GLOBAL SMOOTH AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENENOUS INCOMPRESSIBLE NAVIER-STOKES SYSTEM

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1 GLOBAL SMOOTH AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENENOUS INCOMPRESSIBLE NAVIER-STOKES SYSTEM HAMMADI ABIDI AND PING ZHANG Abstact In this pape, we investigate the global egulaity to 3-D inhomogeneous incompessible Navie-Stokes system with axisymmetic initial ata which oes not have swil component fo the initial velocity We fist pove that the L nom to the quotient of the inhomogeneity by, namely a/ ef = /ρ /, contols the egulaity of the solutions Then we pove the global egulaity of such solutions povie that the L nom of a / is sufficiently small Finally, with aitional assumption that the initial velocity belongs to L p fo some p [,, we pove that the velocity fiel ecays to zeo with exactly the same ate as the classical Navie-Stokes system Keywos: Inhomogeneous Navie-Stokes Equations, axisymmetic flow, ecay ate AMS Subject Classification : 35Q3, 76D3 Intouction In this pape, we consie the global existence of smooth solutions to the following 3-D inhomogeneous incompessible Navie-Stokes equations with axisymmetic initial ata which oes not have swil component fo the initial velocity: t ρ + ivρu =, t, x R + R 3, t ρu + ivρu u u + Π =, ivu =, ρ, u t= = ρ, u whee ρ, u = u, u, u z stan fo the ensity an velocity of the flui espectively, an Π is a scala pessue function Such system escibes a flui that is incompessible but has non-constant ensity Basic examples ae mixtue of incompessible an non eactant flows, flows with complex stuctue eg bloo flow o moel of ives, fluis containing a melte substance, etc A lot of ecent woks have been eicate to the mathematical stuy of the above system Global weak solutions with finite enegy have been constucte by Simon in [3] see also the book by Lions [9] fo the vaiable viscosity case In the case of smooth ata with no vacuum, the existence of stong unique solutions goes back to the wok of Layzhenskaya an Solonnikov in [7] Moe pecisely, they consiee the system in a boune omain Ω with homogeneous Diichlet bounay conition fo u Une the assumption that u W p,p Ω p > is ivegence fee an vanishes on Ω an that ρ C Ω is boune away fom zeo, then they [7] pove Global well-poseness in imension = ; Local well-poseness in imension = 3 If in aition u is small in W p,p Ω, then global well-poseness hols tue Date: Sept, 4

2 H ABIDI AND P ZHANG Lately, Danchin an Mucha [] establishe the well-poseness of in the whole space R in the so-calle citical functional famewok fo small petubations of some positive constant ensity The basic iea ae to use functional spaces o noms that is scaling invaiant une the following tansfomation: ρ, u, Πt, x ρ, λu, λ Πλ t, λx, ρ, u x ρ, λu λx One may check [5, ] an the efeences theein fo the ecent pogesses along this line On the othe han, we ecall that except the initial ata have some special stuctue, it is still not known whethe o not the System has a unique global smooth solution with lage smooth initial ata, even fo the classical Navie-Stokes system N S, which coespons to ρ = in Fo instance, Ukhovskii an Yuovich [4], an inepenently Layzhenskaya [6] pove the global existence of genealize solution along with its uniqueness an egulaity fo N S with initial ata which is axisymmetic an without swil Leonai, Málek, Ne cas an Pokony [8] gave a efine poof of the same esult in [6, 4] The fist autho [] impove the egulaity of the initial ata to be u H In geneal, the global wellposeness of NS with axisymmetic initial ata is still open see [7, 5] fo instance Let x = x, x, z R 3, we enote the cylinical cooinates of x by, θ, z, i e, x, x ef = x + x, θx, x ef = tan x x with [,, θ [, π] an z R, an e ef = cos θ, sin θ,, e θ ef = sin θ, cos θ,, e z ef =,, We ae concene hee with the global existence of axisymmetic smooth solutions to which oes not have the swil component fo the velocity fiel This means solution of the fom: 3 ρt, x, x, z = ρt,, z, Πt, x, x, z = Πt,, z, ut, x, x, z = u t,, ze + u z t,, ze z By vitue of an 3, we fin that ρ, u, Π veifies t ρ + u ρ + u z z ρ =, ρ t u + ρu u + ρu z z u + Π u + z u u =, 4 ρ t u z + ρu u z + ρu z z u z + z Π u z + z u z =, u + u + zu z =, ρ t= = ρ an u, u z t= = u, uz Equation of voticity ω ef = z u u z : we get, by taking z 4 4 3, that 5 t ω +u ω +u z z ω Π u ω + z ρ Equation of Γ ef = ω : in view of 5, one has 6 t Γ + u Γ + u z z Γ + z Π ρ z Π ρ z Π ρ z ω z ρ z Γ z ρ ω + ω/ = ρ Γ + Γ = ρ As fo the classical Navie-Stokes system NS in [6, 4], the quantity Γ will play a cucial ole to pove the global well-poseness of 4 The main esult of this pape states as follows:

