Strong solutions of the compressible nematic liquid crystal flow


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1 Strong solutions of the compressible nematic liqui crystal flow Tao Huang Changyou Wang Huanyao Wen April 9, 11 Abstract We stuy strong solutions of the simplifie EricksenLeslie system moeling compressible nematic liqui crystal flows in a omain R 3. We first prove the local existence of unique strong solutions provie that the initial ata ρ, u, are sufficiently regular an satisfy a natural compatibility conition. The initial ensity function ρ may vanish on an open subset i.e., an initial vacuum may exist). We then prove a criterion for possible breakown of such a local strong solution at finite time in terms of blow up of the quantities ρ L t L x an L 3 t L x. 1 Introuction Nematic liqui crystals are aggregates of molecules which possess same orientational orer an are mae of elongate, rolike molecules. The continuum theory of liqui crystals was evelope by Ericksen [9] an Leslie [3] uring the perio of 1958 through 1968, see also the book by e Gennes [1]. Since then there have been remarkable research evelopments in liqui crystals from both theoretical an applie aspects. When the flui containing nematic liqui crystal materials is at rest, we have the wellknown OssenFrank theory for static nematic liqui crystals, see HarLin Kinerlehrer [13] on the analysis of energy minimal configurations of namatic liqui crystals. In general, the motion of flui always takes place. The socalle EricksenLeslie system is a macroscopic continuum escription of the time evolution of the materials uner the influence of both the flow velocity fiel u an the macroscopic escription of the microscopic orientation configurations of rolike liqui crystals. When the flui is an incompressible, viscous flui, Lin [19] first erive a simplifie Ericksen Leslie equation moeling liqui crystal flows in Subsequently, Lin an Liu [, 1] mae some important analytic stuies, such as the existence of weak an strong solutions an the partial regularity of suitable solutions, of the simplifie EricksenLeslie system, uner the assumption that the liqui crystal irector fiel is of varying length by Leslie s terminology or variable egree of orientation by Ericksen s terminology. When the flui is allowe to be compressible, the EricksenLeslie system becomes more complicate an there seems very few analytic works available yet. We woul like to mention that very recently, there have been both moeling stuy, see Morro [31], an numerical stuy, see Zakharov Vakulenko [39], on the hyroynamics of compressible nematic liqui crystals uner the influence of temperature graient or electromagnetic forces. This paper, an the companion paper [18], aims to stuy the strong solutions of the flow of compressible nematic liqui crystals an the blow up criterions. Department of Mathematics, University of Kentucky, Lexington, KY 456, USA. School of Mathematical Sciences, South China Normal University, Guangzhou, 51631, P. R. China. 1
2 Let R 3 be a omain. We will consier the simplifie version of EricksenLeslie system moeling the flow of compressible nematic liqui crystals in : ρ t + ρu) =, 1.1) ρu t + ρu u + P ρ)) = Lu, 1.) t + u = +, 1.3) where ρ : [, + ) R 1 is the ensity function of the flui, u : [, + ) R 3 represents velocity fiel of the flui, P = P ρ) represents the pressure function, : [, + ) S represents the macroscopic average of the nematic liqui crystal orientation fiel, is the ivergence operator in R 3, an L enotes the Lamé operator: Lu = µ u + µ + λ) iv u, where µ an λ are shear viscosity an the bulk viscosity coefficients of the flui respectively that satisfy the physical conition: µ >, µ + 3λ. 1.4) We refer to the reaers to consult the recent preprint [6] by DingHuangWenZi for the erivation for the system 1.1)1.3) base on energeticvariational approaches. Throughout this paper, we assume that P : [, + ) R is a locally Lipschitz continuous function. 1.5) Notice that 1.1) is the equation of conservation of mass, 1.) is the equation of linear momentum, an 1.3) is the equation of angular momentum. We woul like to point out that the system 1.1)1.3) inclues several important equations as special cases: i) When ρ is constant, the equation 1.1) reuces to the incompressibility conition of the flui u = ), an the system 1.1)1.3) becomes the equation of incompressible flow of namatic liqui crystals provie that P is a unknown pressure function. This was previously propose by Lin [19] as a simplifie EricksenLeslie equation moeling incompressible liqui crystal flows. ii) When is a constant vector fiel, the system 1.1)1.) becomes a compressible Navier Stokes equation, which is an extremely important equation to escribe compressible fluis e.g., gas ynamics). It has attracte great interests among many analysts an there have been many important evelopments see, for example, Lions [7], Feireisl [1] an references therein). iii) When both ρ an are constants, the system 1.1)1.) becomes the incompressible Naiver Stokes equation provie that P is a unknown pressure function, the funamental equation to escribe Newtonian fluis see, for example, Lions [6] an Temam [34] for survey of important evelopments). iv) When ρ is constant an u =, the system 1.1)1.3) reuces to the equation for heat flow of harmonic maps into S. There have been extensive stuies on the heat flow of harmonic maps in the past few ecaes see, for example, the monograph by LinWang [4] an references therein). From the viewpoint of partial ifferential equations, the system 1.1)1.3) is a highly nonlinear system coupling between hyperbolic equations an parabolic equations. It is very challenging to unerstan an analyze such a system, especially when the ensity function ρ may vanish or the flui takes vacuum states. In this paper, we will consier the following initial conition: ρ, u, ) = ρ, u, ), 1.6) t= an one of the three types of bounary conitions:
3 1) Cauchy problem: = R 3, an ρ, u vanish at infinity an is constant at infinity in some weak sense). 1.7) ) Dirichlet an Neumann bounary conition for u, ): R 3 is a boune smooth omain, an u, ν ) =, 1.8) where ν is the unit outer normal vector of. 3) Navierslip an Neumann bounary conition for u, ): R 3 is boune, simply connecte, smooth omain, an u ν, u) ν, ν ) =, 1.9) where u enotes the vorticity fiel of the flui. To state the efinition of strong solutions to the initial an bounary value problem 1.1)1.3), 1.6) together with 1.7) or 1.8) or 1.9), we introuce some notations. We enote f x = f x. For 1 r, enote the L r spaces an the stanar Sobolev spaces as follows: { } L r = L r ), D k,r = u L 1 loc ) : k u L r <, W k,r = L r D k,r, H k = W k,, D k = D k,, { } D 1 = u L 6 : u L <, an satisfies 1.7) or 1.8) or 1.9) for the part of u, H 1 = L D 1, u D k,r = k u L r. Denote an let Q T = [, T ] T > ), Du) = 1 u + u) t ) enote the eformation tensor, which is the symmetric part of the velocity graient. Definition 1.1 For T >, ρ, u, ) is calle a strong solution to the compressible nematic liqui crystal flow 1.1)1.3) in, T ], if for some q 3, 6], ρ C[, T ]; W 1,q H 1 ), ρ t C[, T ]; L L q ); u C[, T ]; D D 1 ) L, T ; D,q ), u t L, T ; D 1 ), ρu t L, T ; L ); C[, T ]; H ) L, T ; H 3 ), t C[, T ]; H 1 ) L, T ; H ), = 1 in Q T ; an ρ, u, ) satisfies 1.1)1.3) a.e. in, T ]. The first main result is concerne with local existence of strong solutions. 3
