 have he same properies as previous A i s and B i s. Now we measure T. From 84 and he previous esimae on g we have T L,T ;C α,p K u +α,p ρk + σ α,p + σ α,p Γ α σ α,p + ρkγ α + k +A 4 B 4, 43 where K is a consan depending on and α, and A 4 and B 4 are as before. Since α <, we can appropriaely choose large Γ > σ α,p + u +α,p and correspondingly small 6 > T > so ha he righ side of 4 and 43 are bounded by Γ 6. Then new Id, τ new, v new P Γ. We show now ha S is a conracion mapping on I for a shor ime. Theorem 9. For given u C +α,p divergence free and σ C α,p, here is a Γ and T >, depending only on u +α,p and σ α,p, such ha he map S is a conracion mapping on I = IΓ, T, ha is S, τ, v S, τ, v P, τ τ, v v P. 44 Proof. Firs from Theorem 8 we can find a Γ and T >, depending only on he size of iniial daa, say N = max{ u +α,p, σ α,p }, 45 6

17 which guaranees ha S maps I o iself. This propery sill holds if we replace T by any smaller T >. In view of he fac ha I is convex, we pu ɛ = ɛ + ɛ, τ ɛ = ɛτ + ɛ τ, ɛ. 46 Then ɛ, τ ɛ, v ɛ I, v ɛ = ɛv + ɛ v, u ɛ, = u, and σ ɛ, = σ. This means ha ɛ =, v ɛ = v v, u ɛ, =, σ ɛ, =. 47 Then from he resuls of Secion 4, we see ha d dɛ V ɛ Lip,T ;C +α,p + v v L,T ;C α,p L,T ;C +α,p + τ τ Lip,T ;C α,p S T Q Γ, ɛ Lip,T ;C +α,p Lip,T ;C +α,p + v v L,T ;C α,p + τ τ Lip,T ;C α,p S T Q Γ, 48 π ɛ Lip,T ;C α,p Lip,T ;C +α,p + v v L,T ;C α,p + τ τ Lip,T ;C α,p S 3T Q 3Γ, where ɛ and π ɛ are defined in, S T, S T, S 3T vanish a he rae of T as T, and Q Γ, Q Γ, Q 3Γ are polynomials in Γ, whose coefficiens depend only on and α. By choosing < T < T small enough, depending on he size of Q i Γs, we conclude he proof. We have obained a soluion o he sysem 6 in he pah space P for a shor ime, ha is, we have, τ, v saisfying v = d d and saisfying 4. We also have Lipschiz dependence on iniial daa, Theorem. Proof. We repea he calculaion of he Theorem 9, bu his ime u ɛ, = u u and σ ɛ, = σ σ. Then we choose T small enough ha S i T Q Γ <. Acknowledgemens The research of P.C. was parially suppored by NSF gran DMS The research of J.L. was parially suppored by a Samsung scholarship. References [] P. Consanin. Lagrangian-Eulerian mehods for uniqueness in hydrodynamic sysems Adv. Mah., 78:67-, 5. [] P. Consanin. Analysis of hydrodynamic models, volume 9 of CBMS-NSF Regional Conference Series in Applied Mahemaics, Sociey for Indusrial and Applied Mahemaics SIAM, Philadelphia, PA, 7. 7