3 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 3 ef Theoem Let a = ρ L L with a L, an thee exist positive constants m, M so that 7 < m ρ M Let u = u e + u z e z H be a solenoial vecto file with u ef an Γ = ω belonging to L Then thee exists a positive time T so that 4 has a unique solution ρ, u on [, T which satisfies fo any T < T 8 ρ L, T R 3, u C[, T ]; H R 3 with u L, T ; H R 3 t t u t t L + ut Ḣ + Πt t L + t t u t t L t < sup t,t ] If T <, thee hols 9 lim at t T L = If we assume moeove that a L ε fo some sufficiently small positive constant ε, we have T =, an u L R + ;H + u L R + ;L + u L R + ;H + tu L R + ;L + Π L R + ;L CG + with ef G = exp C u L + u 6 L u H + u + Γ L L, an a L R + ;L C a L 3 Besies, if u L p fo some p [,, let βp ef = 3 4 p, one has 3 ut L C t βp, ut L C t βp, u t t L + ut Ḣ + Πt L Ct t βp Remak Let us ecall that the eason why one can pove the global well-poseness of classical 3-D Navie-Stokes system with axisymmetic ata an without swil is that Γ ef = ω satisfies t Γ + u Γ + u z z Γ Γ z Γ 3 Γ =, which implies fo all p [, ] that Γt L p Γ L p Nevetheless in the case of inhomogeneous Navie-Stokes system, Γ veifies 6 Then to get a global in time estimate fo Γt L, we nee the smallness conition We emak that in oe to pove the global egulaity fo the axisymmetic Navie-Stokes-Boussinesq system without swil, the authos [3] equie the suppot of the initial ensity ρ oes not intesect the axis Oz an the pojection of suppρ

4 4 H ABIDI AND P ZHANG on the axis is a compact set, which seems stonge than nea the axis Oz Finally since we shall not use the voticity equation 5, hee we o not equie the initial ensity to be close enough to some positive constant We emak that the ecay estimates 3 is in fact pove fo geneal global smooth solutions of, which oes not use the axisymmetic stuctue of the solutions, wheneve u L p fo some p [, In paticula, we get i of the technical assumption in [4] that 3 hols fo p, 6/5 an moeove the poof hee is moe concise than that in [4] Let us complete this section with the notations we ae going to use in this context Notations: Ḣ s esp H s enotes the homogeneous esp inhomogeneous Sobolev space ef with nom given by f Ḣs = R 3 ξ s fξ ξ esp f ef H s = R + ξ s fξ ξ 3 Fo X a Banach space an I an inteval of R, we enote by CI; X the set of continuous functions on I with values in X Fo q [, + ], the notation L q I; X stans fo the set of measuable functions on I with values in X, such that t ft X belongs to L q I Let R+ ef =, R, we enote = f Lq R f q z q Fo a b, we mean that thee is a + unifom constant C, which may be iffeent on iffeent lines, such that a Cb We shall enote by a b o R 3 a b x the L R 3 ef inne pouct of a an b, an finally =, z The global H estimate In this section, we shall pove the a pioi globally in time H estimate fo the velocity of povie that thee hols Befoe poceeing, let us fist ewite the momentum equation of 4 Due to u + u Similaly, one has + zu z = an cul u = ωe θ with ω ef = z u u z, we have u + z u u = z u z + u + z u u = z u z u + z u u = z z u u z = z ω u z + z u z = u z + u z z u + u = z u u z zu u z = ω ω So that we can efomulate the momentum equation of 4 as { ρ t u + ρu u + ρu z z u + Π z ω =, ρ t u z + ρu u z + ρu z z u z + z Π + ω + ω = Local in time H estimate The pupose of this subsection is to pesent the estimate of u L T H with T going to when ε in tening to zeo L enegy estimate

5 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 5 We fist euce fom the tanspot equation of 4 an 7 that m ρt,, z M While by fist multiplying the u equation of 4 by u an then integating the esulting equation ove R + with espect to the measue z, we wite ρu z t ρ + ρu + z ρu x u z t R + R + Π u z + u + z u + u z = R + Wheeas using the tanspot equation an u + z u z = of 4, we fin R + t ρ + ρu + z ρu z = t ρ + u ρ + u z z u + ρ u + z u z =, so that we obtain ρu z + u + z u + u t R + R+ z = Π u z Along the same line, we have ρu z z + t R + R + u z + z u z z = R + R + Π z u z z Hence ue to u + z u z =, we achieve ρ u + u z z + t u + u z + u z = R + Integating the above inequality ove [, t] an using gives ise to u L t L + u L t L + u 3 L t L C u L R + Ḣ enegy estimate By taking L R +, z inne pouct of the u equation of 4 with t u an using integation by pats, we have t R + u + z u + u z + R + ρ t u z = ρ u u + u z z u t u z + R + Similaly we have u z + z u z z + ρ t u z z t R + R + = ρ u u z + u z z u z t u z z + R + R + R + Π t u z Π z t u z z,