4 Theorem 1. Assume that P satisfies 1.5), ρ, ρ W 1,q H 1 L 1 for some q 3, 6], u D D 1, H an = 1 in. If, in aitions, the following compatibility conition Lu P ρ )) = ρ g for some g L, R 3 ) 1.1) hols, then there exist a positive time T > an a unique strong solution ρ, u, ) of 1.1)1.3), 1.6) together with 1.7) or 1.8) or 1.9) in, T ]. We woul like to point out that an analogous existence theorem of local strong solutions to the isentropic 1 compressible NaiverStokes equation, uner the first two bounary conitions 1.7) an 1.8), has been previously establishe by ChoeKim [4] an ChoChoeKim [3]. A byprouct of our theorem 1. also yiels the existence of local strong solutions to a larger class of compressible NavierStokes equations uner the Navierslip bounary conition 1.9), which seems not available in the literature. In imension one, DingLinWangWen [7] have proven that the local strong solution to 1.1) 1.3) uner 1.6) an 1.8) is global. For imensions at least two, it is reasonable to believe that the local strong solution to 1.1)1.3) may cease to exist globally. In fact, there exist finite time singularities of the transporte) heat flow of harmonic maps 1.3) in imensions two or higher we refer the intereste reaers to [4] for the exact references). An important question to ask woul be what is the main mechanism of possible break own of local strong or smooth) solutions. Such a question has been stuie for the incompressible Euler equation or the NavierStokes equation by BealeKatoMaja in their poineering work [1], which showe that the L boun of vorticity u must blow up. Later, Ponce [9] rephrase the BKMcriterion in terms of the eformation tensor Du). When ealing with the isentropic compressible NavierStokes equation, there have recently been several very interesting works on the blow up criterion. For example, if < T < + is the maximum time for strong solution, then i) HuangLiXin [15] establishe a Serrin type criterion: lim T T ivu L 1,T ;L ) + ρu L s,t ;L )) r = for s + 3 r 1, 3 < r ; ii) SunWangZhang [35], an inepenently [15], showe that if 7µ > λ, then lim T T ρ L,T ;L ) = ; an iii) HuangLiXin [16] showe that lim T T Du) L 1,T ;L ) =. When ealing the heat flow of harmonic maps 1.3) with u = ), Wang [36] obtaine a Serrin type regularity theorem, which implies that if < T < + is the first singular time for local smooth solutions, then lim T T L,T ;L ) =. When ealing with the incompressible nematic liqui crystal flow, LinLinWang [5] an Lin Wang [3] have establishe the global existence of a unique almost strong solution for the initialbounary value problem in boune omains in imension two, see also Hong [14] an XuZhang [38] for some relate works. In imension three, for the incompressible nematic liqui crystal flow HuangWang [17] have obtaine a BKM type blowup criterion very recently, while the existence of global weak solutions still remains to be a largely open question. Motivate by these works on the blow up criterion of local strong solutions to the NavierStokes equation an the incompressible nematic liqui crystal flow, we will establish in this paper the following blowup criterion of breakown of local strong solutions uner the bounary conition 1.1) or 1.). Theorem 1.3 Let ρ, u, ) be a strong solution of the initial bounary problem 1.1)1.3), 1.6) together with 1.7) or 1.8). Assume that P satisfies 1.5), an the initial ata ρ, u, ) satisfies 1 i.e. P = aρ γ for some a > an γ > 1. that has at most finitely many possible singular time. 4
5 1.1). If < T < + is the maximum time of existence an 7µ > 9λ, then ) lim ρ L T T,T ;L ) + L 3,T ;L ) =. 1.11) We woul like to make a few comments of Theorem 1.3. Remark 1.4 a) Since we can t yet prove Lemma 4. for the Navierslip an Neumann bounary conition 1.9), it is unclear whether Theorem 1.3 remains to be true uner the bounary conition 1.9). b) In [18], we obtaine a blowup criterion of 1.1)1.3) uner the initial conition 1.6) an the bounary conition 1.7) or 1.8) or 1.9) in terms of u an : if < T < + is the maximum time of existence of strong solutions, then ) lim Du) L T T 1,T ;L ) + L,T ;L ) = +. b) For compressible liqui crystal flows without the nematicity constraint = 1) 3, LiuLiu [] have recently obtaine a Serrin type criterion on the blowup of strong solutions. c) It is a very interesting question to ask whether there exists a global weak solution to the initialbounary value problem of 1.1)1.3) in imensions at least two. In imension one, such an existence has been obtaine by DingWangWen [8]. Now we briefly outline the main ieas of the proof, some of which are inspire by earlier works on the isentropic compressible NavierStokes equations by [3], [35], an [16]. To obtain the existence of a unique local strong solution to ), uner 1.6) an 1.7) or 1.8) or 1.9), we employ the Galerkin s metho that requires us to establish a priori estimate of the quantity ρt) H 1 W 1,q + ut) L + ρu t t) L + ) L, 3 < q 6 for strong solutions ρ, u, ) in the form of a Gronwall type inequality. See Theorem.1. It may be of inepenent interest that we establish W,q estimate for the Lamé equation uner the Navierslip bounary conition, see Lemma 3.1. To prove the blowup criterion 1.11) of Theorem 1.3 in terms of ρ an, a critical step is to establish the L t L q xestimate of ρ. From the continuity equation 1.1), this requires that the Lipschitz norm of velocity fiel u, or ut) L q is boune in L 1 t. This is one in several steps. 1) We show that uner the conition 7µ > 9λ, the boun of ρ L t L + x L 3 ) in equations t L x 1.) an 1.3) can yiel both a high integrability an a high orer estimate of u an, i.e. both ρ 1 5 u L t L 5 + x L ) an u t L5 x L + t L x L t L ) are boune. See Lemma 4.. x ) Base on these estimates from 1), we establish that 3 is boune in L t L x an u is boune in L t Wx 1,q + L t BMO x ). To achieve it, we aapt the approach, ue to SunWangZhang [35], by ecomposing u = w + v, where v H 1 ) solves the Lamé equation Lv = P ρ)). One can prove that v L t BMO x ) by the elliptic regularity theory. The ifficult part is to show that w L t L q x for 3 < q 6. In orer to obtain this estimate, we first establish that ρ u L t L + x t L t L ) an u x L + tt t L x L t L ) are boune by viewing 1.) as an evolution x equation of the material erivative u u t + u u an performing secon orer energy estimates of both equations 1.) an 1.3). Then we employ W,q estimate of the Lamé equation to control w L q. The etails are illustrate by Lemma 4.4 an Corollary the right han sie of equation 1.3) is replace by + f) for some smooth function f : R 3 R 3, e.g. f) = 1). 5