6 6 H ABIDI AND P ZHANG which togethe u + z u z = gives ise to t u + u z + u R z + ρ t u + t u z z + R + = ρ u u + u z z u t u z ρ u u z + u z z u z t u z z R + C ρu u L + ρu z z u L + ρu u z L + ρu z z u z L which along with implies t 4 R + R + + ρ t u L + ρ t u z L, u + u z + u z + t u L + t u z L C u u L + u z z u L + u u z L + u z z u z L The secon eivative estimate of the velocity By taking L R +; z inne pouct of the u equation of with z ω an using integation by pats, one has z ω z = z Π ω z ρ t u + ρu u + ρu z z u z ω z R + R + Similaly taking L R +; z inne pouct of the u z equation of with ω leas to ω z = z Π ω z R + Yet notice that R + R + R + ω z = R + ρ t u z + ρu u z + ρu z z u z ω z = R + R + ω ω + ω ω + ω z + ω z As a consequence, fo Γ given by 6, we obtain ω + z ω + Γ z C u t R L + u z t L + u u L 5 + Along the same line, we have 6 Π L C The combine estimate + u z z u L + u u z L + u z z u z L u t L + u z t L + u u L + u z z u L + u u z L + u z z u z L

7 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 7 Let δ > be a small positive constant, which will be chosen heeafte By summing up 4 with δ leas to t u + u z + u R z + Cδ t u L + t u z L + + δ ω + Γ z + Π L C Taking δ = 4C 7 R + u u L + u z z u L + u u z L + u z z u z L in the above inequality yiels t C R + + u + u z + u z + t u L + t u z L R + ω + Γ z + Π L u u L + u z z u L + u u z L + u z z u z L In oe to cope with the ight han sie tems in 7, we take cut-off functions ϕ C [, an ψ C [, with { { [, /], [/,, 8 ϕ = an ψ = [,, [, /4, an pesent the lemma as follows: Lemma Let f, z be a smooth enough function which ecays sufficiently fast at infinity Then fo ϕ given by 8, one has 9 f 4 ϕ 3 z C f L f L + f L z f L R + Poof It is easy to obseve that an f ϕ f fom which, we infe f 4 ϕ 3 z C R + R C C f ϕ f f + f, z f z = f z f z, R R + R + f f + f f f + f z R + R f z f z z f z f z Applying Höle inequality gives ise to 9

8 8 H ABIDI AND P ZHANG Now let us tun to the estimate of the nonlinea tems in 7 We fist get, by applying Höle s inequality an the -D intepolation inequality, f L 4 R f L R f L R, that u u L u L4 u L4 C ω 4 x u L u L, R 3 whee we use Biot-Savat s law ut, x = 4π R 3 y x e θ ωt, y y x 3 y an the fact that is in A p class see [] fo instance so that = u L4 u 4 4 x C ω 4 x R 3 R 3 Then by vitue of 9 an, we infe ω 4 4 x R 3 Moeove, note that fo any δ >, we wite 4 Γ ϕ z + ω 4 4 ϕ z R + R + u u L C R + Γ L Γ L Γ ϕ z + ωψ L4 Γ L + Γ L Γ L + Γ L + ωψ L ωψ L + ω L ω u C u L L + u L, L + ω L Γ L Γ L + Γ L + ω L ω L + ω L u L u L + u L C δ u L u L + u L ω L + Γ L + δ ω L + ω L + Γ L + Γ L To eal with u z z u L, we split R +u z z u z as R + u z z u z = R + u z z u ϕ z + u z z u ϕ z R +

9 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 9 By applying an convexity inequality, we get fo any δ > u z z u ϕ z 3 R + R + u z z uψ z u z ψ 4 L4 z uψ 4 L4 u z ψ 4 u L z ψ 4 z L uψ 4 z uψ L 4 L C δ u z L u z L + u z L z u L + δ z u L + z u L Befoe poceeing, let us ecall fom of [] that 4 an fom of [4] that u = z Γ z Γ, 5 W = x R W + x R W x x R W ef fo evey axisymmetic smooth function W, an whee R ij = i j By vitue of 9, we infe R + u z z u ϕ z = R + Yet it follows fom 4 an 5 that u z L Γ L, u z 3 u ϕ 3 z ϕ z u z 4 3 u 4 ϕ z 3 z ϕ z R + R + u z L u z L + u z L z u z u L z L u z L + u z L u z L z Theefoe, fo any δ >, we have u L z Γ L an u z L Γ L 6 R + u z z u ϕ z C u z L + u z 4 L u z L Γ L + δ Γ L + Γ L While since u + z u z =, we have 7 u z z u z ϕ z = R + R + u z u + u ϕ z

10 H ABIDI AND P ZHANG Due to 4 an 5, we have u z u R + ϕ z u z 3 ϕ z R + R3 u z 3 3 C u z L Γ L 3 u 3 x z Γ L 6 whee we use Sobolev-Hay inequality fom [6] that RN u q s 8 x s x Cs, q, N, k u q x R N N s N q whee x = x, z R N = R k R N k with k N, < q < N, s q an s < k, ef q = qn s, so that thee hols N q R3 u z 3 3 Wheeas it follows fom 4 that 3 x C u z L u = z Γ + z Γ z Γ Applying Hay s inequality 8 once again yiels R +u z z Γ ϕ z R + Similaly, by applying Lemma, one has u z z Γ ϕ z 9 R + L 6 u z 3 ϕ 3 z 3 z Γ L 6 R3 u z 3 x 3 3 u z L Γ L Γ L u z 4 ϕ 3 z z Γ 4 ϕ 3 z R + R + C u z L u z L + u z L z u z L Γ L Γ + Γ L L z Γ L C δ u z L + u z 4 L u z L Γ L + δ Γ L + Γ L Let W ef = z Γ Then by vitue of 4, we fin x W = R W + x R W x x R W = sin θr W + cos θr W sin θ cos θr W + sin θ R W + cos θ R W sin θ cos θ R W,

11 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM It is easy to obseve that u z sin θr W + cos θr W sin θ cos θr W ϕ z R + u z x z Γ R 3 L u z 6 L Γ L, an it follows fom a simila eivation of 9 that u z sin θ R W + cos θ R W sin θ cos θ R W ϕ 3 z R + R + u z 4 ϕ 3 z R + sin θ R W + cos θ R W sin θ cos θ R W 4 ϕ 3 z C δ u z L + u z 4 L u z L Γ L + δ Γ L + Γ L By esuming the above estimates into 7, we obtain u z z u z ϕ z C δ + u z 6 L u z L Γ L + δ Γ L + Γ L R + Theefoe, by substituting the Estimates 3, 6 an into, we obtain u z z u L C δ + u z 6 L u z L u L + Γ L + + u z 4 L z u L + δ Γ L + z u L + Γ L Note that fo the axisymmetic flow, we have fo < q < i ω L q u L q an ii ω L q + ω L q u L q Thanks to, by esuming the Estimates an into 7 an taking δ to be sufficiently small, we obtain t ut L + u t L + t u L + u + Γ Ḣ L + Π L C δ + u 6 L u L + u L u L + Γ 3 L By applying Gonwall s inequality to 3, we wite + + u z 4 L u L + δ Γ L u L t L + u L t L + tu L t L + u L t Ḣ + Γ L t L + Π L t L C exp C + u 6 L t L u L t L + u L t L u L + u + + u z 4 L L t u L L t L + Γ L t L + Γ L t L,

12 H ABIDI AND P ZHANG fom which an 3, we infe 4 u L t L + u L t L + tu L t L + u L t Ḣ + Γ L t L + Π L t L C exp C u L + u 6 L u H + u + Γ L L t L + Γ L t L The estimate of Γ Let a ef = /ρ Then we get, by taking L inne pouct of 6 with Γ an using integating by pats, that t Γt L + R ρ Γ Γ z Γ z + R ρ + = a Π z Γ z Π Γ z R + a L Π L Γ L Note that at,, z =, by using integation by pats, one has Γ Γ z = ΓΓ z aγγ z R ρ + R + R + = Γ t,, z z + aγ Γ z R R + C a Γ L L Γ 4 ρ L Theefoe ue to, we infe 5 t Γt L + m Γ L C a L Π L + Γ L On the othe han, it follows fom the tanspot equation of 4 that which yiels 6 While note fom [, 8] that t a + u a + u z z a = an a t + a u + a uz z + u a =, a t L a u L exp L t L u L t L Γ L t L3, t 3 4 Γ L t L Γ So that by integating 5 ove [, t], we obtain Γ L t L + Γ L t L Γ L + C a exp Ct 3 L 4 Γ L t L Γ L t L L t L Π L t L + Γ L t L

13 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 3 Resuming the Estimate 4 into the above inequality leas to Γ L t L + Γ L t L Γ L + C a exp Ct 3 L 4 Γ L t L Γ L t L 7 exp C u L + u 6 L u H + u + Γ L L t L + Γ L t L Poposition Let ρ, u, Π be a smooth enough solution of 4 on [, T, which satisfies Let G be given by an ef Γ 4 L t = ln C Γ L C 3 8 a G L Then une the assumption of, one has T t an thee hols 9 3 Γ L t L + Γ L t L Γ L, u L t L + u L t L + tu L t L + u L t Ḣ + Π L t L CG Poof Inee if a L is sufficiently small, we euce fom 7 an 8 that Γ L t L + Γ L t L 3 Γ L Substituting the above estimate into 4 gives ise to 3 blow-up citeia in [5] implies that T t 3 togethe with the The global in time H estimate The goal of this subsection is to pesent the global in time H estimate fo the velocity fiel Towa this, we fist pove such a estimate fo small solutions of, which oes not use the axisymmetic stuctue of the solutions Lemma Let ρ, u, Π be a smooth enough solution of on [, T, which satisfies Then thee exist positive constants η an η, which epen only on u L, so that thee hols 3 ut L + t povie that ut L η t m t ut L + η ut L + Πt L t ut L Poof We fist get, by taking the L inne pouct of the momentum equations of with t u an using integation by pats, that ρ t ut L + t ut L = ρu u t u L ρ L u L 3 u L 6 ρ t u L C u L u L u L + 4 ρ t u L, which gives t ut L ρ t ut L C u L u L u L On the othe han, it follows fom the classical estimates on linea Stokes opeato an { u + Π = ρ t u ρu u, 3 iv u =,