6 3) We show that ρ L Lq is boune by an argument similar to [35] 5. Then we apply W,q estimate of the Lamé equation again to control u L t L an u. See Lemma 4.6, x L t D,q x Corollary 4.7, an Corollary 4.8. It is interesting to notice that uring the proof of both the existence of a unique local strong solutions an the blowup criterion for strong solutions, specific forms of the pressure function P ρ) play no roles an it is the local Lipschitz regularity of P that matters. The paper is written as follows. In, we erive some a priori estimates for strong solutions or approximate solutions via the Galerkin s metho. In 3, we prove both the local existence by the Gakerlin s metho an uniqueness of strong solutions. In 4, we iscuss the blow up criterion of strong solutions an prove Theorem 1.3. Acknowlegement. The first two authors are partially supporte by NSF grant The work is complete uring the visit of thir author to University of Kentucky, which is partially supporte by the secon author s NSF grant 616. The thir author wishes to thank the epartment of Mathematics for its hospitality. A priori estimates In the section, we will erive some a priori estimates for strong or smooth solutions ρ, u, ) to 1.1) 1.3) on a boune omain, associate with the initial conition 1.6) an the bounary conition 1.8) or 1.9), provie that the initial ensity function has a positive lower boun, ρ δ >. All these a priori estimates we will obtain are inepenent of δ > an the size of the omain when = B R R 1) is a ball in R 3, which are the crucial ingreients to prove the local existence of strong solutions to 1.1)1.3) when we allow the initial ata ρ an unboune omain = R 3. Although these estimates may have their own interests, we mainly apply them to the approximate solutions to 1.1)1.3) that are constructe by the Galerkin s metho. Throughout the paper, we enote by C generic constants that epen on ρ W 1,q H 1 L 1, u D D 1, H, an P, but are inepenent of δ >, the solutions ρ, u, ) an the size of omain when = B R R 1) is a ball in R 3. We will also use the obvious notation X1 X k = k Xi for Banach spaces X i, 1 i k an k =, 3. We will use A B to enote A CB for some constant generic C >. Let ρ, u, ) be a strong solution of 1.1)1.3) in, T ] or the approximate solutions ρ m, u m, m ) of 1.1)1.3) constructe by the Galerkin s metho in 3. below). For simplicity, we assume < T 1. For < t < T, set Φt) := sup s t i=1 ρs) H 1 W 1,q + us) L + ρu t s) L + s) H )..1) The main aim of this section is to estimate each term of Φ in terms of some integrals of Φ. In 3 below, we will apply arguments of Gronwall s type to prove that Φ is locally boune. Throughout this section an 3, we will let F to enote the set that consists of monotonic increasing, locally boune functions M from [, + ) to [, + ) with M) =, which are inepenent of δ an the size of. The reaer will see that the exact form of M F is not important an may vary from lines to lines uring the proof of the Lemmas. Now we state the main theorem of this section. 6
7 Theorem.1 There exists M F such that for any < t < T, it hols where [ Φt) exp CMρ, u, ) + C t ] MΦs)) s,.) Mρ, u, ) = 1 + Lu P ρ )) L..3) ρ The proof of Theorem.1 is base on several Lemmas. We may assume P ) =. Observe that 1.5) implies that the Lipschitz norm B P R) := P L [,R]) : [, + ) [, + ) is montonic increasing an locally boune..4) Lemma. energy inequality) There exists M F such that for any < t < T, it hols ρ u + ) t [ x + u + + ] t x C + MΦs)) s..5) Proof. Here we only sketch the proof for the bounary conition 1.9). Multiplying 1.) by u an integrating over, using u = iv u u) an 1.1), an applying integration by parts several times, we obtain 1 ρ u x + µ u + µ + λ) ivu ) x = P ρ)ivu x u x..6) Since is assume to be simply connecte for the bounary conition 1.9), we have see [37]): u L u L + ivu L, u H 1 ) with u ν = on..7) This an 1.4) imply µ u + µ + λ) ivu ) x µ 3 u + ivu ) x 1 C u x..8) By Cauchy inequality, we have P ρ)ivu x 1 C u x + C P ρ) x..9) Multiplying 1.3) by + an integrating over, using integration by parts an the fact that = 1 we obtain 1 x + + x = u x..1) Combining.6),.8),.9), an.1) together, we obtain ρ u + ) x + 1 C u + + ) x C To estimate the right han sie of.11), first observe that by.4) we have 4 P ρ) x..11) ρ L + P ρ) L + P ρ) H 1 W 1,q CΦ + CB P ρ L )Φ MΦ).1) 4 when = B R for R 1, one can the inepenence of C with respect to R as follows: ρ L B R ) max x B R ρ L B 1 x)) C max x B R ρ W 1,q B 1 x)) C ρ W 1,q B R ). 7
8 for some M F. It follows from 1.1) an Sobolev s inequality that P ρ) x = P ρ ) x + C + C C + t t t P ρ)p ρ) ρivu ρ u) x B P ρ L ) P ρ) L 3 ρ L + P ρ) L ρ L ) u L s MΦs)) s C + MΦt)).13) as MΦs)) is increasing an t 1. Substituting.13) into.11) an integrating over [, t] yiels.5). Now we want to estimate ut) H 1 in terms of Φt). Lemma.3 There exists M F such that for < t < T, it hols ut) H 1 MΦt))..14) Proof. By the stanar H estimate of the Lamé equation with respect to the bounary conition 1.7) or 1.8) or 1.9),.1), an Höler s inequality, we have u H 1 Lu L + u L ρu t L + ρu u L + P ρ)) L + L + u L ρ L ρu t L + ρ L u L 6 u L 3 + B P ρ L ) ρ L.15) + L 3 L + u 6 L MΦ)1 + u L u 6 L ) + C 3 L 3 L 6 for some M F. By the interpolation inequality, Sobolev s inequality 5, we obtain u L u 6 L C u 3 3 L u H 1..16) Similar to.16), by.5), we obtain L 3 L 6 L L 6 H 1 H 1 L + H 1 L MΦ).17) for some M F. Substituting.16),.17) into.15), an using.5) an Cauchy s inequality, we have u H 1 1 u H 1 + MΦt)) for some M F. This gives.14) an completes the proof. Now we want to estimate ρu t L. More precisely, we have 5 when = B R for R 1, by simple scalings, one has f L 6 B R ) C R 1 f L B R ) + f L B R )) C f H 1 B R ). 8
9 Lemma.4 There exists M F such tha for any < t < T, it hols t t ρ u t x + u t xs CMρ, u, ) + MΦs)) s..18) Proof. Differentiating 1.) with respect to t, we have 6 ρu tt + ρu u t + ρ t u t + ρ t u u + ρu t u + P ρ)) t =µ + λ) iv u t µ u t ) t + t t I 3 )..19) Multiplying.19) by u t, integrating the resulting equations over, an using 1.1) an integration by parts, we have 1 µ ρ u t x + + λ) iv ut + µ u t ) x = ρuu t u t x ρ t u u u t x ρu t u u t x + P ρ)ρ t iv u t x.) 5 + t + t t I 3 ) : u t x = II i. By Höler s inequality, Sobolev s inequality,.1), an.14), we have i=1 II 1 u t L ρu t L ρu L u t L ρu t L ρ L u H 1 MΦ) u t L for some M F. By 1.1), Höler s inequality, Sobolev s inequality,.1), an.14), we have II = ρu u u u t ) x = ρu u u u t + u u u t + u u u t ) x ρ L u L u 6 L 6 u t L + ρ L u L u 6 L u t L 6.1).) + ρ L u L 6 u L 6 ρu t L MΦ)1 + u t L ) for some M F. For II 3, by.14) we have II 3 ρ L ρu t L u L 3 u t L 6 ρ L ρu t L u 1 L u 1 H 1 u t L MΦ) u t L.3) for some M F. For II 4, by 1.1),.1), an.14) we have II 4 B P ρ L ) ρ t L iv u t L B P ρ L ) ρ L u L + ρ L iv u L ) iv u t L.4) MΦ) u t L 6 here we have use the fact that = 1 I 3), where = ) xi xj is the ientity matrix of orer i,j 3 an I3