14 4 H ABIDI AND P ZHANG that u L + Π L C ρ t u L + ρu u L C ρ t u L + ρ L u L 3 u L 6 C ρ t u L + u L u L u L, so that we obtain fo any η > 33 We enote t ut L + 3m 4 Cη t u L + η C u L u L u L + Π L 34 τ ef = sup { t [t, T ut L η } We claim that τ = T povie that η is sufficiently small η = m 4C an η, we euce fom 33 that η C u L Inee if τ < T, taking t ut L + m t u L + η u L + Π L fo all t [t, τ, which implies ut L + τ t m t ut L + η ut L + Πt L t ut L η This contaict with 34, an thus τ = T This conclues the poof of the lemma Poposition Let ρ, u, Π be the local unique smooth solution of 4 on [, T, which satisfies Then T = an thee hols povie that ε in is sufficiently small Poof It follows fom the eivation of 3 that 35 ρut L + t ut L t = ρ u L, which ensues that fo any positive intege N, thee hols N k= k+ k ut L t ρ u L Thus thee exists k N an some t k, k + such that k + u L τ k N ρ u L an ut L N ρ u L Fo η given by Lemma, taking N so lage that ut L N ρ u L η Then we euce fom Lemma that thee hols 3

15 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 5 On the othe han, in view of 8, we can take a L to be so small that t t Thus by summing up 3 an 3, we obtain fo any t < T, 36 u L t L + tu L t L + u L t L + Π L t L u L,t ;L + tu L,t ;L + u L,t ;L + Π L,t ;L + u L t,t;l + tu L t,t;l + u L t,t;l + Π L t,t;l CG + η, fo G given by an η being etemine by Lemma Then thanks to 36 an the blow-up citeia in [5], we conclue that T = Moeove, by summing up 3 an 36, we achieve This finishes the poof of Poposition 3 Decay estimates of the global solutions of The pupose of this section is to pesent the ecay estimates 3 fo any global smooth solutions of, which oes not use the paticula axisymmetic stuctue of the solutions Lemma 3 Let ρ, u, Π be a smooth enough solution of on [, T, which satisfies Then fo t < T, one has 3 an 3 t ut L + ρu t t L + ut Ḣ + Πt L C ut H ut L, t ρu t t L + u t t L C ut H + ut 4 Ḣ ρut t L + ut 4 L Poof We fist get, by a simila eivation of 33, that t ut L + ρu t L + u Ḣ + Π L C ρu u L CM u L 6 u L 3 CM u Ḣ u L, which gives 3 On the othe han, by taking t to the momentum equation of, we wite ρ t u t + u u t ut + Π t = ρ t u t ρu t u Taking L inne pouct of the above equation with u t an using the tanspot equation of, we obtain 33 t ρu t t L + u t L = ρ t u t x ρ t u u u t x ρu t u u t x R 3 R 3 R 3 By using the tanspot equation of an integation by pats, one has ρ t u t x = ivρu u t x R 3 R 3 = ρu u t u t x R 3 M u L ρu t L u t L,

16 6 H ABIDI AND P ZHANG which togethe with the 3-D intepolation inequality that 34 u L C u Ḣ u Ḣ, implies R 3 ρ t u t x CM u Ḣ u Ḣ ρu t L + 6 u t L Along the same line, we have ρ t u u u t x = R 3 ivρuu u u t x R 3 3 = ρu i i u j j u k u k t x + R 3 ρu i u j i j u k u k t x + R 3 ρu i u j j u k i u k t x R 3 i,j,k= Applying Höle s inequality gives 3 ρu i i u j j u k u k t x M u L u L 3 u L 6 ρu t L R 3 i,j,k= C u L u Ḣ + u Ḣ ρu t L, an an 3 i,j,k= 3 i,j,k= R 3 ρu i u j i j u k u k t x M u L u L ρu t L C u L u Ḣ + ρu t L, R 3 ρu i u j j u k i u k t x M u L 6 u L 6 u t L This yiels R 3 ρ t u u u t x C u L + u Ḣ C u 4 L u Ḣ + 6 u t L Finally it is easy to obseve that ρu t u u t x M u t L 6 u L 3 ρu t L R 3 + u 4 L u Ḣ + ρu t L + 6 u t L C u Ḣ ρu t L + 6 u t L Resuming the above estimates into 33 an using 34 esults in 35 t ρu t t L + u t t L C ut H + ut 4 Ḣ ut Ḣ + ρu t t L