10 for some M F. For II 5, by.5) we have II 5 t u t x u t L L t L u t L H t L.5) u t L L + H 1) t L C + MΦ)) u t L t L for some M F. Substituting.1).5) into.), an using Cauchy s inequality, we have 1 ρ u t x + 1 u t x C 1.6) u t x + MΦ) + C + MΦ)) t L C for some M F, where we have use the following inequality ue to [37]: if i) either is simply connecte an u ν = on or ii) u = on 7, then By.6), we have ρ u t x + 1 C Differentiating 1.3) with respect to x, we have From.9), we have 8 u t L ivu t L + u t L..7) u t x MΦ) + C + MΦ)) t L..8) t = ) u )..9) t L u L + u L + L + 3 L 6 + L L u L + u L 6 L 3 + L L ) 3 + L L H u L + u L H 1 + L L ) 3.3) + H L MΦ) + 1 for some M F. Substituting.3) into.8), an using Cauchy s inequality, we have ρ u t x + 1 u t x MΦ) + C.31) C for some M F. Integrating.31) over, t), an using 1.), an 1.1), we have t t ρ u t x + u t xs C ρ u t x + MΦs))s + C t= CMρ, u, ) + t MΦs))s 7 in fact, in this case, the inequality.7) is an equality. 8 here we also use the Sobolev s inequality: L ) C H ) an the fact that C can be chosen inepenent of R when = B R for R 1. 1
11 for some M F. This completes the proof. As an immeiate consequence of Lemma.4, we obtain an estimate of u L. Lemma.5 There exists M F such that for < t < T, it hols ut) x CMρ, u, ) + t MΦs)) s..3) Proof. By Cauchy s inequality, Lemma.), Lemma.3, an Lemma.4, we have t u t)x = u x + u u t xs t t C + u xs + u t xs CMρ, u, ) + for some M F. This completes the proof. t MΦs)) s Lemma.6 There exists M F such that for < t < T, it hols t } ρt) H 1 W {CMρ 1,q exp, u, ) + C MΦs)) s..33) Proof. It follows from [3] page 49,.11)) that ρt) H 1 W 1,q ρ H 1 W 1,q exp {C t u H 1 D 1,qs }..34) By W,q estimate of the Lamé equation uner either Dirichlet bounary conition 1.8) or the Navierslip bounary conition 1.9) see Lemma 3.1 below), 1.), an Sobolev s inequality, we have u L q ρu t L q + ρu u L q + P ρ)) L q + L q = 4 III i..35) i=1 If q = 6, then by Sobolev s inequality we have III 1 ρ L u t L 6 Φ u t L..36) If q 3, 6), then by Höler s inequality an Sobolev s inequality, we have III 1 ρ L 6q 6 q 6 q 6q u t L 6 ρ L 1 6 q 1 6q ρ L u t L Φ u t L,.37) where we have use the fact that ρx = ρ x. q 3, 6], From.36) an.37), we have that for III 1 Φ u t L..38) 11
12 For III, if q 3, 6], then by similar arguments, Lemma., an Lemma.3, we have for some M F. For III 3 an III 4, if q 3, 6], then we have III Φ u H 1 MΦ).39) III 3 + III 4 CB P ρ L ) ρ L q + H MΦ).4) for some M F. Substituting.38),.39) an.4) into.35), we have u L q Φ u t L + MΦ) u t L + MΦ).41) for some M F. Integrating.41) over, t), an using Cauchy s inequality an.18), we have t u L q CMρ, u, ) + Substituting.14) an.4) into.34), we have ρt) H 1 W 1,q exp {CMρ, u, ) + C for some M F. This completes the proof. t MΦs)) s..4) t } MΦs)) s Lemma.7 There exists M F such that for any < t < T, it hols L + t t L s C + t MΦs)) s..43) Proof. Multiplying.9) by t an integrating over, using integration by parts an ν = on, we obtain t L + 1 [ L = ) u ) ] t x 1 t L + C ) x + C u ) x. Thus we have t L + L ) x + u ) x..44) Similar to the proof of.3), we obtain t L + L MΦ).45) for some M F. Integrating.45) over, t) an applying W, estimate of the equation 1.3), we have L + t t L s L + t MΦs)) s C + t MΦs)) s. This completes the proof. 1
13 Lemma.8 There exists M F such that for < t < T, it hols t t 4 3 L + t L s CMρ, u, ) + MΦs)) s)..46) Proof. Multiplying.9) by t, integrating over, using ν = on an integration by parts, we obtain t L + 1 [ u L = ) ) ] t x = [ u ) ) ] x [ u ) ) ] x. t Now we nee to estimate the secon term of right sie as follows. [ u )] x = [ u t + u t + u t + u t ] x t 4 = IV i. i=1 By Höler s inequality an Sobolev s inequality, we have.47).48) IV 1 u t L L L u t L H MΦ) + u t L.49) for some M F. By Höler s inequality, Sobolev s inequality,.14),.3) an Young s inequality, we obtain IV u L 6 t L 3 L u H 1 t H 1 L u H 1 t L L + u H 1 t L L ε t L + MΦ).5) for some M F. By Höler s inequality, Sobolev s inequality an Cauchy s inequality, we obtain IV 3 u t L 6 L 3 L u t L H 1 L MΦ) + u t L.51) for some M F. By Höler s inequality, Sobolev s inequality,.14) an Cauchy s inequality, we obtain IV 4 u L t L L u H 1 t L L ε t L + MΦ).5) for some M F. Combining.48),.49),.5),.51) an.5), we obtain t [ u )] x ε t L + C u t L + MΦ).53) 13
14 for some M F. By Leibniz s rule an the fact = 1, we have t [ )] x [ t + t + t + t ] x = 4 V i. i=1.54) By Höler s inequality, Sobolev s inequality an.3), Cauchy inequality, an Young inequality, we obtain V 1 L t L L H t L L MΦ),.55) V t L 6 L 3 L t H 1 H 1 Φ t L + t L ) ε t L + MΦ), V 3 L t L L H t L L ε t L + MΦ), V 4 t L 6 L L 3 L t H 1 H H 1 L t H 1MΦ) for some M F. Notice that t L L + L + u 4 L H + u 1 L 6 L H + u 1 L H Φ..56).57).58).59) Thus by.3),.58) an.59), we have V 4 MΦ).6) for some M F. Combining.54),.55),.56),.57) an.6), we have t [ )] x ε t L + MΦ).61) for some M F. Putting.53) an.61) into.47), we obtain t L + 1 L [ u ) ) ] x +4ε t L + C u t L + MΦ).6) for some M F. Integrating.6) over, t), using H k k =, 3) estimate of the elliptic equations, an choosing ε small enough, we have 3 L + t t L s u ) ) x + u ) ) x.63) + 3 L + t u t L s + t MΦs)) s. 14
15 For the first term of right sie of.63), we have u ) ) x u + u ) x = 4 V I i. i=1.64) By Höler s inequality, Nirenberg s interpolation inequality,.5), an Young s inequality, we obtain V I 1 L u L L 1 4 L 3 4 H u L L 3 4 H 1 u L 3 L L u L.65) ε 3 L + C 3 H 1 u L + u 8 L ), V I u L 6 L 3 L u L 1 L 1 H 1 3 L u L L 3 L + u L 1 L 3 3 L ε 3 L + C L u L + u 4 L ),.66) V I 3 3 L 6 L 3 H 1 3 L ε 3 L + C 6 H 1,.67) an V I 4 L L L 1 4 L 3 4 H L 3 L 3 4 H 1 L 3 L L L.68) ε 3 L + C 3 H 1 L + 8 L ). Combining.64),.65),.66),.67) an.68), we obtain u ) ) x 4ε 3 L + C 3 H 1 u L + L ) + C L u L + u 4 L ) +C 6 H L + u 8.69) L ) t ) 4 4ε 3 L + CMρ, u, ) + MΦs)) s for some M F, where we have use Lemma., Lemma.5, an Lemma.7 in the last step. Substituting.69) into.63), choosing ε small enough, an using.18), Cauchy s inequality, Lemma.5 an.43), we have 3 L + t t L s CMρ, u, ) + t ) 4 MΦs)) s for some M F. This completes the proof. Proof of Theorem.1. It is reaily seen that the conclusion follows from.18),.3),.33),.43) an.46). 15
16 3 Proof of Theorem W,p estimate In this subsection, we give a proof of W,p estimate of the Lamé equation on a simply connecte, boune, smooth omain with the Navierslip bounary conition, which is neee in our proof of Theorem 1.. We believe that such an estimate may have its own interest. Lemma 3.1 For any simply connecte, smooth boune omain R 3, 1 < p < +, an f L p, R 3 ), If u H 1 H, R 3 ) is a weak solution of Lu = f in, u ν = u) ν = on. 3.1) Then u W,p ), an there exists C > epening on p,, an L such that ] u [ f C L p L p + u L. 3.) Proof. By the uality argument, we may assume 1 < p. Since u ν = on, it follows from BourguignonBrezis [] that u L p iv u) L p + curl u) L p + u L p. 3.3) Also, since is simply connecte an u) ν = on, it follows from Wahl [37] that curl u) L p C curl u L p + curl u) L p = C curl u) L p 1 [ ] Lu L p + µ + λ) ivu) L p µ ivu) L p + f L p. 3.4) Now we estimate iv u) L p by the uality argument: for p = p p 1, { } ivu) L p C sup ivu) g x : g C, R 3 ), g L p = 1. For any g C, R 3 ), with g L p = 1, by the Helmholtz s ecomposition Theorem see FujiwaraMorimoto [11] an Solonnikov [33]), there exist G C ) W 1,p ) an H C ) L p, R 3 ) such that g = G + H, ivh = in, G ν = g ν on, G W 1,p + H L p C g Lp = C. Thus we have iv u) H x = 16
17 so that iv u) g x = = = = where we have use iv u) G + H) x = iv u) G x iv u) 1 1 f) G x + f G x µ + λ µ + λ µ curl u) G x + 1 f G x µ + λ µ + λ 1 f G x, µ + λ curl u) G =, since iv curl u)) = in an curl u) ν = on. The above inequality implies iv u) g x f L p G L p C f L p. Taking supremum over all such g s, we obtain iv u) L p C f L p. It is clear that this, with the help of 3.3) an 3.4), implies 3.). 3. Existence In this subsection, we will first consier that R 3 is a boune omain, an then employ the Galerkin s metho to obtain a sequence of approximate solutions to 1.1)1.3) uner 1.6) an 1.8) or 1.9) that enjoy a priori estimates obtaine in, which will converge to a strong solution to 1.1)1.3). The existence of strong solutions for the Cauchy problem on R 3 follows in a stanar way from a priori estimates by the omain exhaustion technique, which will be sketche at the en of this subsection. To implement the Galerkin s metho, we take the function space X to be either i) for the Dirichlet bounary conition 1.8), X := H 1 H, R 3 ) an an its finite imensional subspaces as X m := span { φ 1,, φ m}, m 1, where {φ m } X is an orthonormal base of H 1 ), forme by the set of eigenfunction of the Lamé operator uner the bounary conition u = on ; or ii) for the Navierslip bounary conition 1.9), an its finite imensional subspaces as X := { u H, R 3 ) : u ν = u) ν = on }, X m := span { φ 1,, φ m}, m 1, where {φ m } X is an orthonormal base of H 1 ), forme by the set of eigenfunction of the Lamé operator uner the Navierslip bounary conition u ν = u) ν = on. By the W,p  estimate of Lamé equation uner 1.8) or 1.9) see Lemma 3.1), we see that {φ m } W,p ) for any 1 < p < +. 17
18 Now we outline the Galerkin s scheme into several steps. Step 1 moification of initial ata). For δ >, let ρ δ = ρ + δ, δ =, an u δ X be the unique solution of Lu δ P ρ δ )) = ρ δ g in, 3.5) u δ = ; or u δ ν = u δ ) ν = on. 3.6) By the W, estimate of Lamé equation, it is not har to show that u δ X lim u =. δ + Step mth approximate solutions). Fix δ > an 3 < q 6. For m 1 an some < T = T m) < + to be etermine below, we let u m = m u δ, φ k )φ k an look for the triple ρ m C[, T ]; W 1,q H 1 ) u m x, t) = m u m k t)φ kx) C[, T ]; W,q H ) k=1 m C[, T ]; H 3, S )) k=1 solution of the following problem ρ m t + ρ m u m ) =, ρ m u m t, φ k ) + µ u m, φ k ) + µ + λ) u m, φ k ) = ρ m u m u m, φ k ) P ρ m )), φ k ) m m, φ k ) 1 k m), m t + u m m = m + m m, ρ m, u m, m ) = ρ δ, um, ), t= u m, m ν ) [,T =, or u m ν, u m ) ν, m ] ν ) [,T =. ] 3.7) The existence of a solution ρ m, u m, m ) to 3.7) over [, T m)] for some T m) > can be obtaine by the fixe point theorem, similar to that on the compressible NavierStokes equation by Paula [8] see also [4]). Here we only sketch the argument. First, observe that for any given < T < + an u m C[, T ]; W,q H ), it is stanar to show that there exist 1) a solution ρ m C[, T ]; W 1,q H 1 ) of 3.7) 1 along with ρ m t= = ρ δ. ) < t m T, epening on u m an H 3, an a solution m C[, t m ], H 3, S )) of 3.7) 3 along with m t= = an m ν =. [,tm] It is wellknown cf. [8] [4] or Lemma.5 in ) that ρ m x, t) δ exp t ) u m L s >, x, t) Q T. 3.8) 18
19 The coefficients u m k t) can be etermine by the following system of m first orer orinary ifferential equations: 1 k m, m ρ m φ i, φ k ) u m i = F k u m l t), i=1 t u m l ) s, t ; u m k ) = uδ, φ k ), 3.9) where F k enotes the right han sie of 3.7). Since ρ m is strictly positive, the eterminant of the m m matrix ρ m φ i, φ k ) 1 i,k m is positive. Hence we can reuce 3.9) into u m k = G ku m l, b m l, t), ḃ m k = um k ; um k ) = uδ, φ k ), b m k ) =, 3.1) where G k is a regular function of u m l, b m l. Therefore, by the stanar existence theory of orinary ifferential equations, we conclue that there exists a < T m t m an a solution u m k t) to 3.9), which in turn implies the existence of solutions ρ m, m of 3.7) 1 an 3.7) 3 on the same time interval. Step 3 a priori estimates). We will show that there exist < T < + an C >, epening only on the norms given by the regularity conitions on P an the initial ata ρ, u, an, but inepenent of the parameters δ, m, an the size of the omain, such that there exists M F so that for any m 1, φ m, u m, m ) satisfies: Φ m t) exp [ CMρ δ, u δ, δ ) + C t ] MΦ m s)) s, < t T, 3.11) where Φ m t) is efine by.1) with ρ, u, ) replace by ρ m, u m, m ) an Mρ δ, uδ, δ ) is efine by.3) with ρ, u, ) replace by ρ δ, uδ, δ ). Since the argument to obtain 3.11) is almost ientical to proof of Theorem.1, we only birefly outline it here: First, it is easy to see 3.7) hols with φ k replace by u m. By multiplying 3.7) 3 by m + m m ) an integrating over an aing these two resulting equations, we can show that there is a M F such that the energy inequality.5) hols with ρ, u, ), M, an Φ replace by ρ m, u m, m ), M, an Φ m. Secon, since 3.7) implies Lu m = P m ρ m u m + P ρ m )) + m m), 3.1) where P m u) = m i=1 u, φ k)φ k : X X m is the orthogonal projection map, we can check that the same argument as Lemma.3 yiels that exists M F so that u m H 1 MΦ m t)), t T m 3.13) Thir, by ifferentiating 3.1) w.r.t. t, multiplying the resulting equation with u m t, integrating over, an repeating the proof of Lemma.4, we obtain that there exists M F such that for any m 1, ρ m u m t + t u m t C [ Mρ δ, u m, δ ) + t ] MΦ m s)) s. 3.14) Fourth, similar to the proof of Lemma.5 an Lemma.6, we have that there exists M F such that for all m 1, [ t ] u m L C Mρ δ, u m, δ ) + MΦ m s)) s, 3.15) 19
20 an ρ m H 1 W 1,q C exp {C [ Mρ δ, u m, δ ) + t ]} MΦ m s)) s. 3.16) Fifth, by ifferentiating 3.7) 3 w.r.t. x an mutiplying by m t an m t respectively) an integrating over, we can use the same argument as Lemma.7 an Lemma.8 to show that there exists M F such that for all m 1, 3 m L + m L + t t m t L s m t L s C[1 + t CMρ δ, u m, δ ) + MΦ m s)) s], 3.17) t MΦ m s)) s) ) It is reaily seen that combining all these estimates together yiels 3.11) with T replace by T m an u δ replace by um. Step 4 convergence an solution). By the efinition of u δ, M given by.1), an the conition 1.1), we have Mρ δ, u δ, δ ) = 1 + g L, an Mρ δ, u m, δ ) Mρ δ, u δ, δ ) C u m u δ δ Thus there exists N = Nδ) > such that H, as m. Mρ δ, u m, δ ) + g L, m N. 3.19) It follows from 3.19), 3.11), an Gronwall s inequality see, for example, [3] page 63 or [3] Lemma 6) that there exists a small T >, inepenent of δ an m, such that sup Φ m t) C expc g L ), m M. 3.) t T By virtue of 3.), we obtain that for any m M, sup ) ρ m u m t L + ρ m W 1,q H + u m 1 H + m 1 t H + m 1 H t T T ) + u m D + u m,q t L + 4 m L + m t L C expc g L ). 3.1) Base on the estimate 3.1), we can euce that after taking subsequences, there exists ρ δ, u δ, δ ) such that ρ m ρ δ weak in L, T ; W 1,q H 1 ), u m u δ weak in L, T ; D 1 D ), u m u δ weak in L, T ; D,q ), u m t u δ t weak in L, T ; D 1 ), ρ m u m t ρ δ u δ t weak in L, T ; L ), m δ weak in L, T ; D 1 D 3 ) an L, T ; D 4 ), m t δ t in L, T ; H ) an weak in L, T ; H 1 ).