17 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 7 Wheeas it follows fom the classical estimates on linea Stokes opeato an 3 that u Ḣ + Π L C ρu t L + ρu u L C M ρu t L + M u L 6 u L 3 which yiels C ρu t L + u L + u Ḣ, 36 u Ḣ + Π L C ρu t L + u L Substituting 36 into 35 leas to 3 This finishes the poof of the Lemma Coollay 3 Une the assumptions of Lemma 3 an that 37 u L,T ;H + u L,T ;H C, one has fo any t < T, 38 an 39 t ut L + t t u t t L + ut Ḣ + Πt L t C exp CC u H ef = C, t t u t t L + ut + Πt t Ḣ L + t t u t t L t CC + C exp CC + C ef = C Poof We fist get, by multiplying 3 by t, that t ut t L + t ρut t L + ut + Πt Ḣ L ut L + C ut H t ut L Applying Gonwall s inequality an using 35, 37 gives ise to 38 While multiplying 3 by t t esults in t t ρut t L + t t t ut t L t ρu t t L + C ut H + ut 4 Ḣ t t ρut t L + ut 4 L Applying Gonwall s inequality leas to t t ρu t t L + t t t u t t L t C exp C u L t H + t u L t Ḣ u t ρu L t Ḣ t t L t + t ut L L u t L t H + u L t Ḣ u L t, Ḣ fom which, 36-38, we conclue the poof of 39

18 8 H ABIDI AND P ZHANG Poposition 3 Let p [, an βp ef = 3 4 p Then une the assumptions of ef Coollay 3, if we assume futhe that a = ρ L R 3 an u L p R 3, thee hols { C t βp if < p <, 3 ut L C t 3 4 if p =, fo any t < T, whee the constant C epens on a L, C, C an C given by Coollay 3 Poof Motivate by [4], in oe to use Schonbek s stategy in [], we split the phase-space R 3 into two time-epenent egions so that ut L = ξ ût, ξ ξ + St ξ ût, ξ ξ, St c whee St ef M = {ξ : ξ gt} an gt satisfies gt t, which will be chosen late on Then ue to the enegy law 35 of, one has 3 t ρut L + g t ρ ut L Mg t ût, ξ ξ To eal with the low fequency pat of u on the ight-han sie of 3, we ewite the momentum equations of as ut =e t u + t St e t t P u u + a u Π t t whee a ef = ef ρ an P = I iv enotes the Leay pojection opeato Taking Fouie tansfom with espect to x vaiables leas to ût, ξ e t ξ û ξ + which implies that ût, ξ ξ St 3 Thanks to 39, we have 33 t St e t t ξ ξ Fx u u + F x a u Π t t, e t ξ û ξ ξ + g 5 t t F x u ut L ξ t + g 3 t t F x a u Πt L ξ t t F x a u Πt L ξ t t a L t L u Πt L t While it is easy to obseve that t F x u ut L ξ a t L t t t ln t t t ut L t t u L

19 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 9 Note that fo u L p R 3, let q 34 St ef = 4 3 βp = p an p + p =, one has e t ξ û ξ ξ e qt ξ ξ St u L p t βp, q û L p whee we use the Hausöff-Young inequality in the last line so that û L p C u L p Then since gt t, we euce fom 3 that t 35 ût, ξ ξ t βp + t if p < 3, St t βp 3 if p <, In the case when 3 p <, by substituting 35 into 3, we obtain t ρut L + g t ρ ut L g t t βp t βp, fom which, we infe e t g t t ρut L ρ u L + t t e g t t t βp t Taking α > βp an g t = α t in the above inequality leas to ρut L t α + t t α βp t + t α βp, which yiels 3 fo p [3/, In the case when p < 3, by substituting the Estimate 35 into 3, one has t ρut L + g t ρ ut L g t t t 3, which implies e t g t t ρut L ρ u L + t t e g t t t 3 t Taking α > an g t ef = α t in the above inequality esults in which gives ρut L t α + t 36 ut L t 4 t α 3 t + t α, Then by vitue of 36, we wite t F x u ut L ξ t t ut L t t t 37 t t Resuming the Estimates 33, 34 an 37 into 3 esults in 38 ût, ξ ξ t βp + t 3 St t βp if < p < 3, t 3 if p =