21 By the lower semicontinuity, 3.1) implies that for t T, ρ δ, u δ, δ ) satisfies sup ) ρ δ u δ t L + ρ δ W 1,q H + u δ 1 H + δ 1 t H + δ 1 H + t T T ) u δ D + u δ,q t L + 4 δ L + δ t L C expc g L ). 3.) Furthermore, it is straightforwar to check that ρ δ, u δ, δ ) is a strong solution in [, T ] of 1.1) 1.3) uner the initial conition ρ δ, u δ, δ ) = ρ δ, uδ, δ ) an the bounary conition 1.8) or t= 1.9). Since T > is inepenent of δ, ρ δ, u δ, δ ) satisfies 3.), ρ δ ρ in W 1,q H 1, u δ u in D 1 D, an δ =, the same limiting process as above woul imply that after taking a subsequence δ, ρ δ, u δ, δ ) converges weakly in the corresponing spaces) to a strong solution ρ, u, ) of 1.1)1.3) on [, T ] along with 1.6) an 1.8) or 1.9). For the Cauchy problem on R 3, we procee as follows. For R, it is stanar cf. [4]) that there exists R H3 R 3, S ) such that R n outsie B R Now we let u R H1 B R) H B R ) be the unique solution of for some constant n S an lim R H =. 3.3) R R 3 ) Lu R P ρ )) R R = ρ g on B R, u R BR =, 3.4) where g L R 3 ) is given by 1.1). Extening u R to R3 by letting it be zero outsie B R. Then it is not har to show that for any compact subset K R 3, lim u R H u =. 3.5) R 1 K) By the above existence, we know that there exists T >, inepenent of R, an a strong solution ρ R, u R, R ) of 1.1)1.3) on B R [, T ] of 1.1)1.3), uner the initial an bounary conition: ρ R, u R, R ) = ρ, u R, R ); u R R, BR {t=} R ) BR =. 3.6) [,T ] Furthermore, ρ R, u R, R ) satisfies the estimate: sup ) ρ R u R t L + ρ R W 1,q H + u R 1 H + R 1 t H + R 1 H t T T ) + u R D + u R,q t L + 4 R L + R t L C expc g L ), 3.7) with C > inepenent of R. It is reaily seen that 3.7), 3.3), an 3.5) imply that after taking a subsequence, we may assume that ρ R, u R, R ) locally converges weakly in the corresponing spaces) to a strong solution ρ, u, ) of 1.1)1.3) on R 3 [, T ] uner the initial conition 1.6) an the bounary conition 1.7). This completes the proof of Theorem 1.. 1
22 3.3 Uniqueness In this subsection, we will show the uniqueness of the local strong solutions obtaine in Theorem 1.. Let ρ i, u i, i ) i = 1, ) be two strong solutions on, T ] of 1.1)1.3) with 1.6) an either 1.7), or 1.8), or 1.9). Set ρ = ρ ρ 1, u = u u 1, = 1. Then we have ρ t + u 1 )ρ + u ρ + ρivu + ρ 1 ivu =, ρ 1 u t + ρ 1 u 1 u + P ρ ) P ρ 1 )) 3.8) = Lu ρu t + u u ) ρ 1 u u 1, with the initial conition: an the bounary conition: t = + 1 ) 1 + u u 1, ρ, u, ) t= =, x, ) u, =, or u ν, u) ν, ) =. ν ν Multiplying 3.8) by u, integrating over, an using integration by parts, we have 1 µ ρ 1 u x + + λ) ivu + µ u ) x = ρu t + u u ) u x ρ 1 u u u x + P ρ ) P ρ 1 )) ivu x + u + u ) x 1 u x. Observe that P ρ ) P ρ 1 ) B P ρ 1 L + ρ L ) ρ C ρ. Hence, by Höler s inequality an Cauchy s inequality, we have 1 µ ρ 1 u x + + λ) ivu + µ u ) x ρ 3 L u t + u u L 6 u L 6 + u L ρ 1 u x + ρ L ivu L + L L 3 u L 6 + L L u L + 1 L 3 L u L 6 ρ 3 L u t + u u L 6 u L + u W 1,q ρ 1 u x + ρ L ivu L + L u L ɛ u x + C[ ρ u L 3 t + u u L 6 + u W 1,q ρ 1 u x + ρ L + L ].
23 Thus, by choosing ɛ sufficiently small, we have ρ 1 u x + u x C[ ρ L 3 u t + u u L 6 + u W 1,q ρ 1 u x + ρ L + L ]. 3.9) Multiplying 3.8) 1 by ρ, integrating over, an using integration by parts, we have ρ x ρ u ρ x + ρ ivu 1 + ivu ) x + ρ ρ 1 ivu x ρ L ρ L 3 u L 6 + ivu 1 L + ivu L ) ρ x + ρ L ivu L ρ L u L + ivu 1 W 1,q + ivu W 1,q) ρ x ρ L ivu L + u L ) + ivu 1 W 1,q + ivu W 1,q) ρ x ɛ u x + C ɛ ρ L + C ivu 1 W 1,q + ivu W 1,q) ρ x, for any ɛ >. Similarly, we have ρ 3 x ρ 1 u ρ x + ρ 1 L 3 ρ 1 L 3 ρ L u L 6 + ivu 1 L + ivu L ) u L + ivu 1 W 1,q + ivu W 1,q) ρ 1 u L 3 L + ivu 1 W 1,q + ivu W 1,q) Multiplying 3.31) by ρ 1 L 3 ρ 3 ivu1 + ivu ) x + ρ 1 ρ1 ivu x ρ 3 1 x + ρ ivu L 3 L ρ 1 L 6 ρ 3 x ρ 3 x., an using Cauchy s inequality, we have ρ ρ L 3 3 L u L + ivu 1 W 1,q + ivu W 1,q) ρ L 3 ɛ u x + C ɛ ρ + C ivu L 3 1 W 1,q + ivu W 1,q) ρ. L 3 3.3) 3.31) 3.3) Multiplying 3.8) 3 by, integrating over, an using integration by parts an Cauchy s inequality, we have 1 x + x L L + 1 L + L L 6 L 6 + L u L 6 L 3 + L u 1 L L L L + 1 H + L H 1 L + L u L 1 L 1 L 6 + L u 1 H 1 L L L + L u L 1 L + C L + C u L. 3
24 This gives x + x C L + C u L. 3.33) Multiplying 3.9) by 3C, putting the resulting inequality, 3.3) an 3.3) to 3.33), an taking ɛ > small enough, we have 3C ) ρ 1 u L + ρ + ρ L 3 L + L + C u x ρ u L 3 t + u u L + u 6 W 1,q ρ 1 u x + ρ L + L + ivu 1 W 1,q + ivu W 1,q) ρ x + ρ + ivu L 3 1 W 1,q + ivu W 1,q) ρ 3.34) L 3 u t + u u L + u 6 1 W 1,q + u W 1,q + 1 ) 3C ) ρ 1 u L + ρ + ρ L 3 L + L. By 3.34), Gronwall s inequality, an ρ, u, ) =, we have t ρ 1 u L + ρ + ρ L 3 L + L + This yiels u xs =. 3.35) ρ, u, ) =. 3.36) To see =, observe that after substituting 3.36) into 3.8) 3, we have t =, t= =. This implies =. This completes the proof. 4 Proof of Theorem 1.3 Let < T < be the maximum time for the existence of strong solution ρ, u, ) to 1.1)1.3). Namely, ρ, u, ) is a strong solution to 1.1)1.3) in, T ] for any < T < T, but not a strong solution in, T ]. Suppose that 1.11) were false, i.e. lim sup T T ρ L,T ;L ) + T ) 3 L ) = M <. 4.1) The goal is to show that uner the assumption 4.1), there is a boun C > epening only on M, ρ, u,, an T such that [ max ρ W r=,q 1,r + ρ t L r) + ] ρu t L + u H 1) + t H 1 + H ) C, 4.) an sup t<t T ut D 1 + u D,q + t H + H 3 ) C. 4.3) With 4.) an 4.3), we can then show without much ifficulty that T is not the maximum time, which is the esire contraiction. The proof is base on several Lemmas. 4