20 H ABIDI AND P ZHANG With 38, we can epeat the pevious agument to pove 3 fo the emaining case when p [, 3/ This completes the poof of the poposition Poposition 3 Une the assumptions of Poposition 3, thee hols 3 fo any t < T Poof With Poposition 3, we shall use a simila agument fo the classical Navie-Stokes system to eive the ecay estimates fo the eivatives of the velocity see [3] fo instance In fact, fo any s < t < T, we euce fom the enegy equality of that 39 ρut L + t s ut L t = ρus L While multiplying 3 by t s leas to t s ut t L + t s ρut t L + ut L + Πt L ut L + C ut H t s ut L, Applying Gonwall s inequality an using 39 esults in t s ut L exp C u t L t H ut L t s exp CC ρus L In paticula, taking s = t gives ut L C t ut/ L, fom which an 3, we infe fo any t < T { t βp 3 ut L C if < p <, t 5 if p =, Similaly by applying Gonwall s lemma to 3 ove [s, t], we wite 3 t ut L + ρut t L + ut L + Πt L t s exp C u L t H us L exp CC us L Wheeas by multiplying 3 by t s an applying Gonwall s lemma to esulting inequality, we get t s ρu t t L t ρu t t L t + u 4 L s,t;l s exp C u L t H + u L t Ḣ u L t Ḣ exp CC + C us L + u 4 L s,t;l Taking s = t in the above inequality an using 3, we obtain { t u t t t βp L C if < p <, t t 5 if p =

21 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM which togethe with 36 an 3 ensues that { Ct u t t L + ut + Πt t βp if < p <, 3 Ḣ L C t t 5 if p =, fo any t < T With 3 an 3, it emains to pove 3 fo p = As a matte of fact, we fist euce fom 3 that t F x a u Πt L ξ t t a L t L u Πt L t a t L t t 5 4 t C With 33 being eplace by the above inequality, by epeating the poof of Poposition 3, we can pove the fist inequality of 3 fo p = Then epeating the poof of 3, we conclue the poof of the emaining two inequalities in 3 fo p = This finishes the poof of Poposition 3 4 The poof of Theoem The goal of this section is to complete the poof of Theoem In oe to o so, we fist pove the following globally in time Lipschitz estimate fo the convection velocity fiel, which will be use to pove the popagation of the size fo a L Lemma 4 Let ρ, u, Π be a c smooth enough axisymmetic solution of on [, T Then une the assumptions 7 an, we have T =, an thee hols 4 u L R + ;L C, fo some positive constant epening on m, M an u H Poof Une the assumptions of 7 an, we euce fom Poposition that T = an moeove Coollay 3 ensues that 4 sup t ut L + t [, sup t [, t u t t L + ut Ḣ + Πt L t C, t t u t t L + ut + Πt Ḣ L + t t u t t L t C, whee C an C given by 38 an 39 espectively imbeing theoem, we obtain 43 t t u t t L 6 t C In paticula, by using Sobolev On the othe han, in view of 3, we euce fom the classical estimates of linea Stokes opeato that ut L 6 + Πt L 6 C u t t L 6 + ut L ut L 6, which togethe with 34 yiels t t ut L 6 + Πt L 6 t C t t u t t L t + 6 t t ut Ḣ ut 3 t Ḣ

22 H ABIDI AND P ZHANG Yet it follows fom 4 that t t ut Ḣ ut 3 t C C Ḣ C 3 t t t C C C 3, which togethe with 43 ensues that 44 t t ut L 6 + Πt L 6 t CC + C C ef = C 3 By vitue of 4 an 44, we infe ut L t C This gives ise to 4 CC 4 ut Ḣ ut L 6 t t t t t ut 4 L t 6 CC 4 t 3 t t CC 4 C 4 3 t t ut 4 L t 6 Now we ae in a position to complete the poof of Theoem Poof of Theoem The geneal stategy to pove the existence esult to a nonlinea patial iffeential equation is fist to constuct an appopiate appoximate solutions, an then pefom the unifom estimates to these appoximate solution sequence, an finally the existence esult follows fom a compactness agument Fo simplicity, hee we just pesent the a pioi estimates to smooth enough solutions of 4 Given axisymmetic initial ata ρ, u with ρ satisfying 7 an a L L, a L, u H, we euce fom 3 an 5 that thee exists a maximal positive time T so that 4 has a solution on [, T which satisfies fo any T < T, u L T H + u L T H + tu L T L + Π L T L + Γ L T L + Γ L T L C, fom which an Coollay 3, we euce that thee hols 8 An hence the uniqueness pat of Theoem follows fom the uniqueness esult in [] Now if T < an thee hols lim at t T L = C < Let us take δ so small that mcc Then we get, by summing up 3 an mδ 5, that t ut L + u t + mδ Γt L L + t u L + u + Ḣ Π L + Γ L + δ Γ L C δ + u 6 L u L + u L u L + Γ L + + u z 4 L z u L

23 AXISYMMETRIC SOLUTIONS OF 3-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 3 Applying Gonwall s inequality an using 3 leas to u L T L + u L T L + Γ L T L + tu L T L + Π L T L + u L T Ḣ + Γ L T L + Γ L T L C, fo any T < T Theefoe we can exten the solution beyon the time T, which contaicts with the maximality of T Hence thee hols 9 Une the assumption of, we euce fom Poposition that T = an thee hols Moeove, Lemma 4 ensues that u L R + ;L C, which togethe with 6 an u L R + ;L u L R + ;L gives ise to Finally with aitional assumption that u L p fo some p [,, we euce fom Poposition 3 that thee hols the ecay estimate 3 This finishes the poof of Theoem Acknowlegments We woul like to thank Raphaël Danchin an Guilong Gui fo pofitable iscussions on this topic Pat of this wok was one when we wee visiting Moningsie Cente of Mathematics, CAS, in the summe of 4 We appeciate the hospitality an the financial suppot fom the Cente P Zhang is patially suppote by NSF of China une Gant 3737, the fellowship fom Chinese Acaemy of Sciences an innovation gant fom National Cente fo Mathematics an Inteisciplinay Sciences Refeences [] H Abii, Résultats e égulaité e solutions axisymétiques pou le système e Navie-Stokes, Bull Sci Math, 3 8, [] H Abii, T Hmii an S Keaani, On the global well-poseness fo the axisymmetic Eule equations, Math Ann 347, 5 4 [3] H Abii, T Hmii an S Keaani, On the global egulaity of axisymmetic Navie-Stokes-Boussinesq system, Discete Contin Dyn Syst, 9, [4] H Abii, G Gui an P Zhang, Stability to the global lage solutions of the 3-D inhomogeneous Navie- Stokes equations, Comm Pue Appl Math, 64, [5] H Abii, G Gui an P Zhang, Well-poseness of 3-D inhomogeneous Navie-Stokes equations with highly oscillatoy initial velocity fiel, J Math Pues Appl, 9 3, 66-3 [6] M Baiale an G Taantello, A Sobolev-Hay inequality with applications to a nonlinea elliptic equation aising in astophysics, Ach Ration Mech Anal, 63, [7] D Chae an J Lee, On the egulaity of the axisymmetic solutions of the Navie-Stokes equations, Math Z, 39, [8] R Danchin, Axisymmetic incompessible flows with boune voticity, Russian Math Suveys, 6 7, [9] R Danchin, Local an global well-poseness esults fo flows of inhomogeneous viscous fluis, Av Diffeential Equations, 9 4, [] R Danchin an P B Mucha, A Lagangian appoach fo the incompessible Navie-Stokes equations with vaiable ensity, Comm Pue Appl Math, 65, [] R Danchin an P Zhang, Inhomogeneous NavieCStokes equations in the half-space, with only boune ensity, J Funct Anal, 67 4, [] L Gafakos, Classical an moen Fouie analysis, Peason Eucation, Inc, Uppe Sale Rive, NJ, 4

24 4 H ABIDI AND P ZHANG [3] C He an T Miyakawa, On two-imensional Navie-Stokes flows with otational symmeties, Funkcial Ekvac, 49 6, 63-9 [4] T Hmii an F Rousset, Global well-poseness fo the Eule-Boussinesq system with axisymmetic ata, J Funct Anal, 6, [5] H Kim, A blow-up citeion fo the nonhomogeneous incompessible Navie-Stokes equations, SIAM J Math Anal, 37 6, [6] O A Lay zenskaja, Unique global solvability of the thee-imensional Cauchy poblem fo the Navie- Stokes equations in the pesence of axial symmety, Russian Zap Nau cn Sem Leninga Otel Mat Inst Steklov LOMI, 7 968, [7] O Layzhenskaya an V Solonnikov: The unique solvability of an initial-bounay value poblem fo viscous incompessible inhomogeneous fluis, Jounal of Soviet Mathematics, 9 978, [8] S Leonai, J Málek, J Ne cas an M Pokony, On axially symmetic flows in R 3, Z Anal Anwenungen, 8 999, [9] P L Lions: Mathematical topics in flui mechanics Vol Incompessible moels, Oxfo Lectue Seies in Mathematics an its Applications, 3 Oxfo Science Publications The Claenon Pess, Oxfo Univesity Pess, New Yok, 996 [] C Miao an X Zheng, On the global well-poseness fo the Boussinesq system with hoizontal issipation, Comm Math Phys, 3 3, [] M Paicu, P Zhang an Z Zhang, Global well-poseness of inhomogeneous Navie-Stokes equations with boune ensity, Comm Patial Diffeential Equations, 38 3, 8-34 [] M Schonbek, Lage time behavio of solutions to Navie-Stokes equations, Commin P D E, 986, [3] J Simon: Nonhomogeneous viscous incompessible fluis: existence of velocity, ensity, an pessue, SIAM J Math Anal, 99, 93 7 [4] M R Ukhovskii, an V I Iuovich, Axially symmetic flows of ieal an viscous fluis filling the whole space, J Appl Math Mech, [5] P Zhang an T Zhang, Global axisymmetic solutions to thee-imensional Navie-Stokes system, Int Math Res Not, IMRN4, no 3, 6-64 H Abii Dépatement e Mathématiques Faculté es Sciences e Tunis Campus univesitaie 9 Tunis, Tunisia aess: habii@univ-evyf P Zhang Acaemy of Mathematics & Systems Science, an Hua Loo-Keng Key Laboatoy of Mathematics, Chinese Acaemy of Sciences, Beijing 9, CHINA aess: zp@amssaccn