25 Lemma 4.1 Assume 4.1), we have Proof. T x C. 4.4) To see 4.4), observe that 4.1) implies T L M so that T ) 4 x M sup x t<t [ T M ρ P ρ) x + u + ) ] x where we have use.11) in the last step. Applying.11) again, this then implies Since T x = T + T x + [ T 1 + M ) we have, by the conservation of mass an 4.1), T P ρ) B P ρ L ) ρ C ρ, 4 x ρ P ρ) x + u + ) ] x. P ρ) x CT sup t<t ρ L 1 ρ L C. Thus the stanar L estimate yiels 4.4). Following the argument by [35], we let v = L 1 P ρ)) be the solution of the Lamé system: { Lv = P ρ)), v =, or v as x when = R 3 4.5) ). Then it follows from [35] Proposition.1 that v L q C P ρ) L q CB P ρ L ) ρ L q C, 1 < q 6, 4.6) where we have use 4.1) an the conservation of mass in the last step. Denote w = u v, then w satisfies ρw t Lw = ρf, w t= = w = u v, w = or w, as x, 4.7) where F = u u L 1 t P ρ))) = u u + L 1 iv P ρ)u) L 1 P P ρ)ρ)iv u ). Then we have the following estimate. 5
26 Lemma 4. Uner the assumptions of Theorem 1.3, if λ < 7µ 9, then ρ, u, ) satisfies that for any t < T, ρ u 5 + w ) t x w + t ) xs C. 4.8) Proof. The proof of this lemma is ivie into five steps. Step 1. Estimates of w x. Multiplying 4.7) 1 by w t, integrating over, an using integration by parts an Cauchy s inequality, we have µ w + µ + λ) iv w ) x + ρ w t x ρf L + 1 I 3 ) : wx + C For I 1, we have t w x = 3 I i. i=1 4.9) I 1 ρu u L + ρl 1 iv P ρ)u) L + ρl 1 P ρ) P ρ)ρ)iv u) L 3 = I 1j. j=1 4.1) For I 11, by Höler s inequality, 4.1), Sobolev inequality, interpolation inequality, an 4.6), we have I 11 ρ 1 5 u L 5 u L 1 3 ρ 1 5 u L 5 u 4 5 L u 6 5 L 6 ρ 1 5 u L 5 u 4 5 L w 6 5 L 6 + ρ 1 5 u L 5 u 4 5 L v 6 5 L 6 ) ρ 1 5 u L 5 u 4 5 L w 6 5 L + w 6 5 L ) Again by [35] Proposition.1, an 4.7), we have w L ρw t L + ρf L + L. 4.1) Substituting 4.1) into 4.11), an using Young s inequality, we obtain for any ε > I 11 ε ρw t L + ρf L ) +C ρ 1 5 u 5 L 5 u L + w L + L L + 1). 4.13) For I 1 an I 13, by [35] Proposition.1, 4.1),.5), an 1.5), an Sobolev s inequality, we have I 1 P ρ)u L ρu L ρu L C, 4.14) I 13 ρ L 3 L 1 P ρ) P ρ)ρ)iv u) L 6 L 1 P ρ) P ρ)ρ)iv u) L P ρ) P ρ)ρ) u L CB P ρ L ) ρ L u L C u L, 4.15) where we have use the Sobolve inequality when = R 3, an both Sobolve an Poincaré inequalities when is a boune omain. 6
27 Putting 4.13), 4.14) an 4.15) into 4.1), an choosing ε sufficiently small, we obtain I 1 1 ρw t L + C ρ 1 5 u 5 L 5 u L + w L + u L + L L + 1) 1 ρw t L + C ρ 1 5 u 5 L 5 u L + v L + u L + L L + 1) 4.16) 1 ρw t L + C ρ 1 5 u 5 L 5 u L + u L + L L + 1), where we have use 4.6) with q =. For I 3, using Cauchy s inequality, we have I 3 1 t x + C w x 1 t x + C L w x. 4.17) Substituting 4.16) an 4.17) into 4.9), we obtain µ w + µ + λ) iv w ) x + 1 ρ w t x 1 I 3 ) : w x + 1 t L ) +C L w L + L ) + ρ 1 5 u 5 L 5 u L + u L + 1. Step. Estimates of 4.18) ρ u 5 x. Multiplying 1.) by 5 u 3 u, integrating over, an using integration by parts an Cauchy s inequality, we have ρ u 5 x + 5 u 3 µ u + µ + λ) iv u + 3µ u ) x = 5P ρ)iv u 3 u) x ) I 3 iv u 3 u) 15µ + λ)iv u) u u u 45 C ρ u 3 u + u 3 u ) + 5µ + λ) u 3 iv u + 4 µ + λ) u 3 u. By Kato s inequality u u, we have { 15µ 45µ+λ) 4 ) u 3 u 15µ 45µ+λ) 4 ) u 3 u, if µ 3µ+λ) 4 15µ 45µ+λ) 4 ) u 3 u, if µ 3µ+λ) 4 >. Hence we obtain C { ρ u 5 x + 5 min µ, ρ u 3 u x + 4µ u 3 u x). ) 9µ + λ) } 4 u 3 u x 4.19) Since λ < 7µ 9, we have c { := 5 min µ, 4µ ) 9µ + λ) } >. 4.) 4 7
28 Thus by Cauchy s inequality, we have ρ u 5 x + c u 3 u x C ρ u 3 u x + u 3 u x) c [ ] u 3 u x + C ρ u 3 x + 4 u 3 x. Hence by Höler s inequality, Sobolev s inequality, the conservation of mass, 4.1) an Young s inequality, we have ρ u 5 x + c u 3 u x ρ u 3 x + 4 u 3 x ) 3 ) ) 4 ρ u 5 5 x ρ x + u ) x L c [ u 3 u x + C 1 + ρ u 5 x + 5 x) ]. 4 Thus by.5) we have Step 3. Estimates of ρ u 5 x + c 4 u 3 u x ρ u 5 x + 5 x) + 1 ρ u 5 x + 3 L 5 L x. Differentiating 1.3) with respect to x, we obtain 4.1) t + u ) = ). 4.) Multiplying 4.) by 5 3 an integrating by parts over, we have 5 x x [ =5 ) u ) ] 3 x 5 3 ) x u ) x. This, combine with Cauchy s inequality an the fact = since = 1), 4.3) gives 5 x x u 3 ) x 3 L L + L 5 L L u L L 3 L L + u L ) + L 5 L )
29 By 4.6) an 4.4), we have 5 x x 3 L L + w L ) + 3 L + L 5 L. 5 Step 4. Estimates of using Cauchy s inequality, we have 1 x + 4.5) x. Multiplying 4.) by t, integrating by parts over, an t x = ) u ) ) t x ε t L + C u + u ) x ε t L + C [ L L + u ) ] L + u x, 4.6) where we have use 4.3) to estimate 6 x L x. 4.7) For the last term on the right han sie of 4.6), using Nirenberg s interpolation inequality an Cauchy s inequality, we have u x u L 6 L 3 13 ε u 5 L + C 1 3 L 13 3 ε u 5 L + C 8 3 L 6 3 H ε u 5 L + ε 3 L + C 4 L + 6 L + 1) 5ε u 3 u x + ε 3 L + C 4 L + 1). 4.8) By 1.3), H 3 estimate for elliptic equations, an 4.7), we have 3 L t L + u ) L + ) L t L + L u L + ) L + 6 L 6 + u x C [ t L + L u L + ) ] L + u x. Substituting 4.9) into 4.8), an choosing ε sufficiently small, we have u x C[ u 3 u x + 4 L + ε t L + L u L + L ) + 1 ]. 4.9) 4.3) 9
30 Substituting 4.3) into 4.6), using 4.6), an choosing ε sufficiently small, we obtain x + t x C [ L L + u ) ] L + u 3 u x + 4 L + 1 [ C L L + w ) ] L + L + u 3 u x + 4 L ) Step 5. Completion of proof of Lemma 4.. Aing 4.1), 4.18), 4.5) an 4.31) together, an choosing ε sufficiently small, we obtain ρ u 5 + µ w + µ + λ) iv w ) x + 1 ρ w t x + 1 t x 1 u I 3 ) : w x + C[ L + 1 ) ρ u 5 x u L + 3 L + 3 L 5 L 5 + L + w ) L + L 5 L 5 + L + w ) L ] + L + 4 L + 1. This, combine with Cauchy s inequality, implies ρ u 5 + µ w + µ + λ) iv w ) x + 1 ) ρ w t x + t x 1 [ I 3 ) : w x + C u L + 3 L + ) ] u L + L + 3 L + 1) ρ 1 5 u 5 L L + 5 L + w L + 1. Integrating over, t), an using 4.1),.5), we have ρ u 5 + w ) t ρ wt x + + t ) x s where C[ w x + t ) Ks) ρ 1 5 u 5 L 5 + w L + 5 L + 5 L Ks) = us) L + s) L + s) 3 L + 1. By 4.3) an Young s inequality, we have ρ u 5 + w ) t ρ wt x + + t ) x s ] s + 1, 4.3) 1 1 w x + C[ 4 x + w x + t [ 5 t x) + C ) Ks) ρ 1 5 u 5 L 5 + w L + 5 L + 5 L ] s + 1 ) ] Ks) ρ 1 5 u 5 L 5 + w L + 5 L + 5 L s + 1. Thus we obtain ρ u 5 + w ) x + [ C 1 + t t ρ wt + t ) x s ) Ks) ρ 1 5 u 5 L 5 + w L + 5 L + 5 L ] s. 4.33) 3
31 By 4.1),.5) an 4.4), we know t Ks)s C. 4.34) By 4.33), 4.34) an Gronwall s inequality, we obtain that for any t < T, ρ u 5 + w ) t ρ wt x + + t ) x s C. This completes the proof of Lemma 4.. Corollary 4.3 Uner the same assumptions of Lemma 4., we have that for any q 6, sup u L 6 + u L + L q + t L ) + u L,T ;L 6 ) C. 4.35) t<t Proof. Combining 4.6) with 4.8), we get ut) L wt) L + vt) L C. 4.36) The upper boun of sup u L 6 follows from 4.36) an Sobolev s inequality. The boun of t<t sup L q follows from 4.8) an interpolation inequality. For the last term of 4.35), by t<t Sobolev s inequality, 4.6) an 4.8), we have u L,T ;L 6 ) w L,T ;L 6 ) + v L,T ;L 6 ) w L,T ;L ) + w L,T ;L ) + 1 C. By equation 1.3), 4.8) an Höler s inequality, we have sup t L sup L + ) L + u 4 L t<t t<t sup u L 6 L 3) + 1 C. t<t This completes the proof. Lemma 4.4 Uner the same assumptions of Lemma 4., ρ, u, ) satisfies that for any t < T, t ρ ut) + t )t) x + u + tt ) xs C, 4.37) where f is the material erivative: Proof. 1.) as follows, Step 1. Estimates of f := f t + u f. Differentiating 4.38) with respect to t an using 1.1), we have ρ ut) x. By the efinition of material erivative, we can write ρ u + P ρ)) = Lu. 4.38) ρ u t + ρu u + P ρ) t ) + ) t [ ] =L u Lu u) + iv Lu u P ρ)) u ) u. 4.39) 31
32 Multiplying 4.39) by u, integrating by parts over an using the fact u = on, we obtain 1 µ ρ u x + u + µ + λ) iv u ) x = P ρ)) t iv u + u P ρ)) : u) x + µ iv u u) u u)) u x ) 4.4) +µ + λ) iv iv u u) iv u u) u x + u )) : u x + t + t t I 3 ) : u x = 5 J i. By equation 1.1) an 4.1), we have ) J 1 = iv P ρ)u)iv u P ρ)ρ P ρ))iv uiv u + u P ρ)) : u x ) = P ρ)u iv u + P ρ) P ρ)ρ)iv uiv u + P ρ) u) t : u P ρ)u iv u x ) = P ρ) P ρ)ρ)iv uiv u x + P ρ) u) t : u x u L u L. By the prouct rule, we can see iv u u) u u) = k iv u k u) k k u j j u) j k u j k u), so that by integration by parts, we have k J = µ iv u k u) k k u j j u) j k u j k u) ) u x u L u L. 4 Similarly, since we have iv iv u u) iv u u) = k j u j i u i ) k j u i i u j ) i k u i j u j ), J 3 = µ + λ) k j u j i u i ) k j u i i u j ) i k u i j u j ) ) u k x u L u L 4. By Höler s inequality, an Corollary 4.3, we have i=1 J 4 u L L 6 L 6 u L 6 u L L 6, J 5 u t x u L t L L. Putting all these estimates into 4.4), using Young s inequality an Sobolev s inequality,an Lemma 4. an Corollary 4., we have 1 µ ρ u x + u + µ + λ) iv u ) x u L u L + u L u 4 L + u L L 6 + u L t L L µ u L + C u L + u 4 L + 4 H + 1 t L ) L µ u L + C u 4 L L + t L L + 1) 3
33 Thus we obtain ρ u x + µ u x u 4 L L + t L L ) By H 3 estimate of elliptic equations, Lemma 4., Corollary 4.3, an Nirenberg s interpolation inequality, we have Thus we obtain 3 L t L + u L + u L + L + 3 L By the efinition of w, we have t L + u L 6 L 3 + u L 3 L 6 + L 6 L t L + u 1 L u 1 H L 1 H L + C t L + u L + 1 ). 3 L t L + u L ) Lw = ρ u ) By H estimate of the equation 4.43), 4.1), Corollary 4.3, Nirenberg s interpolation inequality, an 4.4), we obtain w L ρ u L + L ρ 1 u L + L 3 L 6 ρ 1 u L + L H 1 ρ 1 u L + L + 1 ρ 1 u L + t L + u L ) By interpolation inequality, Corollary 4.3, 4.6) for q = 6), 4.44), an Cauchy s inequality, we obtain u 4 L u 4 L u 3 L u 6 L 6 u L 6 u L 6 w L 6 + v ) L 6 u L 6 w L + 1 ) ) u L 6 ρ 1 u L + t L + u L + 1 u L 6 ρ 1 u L + u L 6 + t L + u L + 1 u L 6 ρ 1 u L + t L + u L ) Putting 4.45) an 4.4) into 4.41), we have ρ u x + µ u x u L 6 ρ 1 u L + t L L + 1) + u L ) Step. Estimates of t x. Differentiating 1.3) with respect to t, we have tt t = t u ). 4.47) Multiplying 4.47) by tt, integrating by parts over an using ν =, we obtain 1 t x + tt x = t u ) tt x t + t ) 4.48) tt x + u t + u t ) tt x = K 1 + K. 33
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